Singular spectral shift function for Schr\" odinger operators

SINGULAR SPECTRAL SHIFT FUNCTION ¨ FOR SCHRODINGER OPERATORS

arXiv:1608.04184v1 [math.SP] 15 Aug 2016

N. AZAMOV, T. DANIELS Abstract. One aim of this paper is to prove the following theorem. Let H0 = −∆ + V0 (x) be a 3-dimensional Schr¨ odinger operator, where V0 (x) is a bounded measurable real-valued function on R3 . Let V be an operator of multiplication by a bounded integrable real-valued function V (x) and put Hr = H0 + rV for real r. Let Z 1 ξ (s) (φ) = Tr(V φ(Hr )P (s) (Hr )) dr, φ ∈ Cc (R), 0

(s)

where P (Hr ) is the projection onto the singular subspace of Hr . Then ξ (s) defines an absolutely continuous measure whose density ξ (s) (λ) (denoted by the same symbol) is integervalued for a.e. λ. The proof of this theorem makes use of a simple and constructive approach to stationary scattering theory and another aim of this paper is to promote this approach.

August 16, 2016

v7.0

Contents 1. Introduction 1.1. Statement of the main result 1.2. Method of proof: constructive approach to stationary scattering theory 1.3. Singular spectral shift function and resonance index 1.4. Plan of the paper 1.5. Future work 2. Preliminaries 2.1. Miscellaneous 2.2. Resolvent comparable affine spaces of self-adjoint operators 2.3. Spectral shift function 2.4. Rigged Hilbert space and the Limiting Absorption Principle 3. Constructive approach to stationary scattering theory 3.1. Evaluation operator 3.2. Wave matrices 3.3. Wave operators 3.4. Connection with the time-dependent approach 3.5. Scattering matrix 3.6. Scattering operator 4. Singular spectral shift function 4.1. Absolutely continuous and singular spectral shift functions 4.2. Exponential representation of the scattering matrix 4.3. Singular spectral shift function and resonance index References

2 2 3 6 6 7 7 7 11 16 21 28 28 31 34 35 35 37 37 37 41 43 46

2010 Mathematics Subject Classification. Primary 47A40, 47A55; Secondary 47A70, 35P25, 35P05, 47B25, 81U99. Key words and phrases. spectral shift function, singular spectral shift function, Schroedinger operators, resonance index. 1

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N. AZAMOV, T. DANIELS

1. Introduction 1.1. Statement of the main result. The spectral shift function ξ(λ) = ξ(λ; H1 , H0 ) of a pair of self-adjoint operators H0 and H1 acting on a Hilbert space H is a real-valued function which satisfies the Lifshitz-Krein trace formula [L, Kr] Z ∞ φ′ (λ)ξ(λ) dλ (1.1) Tr(φ(H1 ) − φ(H0 )) = −∞

for all test functions φ ∈ Cc∞ (R), provided that the left hand side is well-defined. The spectral shift function (SSF) was first introduced formally by the physicist I. M. Lifshitz in [L] and a rigorous proof of the existence of the SSF for self-adjoint operators with trace class difference V := H1 − H0 was given by M. G. Krein in [Kr]. In [Kr2 ] Krein proved the existence of the SSF for a pair of self-adjoint operators H0 and H1 which satisfy the condition (1.2)

Rz (H1 ) − Rz (H0 ) ∈ L1 (H),

where L1 (H) is the trace class and Rz (H) = (H − z)−1 is the resolvent of H. Adopting terminology which appears in [Y], we say that H0 and H1 are resolvent comparable if (1.2) holds. Later, the existence of the SSF was proved for wider classes of pairs of self-adjoint operators. In [AzS] it was shown that the SSF exists if the product V φ(Hr ) is trace class for all test functions φ, where Hr = H0 + rV. A priori it is often easier to prove the existence of the spectral shift measure ξ, that is, a locally finite measure ξ on the σ -algebra of Borel sets which for all test functions φ satisfies the equality Z ∞

Tr(φ(H1 ) − φ(H0 )) =

φ′ (λ) ξ(dλ).

−∞

Proof of the absolute continuity of the spectral shift measure is often based on reduction to the trace class case. In what follows we shall identify locally finite Borel measures with continuous linear functionals on the linear space of continuous functions with compact support Cc (R), endowed with the standard locally convex inductive topology. In [BS2 ], M. Sh. Birman and M. Z. Solomyak proved that if the perturbation operator V = H1 − H0 is trace class then the spectral shift measure ξ satisfies the formula Z 1 Tr(V φ(Hr )) dr, φ ∈ Cc (R). ξ(φ) = 0

The proof of this formula is quite simple and natural, but the argument behind it does not show the absolute continuity of ξ. Nevertheless, it can be suggested to consider the Birman-Solomyak formula as the definition of the spectral shift measure. From this point of view the argument of [BS2 ] can be considered as a proof of the Lifshitz-Krein trace formula, while Krein’s original argument can be considered as a proof of absolute continuity in the case that V is trace class. An indication of the fundamental nature of the Birman-Solomyak formula is that it represents the spectral shift measure as an integral of the generalised one-form V 7→ ΦH (V )(φ) := Tr(V φ(H))

on some real affine space A of self-adjoint operators. This one-form, called the infinitesimal spectral shift measure, is exact on the corresponding affine space as long as the product V φ(H) is trace class for test functions φ [AzS]. Exactness of the infinitesimal spectral shift measure implies that in the Birman-Solomyak formula one can replace the straight path of integration Hr = H0 + rV by an arbitrary piecewise C 1 path. Further, this path-independence can be used to reduce absolute continuity of the spectral shift measure to the case of trace class V [AzS]. While as it turns out the spectral shift measure ξ is absolutely continuous in all instances it can be defined, the infinitesimal spectral shift measure Φ is obviously not. Therefore, one can consider the decomposition of Φ into its absolutely continuous Φ(a) and singular Φ(s) parts. The absolutely continuous component Φ(a) is absent outside of the common essential spectrum

¨ SINGULAR SSF FOR SCHRODINGER OPERATORS

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of the operators Hj , j = 0, 1, and thus properties of the infinitesimal spectral shift measure imply similar properties of its singular component Φ(s) outside the essential spectrum. Inside the essential spectrum a majority of these properties fail for Φ(s) , including exactness. However, in [Az] it is proved that one of the crucial properties of the SSF, integer-valuedness outside the essential spectrum, holds for a.e. value of the spectral parameter inside the essential spectrum too, if the SSF is replaced by the singular SSF, which is the integral of Φ(s) : Z 1 (s) Tr(V φ(Hr )P (s) (Hr )) dr. ξ (φ) = 0

This is proved under the hypothesis that the perturbation operator V is trace class. The aim of this paper is to prove the following theorem which generalises this result to resolvent comparable pairs of self-adjoint operators. Theorem 1.1. Let H0 be a self-adjoint operator on a Hilbert space H, let V be a bounded self-adjoint operator on H, and let H1 = H0 + V. Assume that H0 and H0 + |V | are resolvent comparable (in which case so are H0 and H1 ). Define the singular spectral shift measure ξ (s) (λ; H1 , H0 ) by the formula Z 1 (1.3) ξ (s) (φ; H1 , H0 ) = Tr(V φ(Hr )P (s) (Hr )) dr, φ ∈ Cc (R), 0

P (s) (H

where r ) is the projection onto the singular subspace of the self-adjoint operator Hr = H0 + rV. Then this measure is absolutely continuous and its density ξ (s) (λ) takes integer values for a.e. λ. Moreover, this theorem holds for any continuous piecewise analytic path Hr connecting the operators H0 and H1 , though the resulting singular spectral shift measure ξ (s) depends on the choice of path. It is well known, see e.g. [S, Theorem B.9.1], that on the Hilbert space L2 (Rν ) for ν 6 3 the premise of Theorem 1.1 holds for a pair of operators H0 = −∆ + V0 and V, where V0 is the operator of multiplication by a bounded measurable real-valued function V0 (x) on Rν and V is the operator of multiplication by a bounded integrable real-valued function V (x). Hence, Theorem 1.1 implies as a corollary the following theorem, which we consider as one of the main results of this paper. Theorem 1.2. Let ν = 1, 2 or 3. Let H0 = −∆ + V0 (x) be a Schr¨ odinger operator acting on L2 (Rν ), where V0 (x) is a bounded measurable real-valued function on Rν . Let V be an operator of multiplication by a bounded integrable real-valued function V (x) and let Hr = H0 + rV for real r. Then the measure ξ (s) defined by formula (1.3) is absolutely continuous and its density ξ (s) (λ) (denoted by the same symbol) is integer-valued for a.e. λ. We remark that the boundedness of the perturbation operator V is not essential for proofs in this paper, but we choose to impose this condition to avoid obscuring the general idea of our method by unnecessary complications of notation. In particular, Theorem 1.2 holds for V0 ∈ L∞ (Rν ) + L1 (Rν ) and V ∈ L1 (Rν ). On the other hand, the restriction ν 6 3 on the dimension of the manifold Rν is essential for the method used in this paper. Theorem 1.2 holds in arbitrary dimensions, but its proof which will appear in [AzD] requires a few significant modifications. 1.2. Method of proof: constructive approach to stationary scattering theory. Two quite independent proofs of Theorem 1.1 can be found in this paper. One of these, which we consider the main proof, follows the argument appearing in [Az]. The other proof uses an argument which appears in [Az2 ] and will be mentioned later in this introduction. The main proof is based on a new approach to stationary scattering theory. Though this approach was developed for the specific purpose of proving Theorem 1.1, we believe that it has its own merit.

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For a resolvent comparable pair of operators H0 and H1 the Birman-Krein formula [BK] holds: (1.4)

det S(λ; H1 , H0 ) = e−2πiξ(λ;H1 ,H0 ) ,

where S(λ; H1 , H0 ) is the scattering matrix of the pair H0 and H1 . Proof of Theorem 1.1 is achieved by showing that the Birman-Krein formula (1.4) still holds, if the SSF ξ in the right hand side is replaced by the absolutely continuous SSF ξ (a) (λ) : (1.5)

det S(λ; H1 , H0 ) = e−2πiξ

(a) (λ;H

1 ,H0 )

.

Similarly to the singular SSF ξ (s) (λ), the function ξ (a) (λ) is defined as the density of the absolutely continuous spectral shift measure ξ (a) , given by the formula Z 1 (a) Tr(V φ(Hr )P (a) (Hr )) dr, ξ (φ) = 0

where P (a) (Hr ) is the projection onto the absolutely continuous subspace of the self-adjoint operator Hr . Since for a.e. λ ξ(λ) = ξ (a) (λ) + ξ (s) (λ), the Birman-Krein formula (1.4) and its modification (1.5) imply that for a.e. λ (1.6)

e−2πiξ

(s) (λ;H

1 ,H0 )

= 1,

and this gives the required result. A detailed review of the approach to scattering theory which is used in the proof of (1.5) was given in the introduction of [Az3 ], and for this reason we will be brief. While the subject of stationary scattering theory is more than fifty years old and is treated in depth in the literature, e.g. [BW, Ku, RS2 , Y, Ag, ASch, BE, BK, BY, I, Ka, KK, Ku2 , Ku3 ], to the best of our knowledge the present exposition contains novel elements. For our purposes its main advantage is that, for a fixed value of the spectral parameter λ, we may vary the coupling constant r in the objects of the theory such as the fibre Hilbert spaces hλ (Hr ), wave matrices w± (λ; Hr , H0 ), and the scattering matrix S(λ; Hr , H0 ). The approach is constructive in the sense that these objects are introduced by explicit formulas for every value of λ from a pre-defined full set. (The full set on which they are defined is the set of points Λ(H0 , F ) ∩ Λ(Hr , F ) where the Limiting Absorption Principle holds for both operators H0 and Hr .) This avoids the notorious modulo-null-set uncertainty resulting from their definition via direct integral decompositions. This modulo null set uncertainty may not be a problem if one works with a fixed pair of operators H0 , H1 , but it becomes a significant hindrance if the perturbed operator H1 needs to be deformed to Hr , r ∈ R. It is often the case that constructive proofs are lengthier and more technical than non-constructive ones, but this is not the case here: all proofs are in fact simpler and more transparent, which can be considered as another advantage. Ultimately, with the aim to prove Theorem 1.1, only this approach to stationary scattering theory is suitable. Proof of the modified Birman-Krein formula (1.5) is based on the following ordered exponential representation of the scattering matrix Z 1 ∗ w+ (λ; H0 , Hr )Zr (λ; F )JZr (λ; F )w+ (λ; Hr , H0 ) dr , (1.7) S(λ; H1 , H0 ) = Texp −2πi 0

where the ingredients in the right hand side are defined as follows: w+ (λ; Hr , H0 ) is the scattering matrix for the pair H0 , Hr , the operators F and J come from a factorisation of the perturbation operator V = F ∗ JF, and the operator Zr (λ; F ) is defined by the formula (1.8)

Zr (λ; F ) = Fλ (Hr )F ∗ ,


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where Fλ (Hr ) is a fibre of a unitary operator F(Hr ) = F which acts from the absolutely continuous subspace H(a) (Hr ) of Hr onto a direct integral of Hilbert spaces Z ⊕ (a) hλ (Hr ) ρ(dλ), (1.9) F : H (Hr ) → σ ˆ Hr

such that for any f ∈ H(a) (Hr ) ∩ dom(Hr ) the equality F(Hr f )(λ) = λF(f )(λ) holds for a.e. λ ∈ σ ˆHr . Here σ ˆHr is a core of the absolutely continuous spectrum of Hr and ρ is a measure which is equivalent to the restriction of the Lebesgue measure to σ ˆHr . In conventional approaches to stationary scattering theory one works with a pair of operators H0 and H1 and this allows many difficulties to be avoided by considering problems a.e., but formula (1.7) requires the simultaneous diagonalisation of a continuous family of operators Hr , r ∈ [0, 1]. Of course, the spectral theorem provides a diagonalisation of a self-adjoint operator, but it is in essence an existence theorem which says nothing about how diagonalisations of different operators Hr are connected to each other. Thus, it is necessary to construct a certain diagonalisation of the absolutely continuous part of a self-adjoint operator. Such a construction was given by T. Kato and S. T. Kuroda in [KK] more than forty years ago, and since then this question apparently has not attracted much attention, partly because this is not a big hindrance in abstract scattering theory as long as one works with a fixed pair of operators. But the paper [KK] also considers a fixed pair of self-adjoint operators, and it remains unclear how the constructive diagonalisation presented there can be used to justify the formula (1.7). A constructive approach to stationary scattering theory which allows a proof of (1.7) was given in [Az] in the case of trace class perturbation operator V. In this paper this method is generalised to the resolvent comparable premise (1.2) and at the same time is simplified. The simplification is partly achieved by working with an abstract auxiliary Hilbert space K instead of the standard Hilbert space ℓ2 , which serves as an ambient Hilbert space for the fibre Hilbert spaces hλ (Hr ) of the representation (1.9). The pair H0 , H0 + V is resolvent comparable if there exists a factorisation V = F ∗ JF, where F is a bounded injective operator from the main Hilbert space H to an auxiliary Hilbert space K and J is a bounded self-adjoint operator on K, such that the operators F Rz (Hr ), r ∈ R, are Hilbert-Schmidt. (If V > 0 is positive, then this is an equivalent condition.) We choose and fix such a factorisation. Under this premise for any r ∈ R the following operator, called a sandwiched resolvent, Tλ+iy (Hr ) = F Rλ+iy (Hr )F ∗ has norm limits Tλ+i0 (Hr ) for a.e. λ. In other words, the Limiting Absorption Principle holds for each operator Hr . We denote by Λ(Hr , F ) the set of all real numbers λ for which the norm limit Tλ+i0 (Hr ) exists. The fibre Hilbert space hλ (Hr ) is defined as a subspace of K by the formula p (1.10) hλ (Hr ) = Im Tλ+i0 (Hr )K. This allows the introduction of the direct integral Z ⊕ hλ (Hr ) dλ. H(Hr ) := Λ(Hr ,F )

Further, we define a linear operator Eλ (Hr ) from the dense manifold F ∗ K in H to hλ (Hr ) by the formula r 1 ∗ Im Tλ+i0 (Hr )ψ. Eλ (Hr )F ψ = π This formula also defines a linear operator E(Hr ) from F ∗ K to H(Hr ).

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Theorem 1.3. The following assertions hold. (i) The linear operator E(Hr ) is bounded and so it admits continuous extension to H. (ii) The operator E(Hr ) maps the absolutely continuous subspace H(a) (Hr ) of Hr isometrically onto H(Hr ) and vanishes on the singular subspace of Hr . (iii) The operator E = E(Hr ) acts on elements of the subspace H(a) (Hr ) ∩ dom(Hr ) by formula E(H0 f )(λ) = λE(f )(λ). If a real number λ belongs to the set Λ(H0 , F ), then it also belongs to all sets Λ(Hr , F ), except the discrete set of those real numbers r which are poles of the meromorphic function r 7→ Tλ+i0 (Hr ). Thus, Theorem 1.3 provides a natural family of explicit diagonalisations of a continuous family of operators, which is necessary to justify (1.7). Once the fiber Hilbert spaces (1.10) are explicitly defined, the wave matrices w± (λ; Hr , H0 ) can be defined as operators acting from hλ (H0 ) → hλ (Hr ) by the formula Dp E p Im Tλ+i0 (Hr )φ, w(λ; Hr , H0 ) Im Tλ+i0 (H0 )ψ = hφ, Im Tλ+i0 (Hr ) (1 + rJTλ+i0 (H0 )) ψi . This gives the last ingredient of the right hand side of (1.7). At this point we stop our discussion of this constructive approach to stationary scattering theory, since, as mentioned above, more details and rationale can be found in [Az3 , Introduction].

1.3. Singular spectral shift function and resonance index. The formula (1.6) asserts that the singular SSF ξ (s) is a.e. integer-valued. There exist a few interpretations of the integer ξ (s) (λ). One interpretation is that ξ (s) (λ) measures the difference in winding numbers of the eigenvalues of the scattering matrix S(λ; H1 , H0 ), as it is continuously deformed to the identity operator on the fibre Hilbert space hλ in two different and natural ways. The first way is to send the imaginary part of λ from zero to +∞ and the second way is to send H1 to H0 . A second interpretation of the integer ξ (s) (λ) is the total resonance index. The resonance index of a real pole rλ of the meromorphic function r 7→ (1 + rTλ+i0 (H0 )J)−1 is the number N+ − N− , where N± is the number of eigenvalues of Tλ+iy (H0 )J in C± , y > 0, which converge to the eigenvalue −r −1 of Tλ+i0 (H0 )J as y → 0+ . The total resonance index of the pair H0 , H1 is the sum of resonance indices of poles of the function r 7→ (1+rTλ+i0 (H0 )J)−1 from [0, 1]. These interpretations of the singular SSF for trace class perturbations were given in [Az] and [Az2 , Az3 ]. In subsection 4.3 of this paper we prove the equality of the singular SSF and the total resonance index for perturbations satisfying (1.2), closely following the argument of [Az2 , Az3 ]. Since the total resonance index is integer-valued, this provides a second proof of Theorem 1.1, which does not depend on the constructive approach to stationary scattering theory given in §3. 1.4. Plan of the paper. In §2.1 and §2.2 we collect some standard material from function and operator theories which will be used later. This material includes the functional calculus for self-adjoint operators given by the Helffer-Sj¨ ostrand formula. In §2.3 we give an exposition of the theory of the SSF for pairs of operators which satisfy the condition (1.2). Starting from the Birman-Solomyak formula as the definition of the spectral shift measure, we show path-independence of the right hand side of this formula by proving exactness of the infinitesimal spectral shift measure Φ. Then we prove Krein’s trace formula, the invariance principle, and the absolute continuity of the spectral shift measure. Proof of the absolute continuity consists in reducing the matter to the trace class case using the invariance principle and then alluding to the classical theorem of Krein. The SSF is introduced as the density of the spectral shift measure. Finally, we prove a formula for the Poisson integral of the spectral shift measure. In this subsection we follow mainly [AzS]. In §2.4 we mainly discuss the Limiting Absorption Principle for pairs of operators which satisfy the condition (1.2). In §3 we give an exposition of the constructive approach to stationary scattering theory discussed above.


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In §4 we prove the main result of this paper, Theorem 1.1, and in §4.3 we discuss the connection of the singular SSF with the resonance index. 1.5. Future work. The resolvent comparable premise (1.2) covers Schr¨ odinger operators in dimensions up to three. There is a work in progress [AzD], where the main result of this paper will be extended to cover the case of ν -dimensional Schr¨ odinger operators for arbitrary ν.

2. Preliminaries 2.1. Miscellaneous. The facts collected in this subsection can be found e.g. in the books [Sch, BS, Y, DS]. 2.1.1. Measures. By a (positive) measure µ we will mean a locally-finite measure on the σ algebra of Borel subsets of the real line R. The Riesz-Markov theorem allows to identify a R measure µ with the positive linear functional φ 7→ φ(x) µ(dx) on the space of compactly supported continuous functions Cc (R). Equipping the space Cc (R) with the inductive limit topology, we may also refer to a continuous linear functional µ ∈ Cc (R)′ as a (complex-valued, unbounded) measure. Each such complex measure µ admits a decomposition µ = µ1 − µ2 + iµ3 − iµ4 , where each µj is a positive measure. By the usual default, ‘measure’ means ‘positive measure’. We denote the Lebesgue measure of a Borel set X by |X|. Null sets and full sets always refer to Lebesgue measure, as do absolute continuity, singularity, and the abbreviation a.e. A support of a measure µ is any Borel set whose complement has zero µ -measure. A support X is called minimal if for any other support X ′ the difference X \ X ′ is a null set. For the resolvent function we will use the notation Rz (x) := (x − z)−1 ,

where z ∈ C.

If for some (hence any) nonreal z, the imaginary part of the resolvent function is integrable with respect to a measure µ, then the Poisson integral of µ is defined by Z 1 Im Rz (x) µ(dx). Pµ (z) := π R

With z = λ + iy, this is the convolution µ ∗ Py (λ) of the measure µ with the Poisson kernel Py (λ) = y(π(λ2 + y 2 ))−1 . Since the Poisson kernel converges in the sense of distributions to the delta function as y → 0+ , the Poisson integral µ ∗ Py converges in the sense of distributions to µ : Z µ(φ) = lim µ ∗ Py (φ) = lim y→0+

y→0+

φ(λ)Pµ (λ + iy) dλ

R

∀φ ∈ Cc∞ (R).

On the other hand, taking the pointwise limit of the Poisson integral as y → 0+ recovers the absolutely continuous part of the measure, according to the following theorem, see e.g. [D] and [Sch, Appendices]. Theorem 2.1. Let µ be a measure such that (1 + x2 )−1 ∈ L1 (µ) and let µ = µ(a) + µ(s) be its Lebesgue decomposition, where dµ(a) = f dλ. Then the density (Radon-Nikodym derivative) f of its absolutely continuous part is equal a.e. to the limit of its Poisson integral: f (λ) = lim Pµ (λ + iy) y→0+

for a.e λ ∈ R.

In other words, µ

(a)

(φ) =

Z

R

φ(λ) lim Pµ (λ + iy) dλ y→0+

∀φ ∈ Cc∞ (R).

In particular, the Poisson integral of µ has a finite limit at a.e. point of the real axis. Moreover, the set of points where the limit fails to converge is a minimal support of the singular part of µ.

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2.1.2. Spectral measures. By a Hilbert space we mean a complex separable Hilbert space with scalar product linear in the second argument. By an operator H we mean a linear operator acting between Hilbert spaces. Occasionally, we identify a bounded operator A ∈ B(H1 , H2 ) from one Hilbert space H1 to another H2 with its continuous sesqui-linear form H2 × H1 ∋ (f, g) 7→ hf, Agi ∈ C. The spectral measure E of a self-adjoint operator H is defined by µf,g (·) = hf, E(·)gi , f, g ∈ H. The absolutely continuous subspace H(a) of H with respect to H consists of those vectors f for which µf := µf,f is absolutely continuous. The singular subspace H(s) is defined similarly. The Hilbert space H decomposes into the direct sum H(a) ⊕ H(s) of subspaces invariant under H. Then the absolutely continuous H (a) and singular H (s) parts of the operator H are its restrictions to H(a) and H(s) respectively. The absolutely continuous spectrum of H, denoted σac (H), is the spectrum of H (a) . The singular spectrum σs (H) is that of H (s) . The projections P (a) and P (s) onto H(a) and H(s) respectively can be expressed as P (a) = E(Za ) and P (s) = E(Zs ) for some (non-unique) Borel sets Za and Zs . In this case Za supports the absolutely continuous part of each of the measures µf,g , and Zs supports the singular part of each µf,g . E (a) = P (a) E and E (s) = P (s) E are respectively called the absolutely continuous and singular parts of the spectral measure E. The resolvent of H is the Cauchy-Stieltjes transform of its spectral measure: For any z ∈ ρ(H) = C \ σ(H) and any f, g ∈ H, Z Rz (λ) d hf, E(λ)gi . hf, Rz (H)gi = R

The imaginary part of the resolvent is the Poisson integral: Z

1 −1 Im Rz (λ) d hf, E(λ)gi . f, π Im Rz (H)g = π R Theorem 2.1 gives

Corollary 2.2. For any f, g ∈ H the set of points λ where there exists a finite limit

lim f, π −1 Im Rλ+iy (H)g y→0+

(a)

(s)

is a full set and a support of µf,g , whose complement is a minimal support of µf,g . For any bounded Borel function φ, we have Z D E 1 φ(λ) lim hf, Im Rλ+iy (H)gi dλ. f, φ(H)P (a) (H)g = π R y→0+ 2.1.3. Schatten ideals. (See for example [GK].) Within the Banach space of bounded operators B(H, K) are the Schatten ideals Lp (H, K), 1 6 p 6 ∞ of compact operators T such that !1/p ∞ X p sn (T ) kT kp := < ∞, n=1

where sn (T ) are the eigenvalues of |T | written in decreasing order. When p = ∞ this coincides with the norm of B(H, K) and we drop the subscript. Each Lp (H) = Lp (H, H) is a Banach space with the above norm, which for any T ∈ Lp (H) satisfies kT kp > kT k, kT ∗ kp = kT kp , and for any bounded operators A, B, (2.1)

kAT Bkp 6 kAkkT kp kBk.

This implies that the multiplication B(H) × Lp (H) × B(H) ∋ (A, T, B) 7→ AT B ∈ Lp (H) is continuous. In fact this multiplication is still continuous with the so*-topology on B(H). This follows from


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Lemma 2.3. [Y, Lemma 6.1.3] Let p > 1. Suppose A ∈ Lp (H) and Tn → 0 in the so*-topology as n → ∞. Then kTn Akp → 0 and kATn kp → 0. For compact operators we will need the analytic Fredholm alternative: Theorem 2.4. [RS, Theorem VI.14] Let G be an open connected subset of C and let T : G → L∞ (K) be a holomorphic family of compact operators on K. Then −1 ∈ σ(T (z)) either for all z ∈ G, or for only those z from the discrete set R := {z ∈ G : ∃φ 6= 0 (1 + T (z))φ = 0}.

Further, in the latter case, the operator valued function z 7→ (1 + T (z))−1 is meromorphic in G and the set of its poles is R. Of the Schatten ideals, apart from compact operators L∞ , we are only concerned with trace class operators L1 and Hilbert-Schmidt operators L2 . The trace of T ∈ L1 (H) is defined by Tr(T ) =

∞ X

n=1

hφn , T φn i ,

where (φn ) is an orthonormal basis in H. Lidskii’s theorem (see [GK], [BS] or [S2 ]) asserts that the trace is equal to the sum of eigenvalues counting multiplicities. The trace is a continuous linear functional on L1 (H) which satisfies the equality Tr(AB) = Tr(BA) whenever both AB and BA are trace class. The proof of this cyclic property is well-known at least for bounded A and B, in which case it follows from Lidskii’s theorem and the fact that AB and BA have common non-zero eigenvalues with coinciding algebraic multiplicities: (2.2)

σAB ∪ {0} = σBA ∪ {0}, including algebraic multiplicities.

If an operator T on H is trace class, then the determinant of 1 + T can be defined as the product of its eigenvalues ∞ Y (1 + λj ), det(1 + T ) = j=1

where λj are the eigenvalues of T counting multiplicities. The determinant det(1 + T ) varies continuously with T ∈ L1 (H). If A and B are Hilbert-Schmidt operators, then the product AB is trace class and kABk1 6 kAk2 kBk2 . This is (a special case of) Hölder’s inequality. The square root of a positive √ trace√class operator is Hilbert-Schmidt. Further, if 0 > An ∈ L1 and An → A in L1 , then An → A in L2 .

2.1.4. Ordered exponential. For more information see e.g. [Az, Appendix]. Consider the initial value problem 1 dX(t) = A(t)X(t), X(a) = 1, dt i where A : [a, b] → Lp (H), 1 6 p 6 ∞, is a continuous path and the derivative is taken in Lp (H). The unique solution X(t) of this problem is the (left) ordered exponential: Z t Z tk Z t2 Z ∞ X 1 t 1 dt1 A(tk ) . . . A(t1 ), dt dt . . . A(s) ds := 1 + (2.3) X(t) = Texp k k−1 i a ik a a a k=1

where 0 6 t1 6 . . . 6 tk 6 t. If p = 1 then Z t Z t 1 1 (2.4) det Texp A(s) ds = exp Tr(A(s)) ds . i a i a

10


2.1.5. Direct integral of Hilbert subspaces. For more on this topic see e.g. [BS, §7.1]. Let Λ be a Borel subset of R and suppose hλ is a closed subspace of the Hilbert space K for each λ ∈ Λ. Let Pλ ∈ B(K) be the orthogonal projection onto hλ . The field of Hilbert spaces {hλ : λ ∈ Λ} is said to be measurable if λ 7→ Pλ is weakly measurable, i.e. for all φ, ψ ∈ K the function Λ ∋ λ 7→ hφ, Pλ ψi is (Lebesgue) measurable. In this case, the direct integral Z ⊕ hλ dλ Λ

is the closed subspace of L2 (Λ, K), which consists of all square-integrable functions λ 7→ f (λ) such that f (λ) ∈ hλ for a.e. λ ∈ Λ.

2.1.6. Helffer-Sj¨ ostrand formula. The Helffer-Sj¨ ostrand formula is a functional calculus for selfadjoint operators based on the Cauchy-Green formula. It was initially constructed by E. M. Dyn’kin [Dy] and it is based on the notion of almost analytic extension introduced by L. Hörmander. In the material of this subsection we follow [DS]. Let φ ∈ Cc1 (C), by which we mean φ : C → C such that φ(x, y) := φ(x + iy) ∈ Cc1 (R2 ). Then (see for example [Ru, Lemma 20.3]) Z 1 1 ¯ (2.5) φ(ζ) = ∂φ(z)R where z = x + iy and ∂¯ = (∂x + i∂y ). z (ζ) dx dy, π R2 2 1 ˜ A C -function φ on C is called an almost analytic extension of a function φ on the real line ˜ if it is almost analytic near the real axis, in the sense that |∂¯φ(z)| = O(|y|) as |y| → 0, and it ˜ extends φ, i.e. φ|R = φ. Lemma 2.5. If φ ∈ Ccp+1 (R), p > 1, then there exists an almost analytic extension φ˜ ∈ Cc1 (C), ˜ such that ∂¯φ(z) = O(y p ) as y → 0. Proof. Put ˜ φ(z) = χ(y)

p X 1 (iy)k φ(k) (x), k! k=0

where χ is a test function equal to 1 in a neighbourhood of 0. Clearly, the function φ˜ is of class Cc1 (C) and it extends φ. Further, p

iχ′ (y) X 1 χ(y) 1 ˜ (iy)p φ(p+1) (x) + (iy)k φ(k) (x) ∂¯φ(z) = 2 p! 2 k! k=0

˜ which implies that |∂¯φ(z)| 6 const.|y|p for small |y|.

We shall assume that an almost analytic extension of φ satisfies the conclusion of this lemma, that is, if φ is of class Cc3 (R), then an analytic extension of φ is to be O(y 2 ). Proof of the following theorem is taken from [DS, Chapter 8]. Theorem 2.6. If H is a self-adjoint operator on a Hilbert space H and if φ ∈ Cc2 (R), then Z 1 ˜ ∂¯φ(z)R (2.6) φ(H) = z (H) dx dy, π R2 where φ˜ is an almost analytic extension of φ and the integral is a norm integral. Further, if φ ∈ Cc3 (R), then Z 1 2 ′ ˜ ∂¯φ(z)R (2.7) φ (H) = − z (H) dx dy. π R2 Proof. Note that the integrand on the right hand side of (2.6) is defined and is continuous in the open set y 6= 0. The almost analyticity of φ˜ combined with the estimate kRz (H)k 6 |y|−1 implies that −1 ˜ ¯˜ k∂¯φ(z)R 6 const . as |y| → 0. z (H)k 6 |∂ φ(z)||y|


11

Hence, the integrand on the right hand side of (2.6) is compactly supported, bounded, and almost everywhere continuous. Denoting the right hand side of (2.6) by Q, for any two vectors f, g ∈ H we get Z 1 ˜ hf, Rz (H)gi dx dy hf, Qgi = ∂¯φ(z) π R2 Z Z 1 ¯ ˜ = ∂ φ(z) Rz (λ) d hf, E(λ)gi dx dy, π R2 R where E is the spectral measure of H. Fubini’s theorem gives Z Z 1 ˜ hf, Qgi = ∂¯φ(z)R z (λ) dx dy d hf, E(λ)gi . R π R2 Combining this with (2.5) gives hf, Qgi =

Z

R

φ(λ) d hf, E(λ)gi = hf, φ(H)gi

and this implies (2.6). With some straightforward changes the same reasoning proves the equality (2.7), provided φ belongs to the class Cc3 (R). By Lemma 2.5, this condition ensures that the mapping z 7→ 2 ˜ ∂¯φ(z)R z (H) is bounded. The number version of (2.7) follows easily from (2.5): φ(λ) − φ(µ) λ−µ Z 1 ˜ Rz (λ) − Rz (µ) dx dy ∂¯φ(z) = lim µ→λ π R2 λ−µ Z 1 ˜ ∂¯φ(z)R = − lim z (λ)Rz (µ) dx dy µ→λ π R2 Z 1 2 ˜ =− ∂¯φ(z)R z (λ) dx dy. π R2

φ′ (λ) = lim

µ→λ

Other details are omitted.

2.2. Resolvent comparable affine spaces of self-adjoint operators. Throughout the rest of this paper, H and K will denote Hilbert spaces. H is the main Hilbert space, while K is the auxiliary Hilbert space which will appear more frequently later. The first and second resolvent identities Rz (H) − Rw (H) = (w − z)Rz (H)Rw (H) = (w − z)Rw (H)Rz (H),

Rz (H0 ) − Rz (H1 ) = Rz (H1 )(H1 − H0 )Rz (H0 ) = Rz (H0 )(H1 − H0 )Rz (H1 )

will be used very many times. The first one holds for any z and w from the resolvent set ρ(H) of the operator H. The second one holds if z ∈ ρ(H1 ) ∩ ρ(H0 ) and dom H1 = dom H0 . This is satisfied if z is nonreal, H0 is self-adjoint, and the difference V = H1 − H0 is relatively compact with respect to H0 . With H0 self-adjoint, the last condition (abbreviated “V is H0 compact”) means that dom H0 ⊂ dom V and the operator V Rz (H0 ) is compact for some, hence any, nonreal z. (“Hence any” because the compact operators form an ideal of the ring of bounded operators and (Rw /Rz )(H0 ) is bounded for any nonreal z, w. ) In this case, by the Kato-Rellich theorem, the operator H1 = H0 + V is closed and has the same domain as H0 . And further, if the perturbation V is symmetric, then H1 is self-adjoint. We will assume in addition that V is bounded. Lemma 2.7. Let H0 and V be self-adjoint, with V bounded and H0 -compact, and let z be nonreal. The compact operators Rz (H0 )V and V Rz (H0 ) do not have nonzero real eigenvalues.

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This well known lemma ensures that the operators 1+V Rz (H0 ) and 1+Rz (H0 )V are invertible and from the second resolvent identity we obtain the important equalities Rz (H1 ) = Rz (H0 )(1 + V Rz (H0 ))−1 = (1 + Rz (H0 )V )−1 Rz (H0 ). It follows that V is also relatively compact with respect to H1 , or for that matter any other self-adjoint operator H such that the difference H − H0 is H0 -compact. By a relatively compact affine space of self-adjoint operators on H, we will mean an affine space A of self-adjoint operators on H over a real vector space A0 of bounded self-adjoint operators on H, such that any direction V ∈ A0 is relatively compact with respect to any H ∈ A. For this to hold it is enough for V Rz (H0 ) to be compact for any V ∈ A0 , some H0 ∈ A, and some nonreal z, by the resolvent identities. Proposition 2.8. If A is a relatively compact affine space then all operators from A have the same domain and the same essential spectrum. Proof. These assertions follow respectively from the Kato-Rellich theorem and Weyl’s theorem, see e.g. [RS3 , Corollary 2 of Theorem XIII.14]. The common domain of operators from A will be denoted D and the common essential spectrum by σess . Two self-adjoint operators H0 , H1 will be called resolvent comparable if the difference of their resolvents is trace class: (2.8)

Rz (H1 ) − Rz (H0 ) ∈ L1 (H).

Resolvent comparability is an equivalence relation. Note that if (2.8) holds for some nonreal z, then it holds for any nonreal z. We say that a relatively compact affine space A of self-adjoint operators on H is resolvent comparable if any pair of operators from A satisfies (2.8). Using the resolvent identities, it is equivalent to require that for any V from A0 , the operator Rz (H0 )V Rz (H0 ) is trace class for some H0 ∈ A and some nonreal z. Let H0 be self-adjoint and let V be bounded and self-adjoint. If V is H0 -compact, then the straight line Hr = H0 + rV, r ∈ R forms a relatively compact affine space. If the pair H0 , H0 + V also satisfies the hypothesis (2.8), then the same line is a resolvent comparable affine space. Further, if we have a finite collection V1 , . . . , Vn ∈ Bsa (H) of H0 -compact operators, then A = H0 + A0 , where A0 := spanR {V1 , . . . , Vn }, is a relatively compact affine space. And if in addition Rz (H0 )Vj Rz (H0 ) ∈ L1 (H), for j = 1, . . . , n, then A is a resolvent comparable affine space. Suppose a bounded operator F : H → K is H0 -compact and set A0 := F ∗ Bsa (K)F. Then A = H0 + A0 is a relatively compact affine space. If F is such that the operator F Rz (H0 ) is Hilbert-Schmidt for some (hencepany) nonreal z, then A is a resolvent comparable affine space. In particular, we can put F = |V |. Let E be any spectral projection of V and consider the equality p p (2.9) Rz¯(H0 )E|V |Rz (H0 ) = (E |V |Rz (H0 ))∗ (E |V |Rz (H0 )). p It follows that the operator (2.9) is trace class if and only if the operator E |V |Rz (H) is Hilbert-Schmidt. Therefore, for example, the pair H0 , H0 + |V | is resolvent comparable if and p only if |V |Rz (H0 ) ∈ L2 (H). And if this holds, then in addition, the pair H0 , H0 + EV is resolvent comparable for any spectral projection E of V. In particular, the pairs H0 , H0 + V+ and H0 , H0 + V− are resolvent comparable and hence so is the pair H0 , H0 + V. We will now consider the motivating example, namely, low-dimensional Schr¨ odinger operators: If H0 = −∆ + V0 is a Schr¨ odinger operator on L2 (Rν ), ν = 1, 2, or 3, where V0 is the operator of multiplication by a bounded measurable real-valued function which we denote by the same symbol, and if V is the operator of multiplication by an integrable real-valued function, then the pair of self-adjoint operators H0 and H0 + V is resolvent comparable. This can be derived


13

from results which address the following kind of questions. Let H be a Schr¨ odinger operator and let f and g be two functions. When is the operator (2.10)

g(x)f (H)

Hilbert-Schmidt? These kind of questions are discussed in detail in B. Simon’s book [S2 ] in the case of the free Hamiltonian H = −∆. For some other Schr¨ odinger operators H = −∆ + V, Schatten ideal properties of the operator (2.10) can be reduced to the case of the free Hamiltonian with the aid of the second resolvent identity; this is a well-known method which is also used in the proof of the Limiting Absorption Principle for Schr¨ odinger operators with short range potentials, see e.g. [Ag]. For general Schr¨ odinger operators H = −∆ + V, Schatten ideal properties of the product (2.10) are surveyed in [S]. For our purposes, Corollary 2.11 below is easily implied by [S, Theorem B.9.1], however we will use the above mentioned method involving the second resolvent identity, beginning with the more basic result: Theorem 2.9. [S2 , Theorem 4.1] If f, g ∈ L2 (Rν ), then the operator g(x)f (−i∇) is HilbertSchmidt. The following corollary is an application of Theorem 2.9 with f (x) := (x2 − z)−1 ; a function which is square-integrable only when ν 6 3. This is the reason that the method used in this paper works for Schr¨ odinger operators only in dimensions up to 3 (the same restriction on dimension appears in [S, Theorem B.9.1]). Corollary 2.10. If g ∈ L2 (Rν ), ν = 1, 2, or 3, then the operator g(x)Rz (−∆) is HilbertSchmidt. Corollary 2.11. Let H0 = −∆ + V0 be a Schr¨ odinger operator on L2 (Rν ), where V0 ∈ ν ν ν L∞ (R ), p and let V ∈ L∞ (R ) ∩ L1 (R ). If ν = 1, 2, or 3, then for any nonreal z the |V |Rz (H0 ) is Hilbert-Schmidt. Further, there is a resolvent comparable affine space operator containing H0 and H0 + V, as well as H0 + EV where E is any spectral projection of V. p Proof. Since V ∈ L1 (Rν ), we have F := |V | ∈ L2 (Rν ). From Corollary 2.10, it follows that the operator F Rz (−∆) is Hilbert-Schmidt. Then using the second resolvent identity, the operator F Rz (H0 ) = F Rz (−∆)(1 + V0 Rz (H0 )) is also Hilbert-Schmidt. Further, let A0 = F Bsa (H)F, and A = H0 + A0 . Since F Rz (H0 ) is Hilbert-Schmidt, for any F JF ∈ A0 the operator Rz (H0 )F JF Rz (H0 ) is trace class. It follows that A is a resolvent comparable affine space of self-adjoint operators, when A0 is given the norm kF JF kF := kJk. The fact that (2.11) is continuous is not difficult to check and this is done in §2.4. Finally, any spectral projection E of V commutes with F and hence EV = F EF ∈ A0 . If A has infinite dimension, then we should specify a topology. This is done below, although these concerns are of little consequence if one is primarily interested in pairs of operators H0 , H0 + V. Given an affine space A of self-adjoint operators which is (A) relatively compact, or (B) resolvent comparable, we (from now on) make the assumption that the real vector space A0 is equipped with a norm such that, for some nonreal z and some H ∈ A, the mapping (2.11)

A0 × A0 × A0 ∋ (V1 , V2 , V3 ) 7→ Rz (H + V1 )V2 Rz (H + V3 ) ∈ Lp (H),

is continuous, where in case (A) p = ∞, while in case (B) p = 1. Either case (A) or (B) implies that the resolvent Rz (H) depends norm-continuously on H ∈ A. This can be seen by taking V1 = V2 and V3 = 0. From this fact and the first resolvent identity, it follows that if (2.11) is continuous for some nonreal z, then it is continuous for any nonreal z. On the other hand, replacing V1 by V1 + V and V3 by V3 + V for a fixed V shows that if (2.11) is continuous for some H ∈ A, then it is continuous for any. The map (2.11) is continuous in either of the following situations: (i) both V 7→ V Rz (H) ∈ B(H) and V 7→ Rz (H)V Rz (H) ∈ Lp (H) are continuous for some (hence any) nonreal z and some (hence any) H ∈ A, where p = ∞ in case (A) and p = 1 in case (B); (ii) there exists

14


a bounded operator F, such that each direction V ∈ A0 admits a decomposition of the form V = F ∗ JF, where J is some bounded self-adjoint operator, and F is such that F Rz (H) ∈ Lp for some (hence any) nonreal z and some (hence any) H ∈ A, where p = ∞ in case (A) and p = 2 in case (B). In this situation we equip A0 with the norm kF ∗ JF kF := kJk. The first of these (i) is not difficult to check using the second resolvent identity. The second (ii) is checked in §2.4 and also uses the second resolvent identity. In case (A), the usual norm which A0 inherits from B(H) gives (i). Finite-dimensional affine spaces A are always cases of (i). And with F being bounded, (ii) implies (i). However (ii) does not depend on (i) in the sense that (ii) directly implies the continuity of (2.11). Proposition 2.12. If A is a relatively compact affine space of self-adjoint operators, then for any nonreal z and for any φ ∈ Cc (R) the mappings A ∋ H 7→ Rz (H) ∈ B(H) and A ∋ H 7→ φ(H) ∈ B(H) are continuous. Proof. Norm-resolvent continuity (i) follows from the continuity of (2.11) and has already been noted. And (i) implies (ii), which can be seen e.g. from [RS, Theorem VIII.20(a)]. Lemma 2.13. Let A be a resolvent comparable affine space of self-adjoint operators on H. Let Hr be a continuously differentiable path in A, with derivative H˙ r . For any nonreal z, any H ∈ A, and any φ, ψ ∈ Cc (R), the mappings (i) r 7→ Rz (Hr ) − Rz (H),

(ii) r 7→ −Rz (Hr )H˙ r Rz (Hr ),

(iii) r 7→ φ(Hr )H˙ r ψ(Hr )

are continuous paths in L1 (H). The first one (i) is continuously differentiable and its continuous derivative is (ii). This lemma also holds for relatively compact affine spaces, if L1 (H) is replaced by L∞ (H). The proof is identical. Moreover, this lemma and the next one lead to Proposition 2.15 and there are analogues of each of these in the relatively compact case which will not be used in this paper. Proof. Consider the second resolvent identity (2.12)

Rz (A) − Rz (B) = Rz (A)(B − A)Rz (B)

Putting A = Hr and B = H, it follows from the continuity of (2.11) that the difference of resolvents Rz (Hr ) − Rz (H) continuously depends on r ∈ [0, 1]. Now let A = Hr and B = Hs . Since Hr is continuously differentiable, dividing (2.12) by r − s and sending s → r gives d Rz (Hs ) = −Rz (Hr )H˙ r Rz (Hr ) ∈ L1 (H). ds s=r The mapping r 7→ H˙ r ∈ A0 is continuous and therefore so is the function (ii). Finally, since Hr is continuous, the function r 7→ φ(Hr )(Hr − z) is norm-continuous for any φ ∈ Cc (R) by Proposition 2.12. Therefore, r 7→ φ(Hr )H˙ r ψ(Hr ) = φ(Hr )(Hr − z)Rz (Hr )H˙ r Rz (Hr )(Hr − z)ψ(Hr ) ∈ L1 (H), is continuous for any φ, ψ ∈ Cc (R).

Lemma 2.14. Let A and Hr be as in Lemma 2.13. For any H ∈ A, for any z = x + iy within a subset of C of the form K \R, where K is compact, and for any r within the compact interval [0, 1], there holds the estimate (2.13)

kRz (Hr ) − Rz (H)k1 6 const.|y|−2 .

Further, for any z ∈ K \ R, and any r, s ∈ [0, 1],

Rz (Hr ) − Rz (Hs )

6 const.|y|−2 .

(2.14)

r−s 1

If s = r, the operator on the left hand side is to be interpreted as the derivative −Rz (Hr )H˙ r Rz (Hr ).


15

Proof. Using the first resolvent identity, we get Rz (A) − Rz (B) = Rz (A)(B − A)Rz (B)

= (1 + (z − w)Rz (A))Rw (A)(B − A)Rw (B)(1 + (z − w)Rz (B)),

where w is a fixed nonreal number. For A, B ∈ A, it follows that this difference of resolvents depends continuously in L1 (H) on z ∈ C \ R, since z 7→ 1 + (z − w)Rz (H) is norm-continuous away from the real line for any self-adjoint H. With z = x + iy, using the estimate kRz (H)k 6 |y|−1 , we obtain |x| + |w| 2 (2.15) kRz (A) − Rz (B)k1 6 2 + kRw (A)(B − A)Rw (B)k1 . |y| With A = Hr and B = H, according to Lemma 2.13 there exists a constant c such that |x| + |w| 2 , kRz (Hr ) − Rz (H)k1 6 c 2 + |y|

for any r ∈ [0, 1]. The required estimate (2.13) follows for z within K \ R. To prove (2.14), put A = Hr , B = Hs , and consider (2.15) divided by s − r. By Lemma 2.13, the divided difference Hr − Hs Rw (Hr ) − Rw (Hs ) = −Rw (Hr ) Rw (Hs ) r−s r−s is an L1 -continuous function of two variables r, s. Therefore there exists a constant c such that

2

Rz (Hr ) − Rz (Hs )

6 c 2 + |x| + |w| ,

r−s |y| 1 for any r, s ∈ [0, 1]. From this follows (2.14).

Proposition 2.15. Let A be a resolvent comparable affine space of self-adjoint operators on H. If {Hr } is a continuously differentiable path in A, then for any φ ∈ Cc3 (R) the map r 7→ φ(Hr ) − φ(H) ∈ L1 (H)

is continuously differentiable for any H ∈ A. Moreover, its trace class derivative is Z 1 dφ(Hr ) ˜ ˙ =− ∂¯φ(z)R (2.16) z (Hr )Hr Rz (Hr ) dx dy, dr π R2 where φ˜ is an almost analytic extension of φ, z = x + iy, and ∂¯ = 12 (∂x + i∂y ). The integral exists in L1 (H). Proof. For any A, B ∈ A, from the Helffer-Sj¨ ostrand formula (2.6) we have Z 1 ˜ (2.17) φ(A) − φ(B) = ∂¯φ(z)(R z (A) − Rz (B)) dx dy. π R2 ˜ Since |∂¯φ(z)| = O(|y|2 ) as |y| → 0, it follows from Lemma 2.14 that the integrand (2.18)

˜ (x, y) 7→ ∂¯φ(z)(R z (A) − Rz (B)) ∈ L1 (H),

is bounded. Therefore, the integral exists in the trace class norm. To show that the mapping r 7→ φ(Hr ) − φ(H) ∈ L1 (H) is continuous, let A = Hr and B = H. By Lemma 2.14, the family of mappings (2.18) for r ∈ [0, 1] is dominated. It follows that (2.17) is L1 -continuous as a function of r. To find the derivative, let A = Hr and B = Hs and consider (2.17) divided by r − s. Again using Lemma 2.14, the family of integrands ˜ Rz (Hr ) − Rz (Hs ) ∈ L1 (H) (x, y) 7→ ∂¯φ(z) r−s for r, s ∈ [0, 1] is dominated. Therefore the divided difference Z 1 φ(Hr ) − φ(Hs ) Hr − Hs ˜ =− Rz (Hs ) dx dy ∈ L1 (H) ∂¯φ(z)R z (Hr ) r−s π R2 r−s

16


is continuous. Sending s → r gives (2.16).

The integral on the right hand side of (2.16) is a double operator integral. In the language of the theory of multiple operator integrals presented in [ACDS], the equation (2.16) reads dφ(Hr ) = TφH[1]r ,Hr H˙ r , dr

where φ[1] is the first divided difference of φ. A Birman-Solomyak-representation of a function f of n + 1 variables is an integral representation of the form Z α0 (λ0 , s) . . . αn (λn , s) dν(s), f (λ0 , . . . , λn ) = S

where (S, ν) is a finite measure space and α0 , . . . , αn are bounded Borel functions on R × S. A multiple operator integral is then defined by Z H0 ,...,Hn α0 (H0 , s)V1 α1 (H1 , s)V2 . . . Vn αn (Hn , s) dν(s), (V1 , . . . , Vn ) := Tf S

where H0 , . . . , Hn are self-adjoint operators and V1 , . . . , Vn are bounded operators. This definition does not depend on the BS-representation of the function f (see [ACDS] for more details). In (2.16), we are using the BS-representation Z [1] α0 (λ0 , s)α1 (λ1 , s) dν(s), φ (λ0 , λ1 ) = S

˜ dx dy. where s = (x, y), z = x + iy, αj (λj , s) = Rz (λj ) for j = 1, 2, and dν(s) = −π −1 ∂¯φ(z)

2.3. Spectral shift function. This subsection is devoted to establishing an analogue of the Birman-Solomyak formula for resolvent comparable operators. This is a classical folklore result, but nevertheless we present here its proof which to the best of our knowledge has novel aspects. This proof uses the method of the paper [AzS], where the SSF is defined for so-called tracecompatible operators, which satisfy V φ(Hr ) ∈ L1 (H) for any test function φ. With some adjustments, the arguments of [AzS] are reproduced below. 2.3.1. Spectral shift measure. Throughout the rest of this subsection, A will denote a resolvent comparable affine space of self-adjoint operators on a Hilbert space H. Note that if H1 and H0 are resolvent comparable, then 1φ (Hr )V φ(Hr ) ∈ L1 (H), where 1φ is a function of class Cc (R) which is equal to 1 on the support of φ (see Lemma 2.13). For H ∈ A, V ∈ A0 , the measure defined by (2.19)

ΦH (V )(φ) = Tr(1φ (H)V φ(H))

φ ∈ Cc (R)

is called the infinitesimal spectral shift measure. The right hand side of (2.19) does not depend on the choice of a function 1φ . Indeed, suppose ˜1φ is another function equal to 1 on the support of φ. Then the trace class operators 1φ (H)V φ(H) and ˜ 1φ (H)V φ(H) have the same nonzero eigenvalues (counted with algebraic multiplicities), since by (2.2) they both share their nonzero eigenvalues with V φ(H). Therefore, having the same eigenvalues, they have the same trace by Lidskii’s theorem. We will sometimes write the infinitesimal spectral shift measure as ΦH (V )(φ) = Tr(V φ(H)). More generally, if the eigenvalues of a compact operator A are summable, then let Tr(A) denote their sum, whether or not A is trace class. In particular this will be used is when the operator A has a decomposition A = BC, where B and C are bounded operators and the product CB is trace class. As a rule, this formal notation will not be used in a proof. The notation ΦH (V )(φ) is intended to suggest a generalised 1-form on the affine space A. Indeed, for any test function φ, Φ(φ) is a 1-form, while for a point H ∈ A and a tangent direction V ∈ A0 , ΦH (V ) is a generalised function, in particular a real measure. We will not need any deep theory of such objects.


17

It is well-known and easily shown that the infinitesimal spectral shift ΦH (V ) is a real measure, which is positive if V is. Any compactly supported bounded Borel function φ is integrable w.r.t. ΦH (V ) and the integral ΦH (V )(φ) is given by (2.19). Let H1 , H0 ∈ A and let Hr be a continuously differentiable path from H0 to H1 . Then the spectral shift measure is defined as the integral of the infinitesimal spectral shift along the path Hr : Z 1 Tr 1φ (Hr )H˙ r φ(Hr ) dr φ ∈ Cc (R). (2.20) ξ(φ; H1 , H0 ) = 0

The spectral shift ξ is indeed a measure: using Lemma 2.13, it’s easy to show that if a sequence (φn ) ⊂ Cc (R) converges to zero in the inductive limit topology, then the convergence | Tr(1φn (Hr )V φn (Hr ))| → 0 is locally uniform with respect to r. What’s more, any compactly R supported bounded Borel function φ is integrable with respect to ξ and the integral ξ(φ) = φ dξ is given by the same formula (2.20).

Theorem 2.16. The spectral shift measure is independent of the path Hr connecting H0 to H1 .

Proof. To prove path-independence we show that the 1-form Φ(φ) is exact for any test function φ, in other words, there are 0-forms θ φ such that d φ φ ΦH (V )(φ) = dθH (V ) = θH+sV . ds s=0

φ For this purpose we define θH to be the spectral shift measure (i.e. the integral of the infinitesimal spectral shift) along the straight path Hr from some H0 to H (any fixed choice of H0 will do). Let W = H − H0 and Hr = H0 + rW. Unpacking definitions, d φ φ dθH (V ) = θH+sV ds s=0 Z 1 1 (2.21) Tr[1φ (Hr + rsV )(W + sV )φ(Hr + rsV ) − 1φ (Hr )W φ(Hr )] dr. = lim s→0 0 s

The limit and the integral can be interchanged here, because the integrand of (2.21) continuously depends on 0 6 r, s 6 1. This is more easily seen if the operator appearing in the trace is rewritten: 1 1φ (Hr + rsV )(W + sV )φ(Hr + rsV ) − 1φ (Hr )W φ(Hr ) s = 1φ (Hr + rsV )V φ(Hr + rsV ) 1φ (Hr + rsV ) − 1φ (Hr ) W φ(Hr + rsV ) s φ(Hr + rsV ) − φ(Hr ) + 1φ (Hr )W . s

+

By Proposition 2.15, assuming that 1φ is chosen from Cc3 (R), this operator varies continuously in L1 (H) with 0 6 r, s 6 1. Moreover, as s → 0 it converges to (2.22)

Ar := 1φ (Hr )V φ(Hr ) +

d1φ (Hr + rsV ) dφ(Hr + rsV ) W φ(Hr ) + 1φ (Hr )W . ds ds s=0 s=0

Therefore, interchanging the limit and the integral in (2.21) gives the equality Z 1 φ Tr(Ar ) dr. dθH (V ) = 0

18


Consider the trace of the last term of (2.22). Using the formula (2.16) from Proposition 2.15, we get Z dφ(Hr + rsV ) 1 ¯ ˜ Tr 1φ (Hr )W ∂ φ(z)Rz (Hr )rV Rz (Hr ) dx dy = Tr 1φ (Hr )W − ds π R2 s=0 Z 1 ˜ Tr(1φ (Hr )W Rz (Hr )rV Rz (Hr )) dx dy ∂¯φ(z) =− π R2 Z r ˜ Tr(Rz (Hr )W Rz (Hr )V 1φ (Hr )) dx dy ∂¯φ(z) =− π R2 Z 1 ¯ ˜ = r Tr − ∂ φ(z)Rz (Hr )W Rz (Hr ) dx dy V 1φ (Hr ) π R2 dφ(Hr ) = r Tr V 1φ (Hr ) . dr Here the interchanges of the trace and the integral are justified since by Proposition 2.15 the integrals exist and are taken in L1 (H). Similarly we obtain d1φ (Hr + rsV ) d1φ (Hr ) Tr . W φ(Hr ) = r Tr φ(Hr )V ds dr s=0

Therefore, the trace of the operator (2.22) is equal to d (φ(Hr )V 1φ (Hr )) . Tr(Ar ) = Tr(1φ (Hr )V φ(Hr )) + r Tr dr

Note that the function r 7→ φ(Hr )V 1φ (Hr ) is L1 -continuously differentiable, and we have d d d (φ(Hr )V 1φ (Hr )) = Tr (φ(Hr )V 1φ (Hr )) = Tr (1φ (Hr )V φ(Hr )) . Tr dr dr dr φ Returning to dθH (V ), it is thus equal to Z 1 φ Tr(Ar ) dr dθH (V ) = 0 Z Z 1 Tr(1φ (Hr )V φ(Hr )) dr + =

0

0

1

r

d Tr (1φ (Hr )V φ(Hr )) dr. dr

Integrating the last term by parts gives φ dθH (V ) = Tr (1φ (H)V φ(H)) = ΦH (V )(φ),

completing the proof.

For any H1 , H0 ∈ A with H1 − H0 = V, the spectral shift measure can be written, using the notation introduced on p. 16, as Z 1 Tr(V φ(Hr )) dr φ ∈ Cc (R). ξ(φ; H1 , H0 ) = 0

Note that the spectral shift measure is real-valued and positive if V > 0. It is also additive in the sense that if H0 , H1 , H2 ∈ A then ξ(φ; H2 , H0 ) = ξ(φ; H2 , H1 ) + ξ(φ; H1 , H0 ) ∀φ ∈ Cc (R). Thus we can consider the spectral shift along any piecewise continuously differentiable path in A. To show that the spectral shift measure satisfies Krein’s trace formula (1.1) and obeys the invariance principle, we use the following


19

Proposition 2.17. Let Hr be a continuously differentiable path in a resolvent comparable affine space and let φ ∈ Cc3 (R). Then, for the mapping r 7→ φ(Hr ), the chain rule holds under the trace: dφ(Hr ) (2.23) Tr f (Hr ) = Tr 1φ (Hr )H˙ r φ′ (Hr )f (Hr ) , dr

for any bounded measurable function f.

Proof. The left hand side of (2.23) is equal to dφ(Hr ) Tr 1φ (Hr ) f (Hr ) dr r) from Proposition 2.15, we get Now using the trace class integral formula (2.16) for dφ(H dr Z dφ(Hr ) 1 ˜ ˙ Tr f (Hr ) = Tr 1φ (Hr ) − ∂¯φ(z)R z (Hr )Hr Rz (Hr ) dx dy f (Hr ) dr π R2 Z 1 ˜ Tr 1φ (Hr )Rz (Hr )H˙ r Rz (Hr )f (Hr ) dx dy =− ∂¯φ(z) π R2 Z 1 ˜ Tr 1φ (Hr )H˙ r R2 (Hr )f (Hr ) dx dy =− ∂¯φ(z) z π R2 Z 1 2 ˜ ∂¯φ(z)R = Tr 1φ (Hr )H˙ r − z (Hr ) dx dy f (Hr ) . π R2

The operator appearing in the brackets within the trace is φ′ (Hr ) by Theorem 2.6, and so the proof is complete. Krein’s trace formula (1.1) can now be obtained by integrating (2.23) with f = 1. Corollary 2.18. Let H1 , H0 be a pair of self-adjoint operators from a resolvent comparable affine space. Then the spectral shift measure satisfies the formula for any φ ∈ Cc3 (R).

Tr(φ(H1 ) − φ(H0 )) = ξ(φ′ ; H1 , H0 )

It follows that in the case of trace class perturbations, the spectral shift measure is absolutely continuous and its density coincides with the classical SSF. Theorem 2.19. Let A be a resolvent comparable affine space, let H1 , H0 ∈ A, and let φ ∈ Cc3 (R) be real-valued. If ξ and ξφ are the spectral shift measures of the pairs H1 , H0 and φ(H1 ), φ(H0 ) respectively, then for any compactly supported and bounded Borel function g ξφ (g) = ξ(g ◦ φ · φ′ ). Proof. Since the difference φ(H1 ) − φ(H0 ) is trace class, the pair of self-adjoint operators φ(H1 ), φ(H0 ), are certainly resolvent comparable. Using Proposition 2.17, we have dφ(Hr ) g(φ(Hr )) = Tr 1φ (Hr )H˙ r φ′ (Hr )g(φ(Hr )) . Tr dr

Since by Theorem 2.16 the spectral shift measure is path-independent, integrating both sides of this equality over r ∈ [0, 1] gives the required equality.

2.3.2. Absolute continuity of spectral shift measure. Now we are in position to prove absolute continuity of the spectral shift measure for resolvent comparable pairs of operators. The proof relies on the invariance principle for the spectral shift measure (Theorem 2.19) and reduces the question of absolute continuity of the spectral shift measure for a resolvent comparable pair to that of a pair with trace class difference.

20


Theorem 2.20. Let A be a resolvent comparable affine space of self-adjoint operators and let H1 , H0 ∈ A. Suppose there exist positive V1 , V2 ∈ A0 such that V1 − V2 = V = H1 − H0 . Then the spectral shift measure ξ(·; H1 , H0 ) is absolutely continuous. Proof. Since the spectral shift measure is additive, if the spectral shifts from H0 to H0 + V1 and from H0 +V1 to H1 are absolutely continuous, then so is the spectral shift from H0 to H1 . Thus we can assume without loss of generality that the perturbation V is positive, so that the spectral shift measure is positive. Let µ be the singular part of the spectral shift measure. We will show that µ is translation invariant. It therefore must be a multiple of Lebesgue measure, leaving µ = 0 as the only possibility. Fix a small positive number ǫ and let E be a Borel subset of [ǫ, 1 − ǫ] with Lebesgue measure 0. To demonstrate translation invariance of µ, without loss of generality it is enough to show that µ(E) = µ(E + a) for any a ∈ R. For any a, b ∈ R with b − a > 2, we consider a test function fa,b , whose graph looks like a smoothed isosceles trapezium with height 1 and slopes of the sides equal to ±1, stretched over the interval [a, b].

a

a+1

b−1

b

More precisely, we choose fa,b ∈ Cc∞ (R) subject to the constraints: fa,b (λ) = λ − a on the −1 ([ǫ, 1 − ǫ]) = interval (a + ǫ, a + 1 − ǫ); fa,b (λ) = b − λ on the interval (b − 1 + ǫ, b − ǫ); and fa,b [a + ǫ, a + 1 − ǫ] ∪ [b − 1 + ǫ, b − ǫ], in other words, except as already specified, fa,b does not take values in [ǫ, 1 − ǫ]. Let 1A denote the indicator function of a Borel set A. Note that by ′ =1 construction 1E ◦ fa,b · fa,b a+E − 1b−E . Using Theorem 2.19, we get (2.24)

′ ξfa,b (1E ) = ξ(1E ◦ fa,b · fa,b ) = ξ(1a+E ) − ξ(1b−E ) = µ(1a+E ) − µ(1b−E ).

By Proposition 2.15 and Corollary 2.18, the spectral shift measure ξfa,b is absolutely continuous. Thus the left hand side of (2.24) is 0. Hence choosing b such that b > 2 and b − a > 2, we conclude that µ(1a+E ) = µ(1b−E ) = µ(1E ), which completes the proof. The spectral shift function is the locally integrable density of the spectral shift measure. The function and the measure are denoted by the same symbol. We have Z φ(λ)ξ(λ; H1 , H0 ) dλ, ξ(φ; H1 , H0 ) = R

for any compactly supported bounded Borel function φ. The spectral shift function (SSF) may be represented as the limit of its Poisson integral:

Proposition 2.21. Let A be a resolvent comparable affine space of self-adjoint operators and let H0 , H1 ∈ A be as in Theorem 2.20. Then the SSF ξ of the pair H0 , H1 belongs to L1 (R, (1 + x2 )−1 dx) and the Poisson integral Pξ (z) of ξ is equal to Z y 1 Tr Rz (Hr )H˙ r Rz¯(Hr ) dr, (2.25) π 0 where y = Im z and Hr is any continuously differentiable path in A from H0 to H1 . Proof. Path-independence of the spectral shift measure (Theorem 2.16) allows us to assume that the difference H1 − H0 is non-negative, and therefore so is the spectral shift measure ξ. Let z = λ + iy be a nonreal complex number with positive imaginary part y and let 1n be the indicator of the interval [−n, n]. Since the increasing sequence of non-negative functions 1n (x) · Im Rz (x) converges pointwise to the positive function Im Rz (x), the monotone convergence theorem implies the equality Z Z 1n (x) · Im Rz (x) ξ(dx), Im Rz (x) ξ(dx) = lim R

n→∞ R


21

whether or not the limit is finite. From the definition of the spectral shift measure ξ and the identity Im Rz = yRz¯Rz it follows that the right hand side of the last equality is equal to Z 1 Tr 1n (Hr )H˙ r 1n (Hr ) Im Rz (Hr ) dr (2.26) lim n→∞ 0 Z 1 Tr 1n (Hr )Rz (Hr )H˙ r Rz¯(Hr )1n (Hr ) dr. = y lim n→∞ 0

Since the sequence of operators 1n (Hr )Rz (Hr )H˙ r Rz¯(Hr )1n (Hr ) converges to Rz (Hr )H˙ r Rz¯(Hr ) in the trace class norm as n → ∞, the sequence of integrands of the right hand side of (2.26) converges pointwise to the continuous function r 7→ Tr(Rz (Hr )H˙ r Rz¯(Hr )), which dominates the sequence. Hence, by dominated convergence, Z 1 Z Tr Rz (Hr )H˙ r Rz¯(Hr ) dr < ∞. Im Rz (x) ξ(dx) = y R

Since Pξ (z) = π −1

R

0

Im Rz (x) ξ(dx), this completes the proof.

The Poisson integral Pξ (z) of the SSF we denote by ξ(z; H1 , H0 ) and we call it the smoothed SSF. Since the Poisson integral of a function of class L1 (R, (1 + x2 )−1 dx) converges a.e. to the function itself as y → 0+ , Proposition 2.21 implies the following equality for the SSF: Z y 1 ξ(λ; H1 , H0 ) = lim Tr Rλ+iy (Hr )H˙ r Rλ−iy (Hr ) dr, a.e. λ ∈ R. y→0+ π 0

2.4. Rigged Hilbert space and the Limiting Absorption Principle. The setting of a Hilbert space alone is not sufficient for our purposes. By a rigged Hilbert space (H, F ) we mean a Hilbert space H along with a closed operator F from H to another Hilbert space K, with trivial kernel and cokernel. In other words, F is an injective operator and K = ran F . The operator F is called a rigging operator. The partial isometry U : H → K in the polar decomposition F = U |F | is unitary since ker(U ) = ker(F ) = {0} and ran U = ran F = K, so we may naturally identify H and K. Although K being different from H is thus technically unnecessary, it is convenient. There will be an assumed relationship between a rigging operator and a perturbation operator (see p. 23). Because in this paper we are dealing only with bounded perturbations of a selfadjoint operator, we only need to consider bounded rigging operators. For this reason a rigging operator will always be assumed to be bounded, unless specified otherwise. However, this paper has been written in such a way as to depend minimally (within reason) on this assumption. The following information is included to give an idea of the connection to other definitions of rigged Hilbert space, but will not be further used. Suppose that F is a (bounded) rigging operator. The dense linear subspace ran F ∗ of H can be endowed with an inner product, pushed from K, that makes F ∗ isometric. With this inner product ran F ∗ is complete and so a Hilbert space in its own right, which we denote H+ . This Hilbert space is continuously embedded in H. The completion of H in the inner product (f, g) 7→ hF f, F gi we denote H− . Whether or not a rigging operator F is bounded, the spaces H± can be succinctly defined as the completions of the dense linear subspace dom F ∩ ran F ∗ in the inner products

(2.27) hf, gi± = |F |∓1 f, |F |∓1 g H .

The operator F ∗ can be considered as (or, if F is unbounded, extended to) a unitary operator identifying K and H+ , and the operator F can be extended to a unitary operator identifying H− and K. If F is bounded, there are continuous and dense embeddings: H+ ֒→ H ֒→ H− . Recall that for an operator F to be relatively compact with respect to a self-adjoint operator H means that for some, hence any, nonreal z the operator F Rz (H) is compact. Likewise, we will say that F is relatively Hilbert-Schmidt with respect to H, or simply H -HilbertSchmidt, if for some, hence any, nonreal z the operator F Rz (H) is Hilbert-Schmidt.

22


Proposition 2.22. Let H be a self-adjoint operator and let F be a rigging operator. Then for any nonreal z, the operator F Rz (H) is compact if and only if the operator F Rz (H)F ∗ is compact. Proof. “Only if” is obvious since we are assuming F to be bounded. “If” (shown in [Az3 , Lemma 2.5.1]): Since F Rz (H)F ∗ is compact, so is its adjoint F Rz¯(H)F ∗ . Hence, so is the difference F Rz (H)F ∗ − F Rz¯(H)F ∗ = (z − z¯)F Rz (H)Rz¯(H)F ∗ = (z − z¯)(F Rz (H))(F Rz (H))∗ ,

where we have used the first resolvent identity. If follows that F Rz (H) is compact. Indeed, in general if A∗ A = |A|2 is compact, then so is its square root |A| and so is A. The sandwiched resolvent will be denoted Tz (H) := F Rz (H)F ∗ . This notation does not indicate the dependence on the rigging operator; it is used only in the context of a fixed rigged Hilbert space (H, F ). Proposition 2.23. Let H be a self-adjoint operator on a Hilbert space H and let F : H → K be a rigging operator. For any nonreal z, the operator F Rz (H) is Hilbert-Schmidt if and only if the operator Im Tz (H) is trace class. Proof. Let z = λ + iy. The operator y −1 Im Tz (H) = (F Rz (H))(F Rz (H))∗ is trace class if and only if F Rz (H) is Hilbert-Schmidt; the product of two Hilbert-Schmidt operators is trace class and if AA∗ = |A∗ |2 is trace class then |A∗ | and hence A is Hilbert-Schmidt. Suppose H is self-adjoint and a rigging operator is H -Hilbert-Schmidt. If φ is a bounded Borel function with compact support, then for nonreal z the operator (H − z)φ(H) is bounded, hence the operator F φ(H) = F Rz (H)(H − z)φ(H) is Hilbert-Schmidt. Proposition 2.24. Let H be a self-adjoint operator on a Hilbert space H and let F : H → K be a rigging operator. The following six assertions are equivalent. (A) The operator F φ(H)F ∗ is trace class for (i) any φ ∈ Cc∞ (R); (ii) any compactly-supported bounded Borel function φ ∈ Bc (R); (iii) any φ which is the characteristic function of a bounded interval. (B) The operator F φ(H) is Hilbert-Schmidt for (i); (ii); (iii). Proof. For either (A) or (B), obviously (ii) implies (i) and (iii). (A.i) implies (B.ii): For any bounded Borel function φ with compact support, there is ψ ∈ Cc∞ (a smooth bump function equal to 1 on the support of φ ) such that 0 6 ψ 6 1 and φ = ψφ. Then by the functional calculus and the ideal property of the Hilbert-Schmidt class, F φ(H) = F ψ(H)φ(H) ∈ L2 (K, H). To√ show (A.i) √ implies (A.ii) we modify this argument as follows. Since the operator F ψ(H)F ∗ = √ ∗ Therefore (F√ ψ(H))(F ψ(H)) is trace class, it follows that F ψ(H)√is Hilbert-Schmidt. √ F ψ(H)φ(H) is Hilbert-Schmidt and hence F φ(H)F ∗ = F ψ(H)φ(H) ψ(H)F ∗ is trace class. Interpreting ψ as the characteristic function of a bounded interval containing the support of φ shows that (B.iii) implies (B.ii) and (A.iii) implies (A.ii). Finally, (A.iii) and (B.iii) are equivalent since F E(∆)F ∗ = (F E(∆))(F E(∆))∗ is trace class iff F E(∆) is Hilbert-Schmidt, E being the spectral measure of H. Proposition 2.25. If H is a self-adjoint operator with spectral measure E on a Hilbert space H and if F : H → K is a H -Hilbert-Schmidt rigging operator, then the sequence of trace class operators F E([−N, N ]) Im Rz (H)F ∗ converges to Im Tz (H) as N → ∞ in the trace class norm. Proof. Since E([−N, N ]) → 1 in the so∗ -topology, by Lemma 2.3 F Rz (H)E([−N, N ]) → F Rz (H)

and

E([−N, N ])Rz¯(H)F ∗ → Rz¯(H)F ∗

in the classes L2 (H, K) and L2 (K, H) respectively. Hence, Hölder’s inequality completes the proof.


23

Let an affine space A be (A) relatively compact, or (B) resolvent comparable. A is said to be compatible with a rigging operator F, if firstly any perturbation V ∈ A0 admits a decomposition of the form V = F ∗ JF, where J is some bounded self-adjoint operator on the auxiliary Hilbert space K, and secondly F is H -compact in case (A), or H -Hilbert-Schmidt is case (B), for some, hence any, H ∈ A. If a relatively compact or resolvent comparable affine space A is compatible with a rigging operator F, it is indicated by writing A(F ). We also write the real vector space of perturbations as A0 (F ) and equip it with the F -norm kF ∗ JF kF := kJk. Proposition 2.27 below shows that the F -norm is such that the mapping (2.11) is continuous. A sidenote about the F -norm: it would be equivalent to view operators from A0 (F ) as acting from H− to H+ with the norm of B(H− , H+ ) (see (2.27)). For any (bounded) rigging operator F there is a largest real vector space A0 (F ) generated by F, namely A0 (F ) = F ∗ Bsa (K)F. This is a real Banach space which is isomorphic to the Banach space Bsa (K). Moreover, if H is a self-adjoint operator on H, such that the rigging F is (A) H -compact, or (B) H -Hilbert-Schmidt, then A(F ) = H + F ∗ Bsa (K)F is a (A) relatively compact, or (B) resolvent comparable, affine space; it is the largest such affine space which contains H and is compatible with the rigging F. It is with the case of an unbounded rigging in mind that we do not set A0 (F ) = F ∗ Bsa (K)F by definition. If two self-adjoint operators H0 , H1 differ by a perturbation operator of the form F ∗ JF = H1 − H0 , there is a sandwiched version of the second (but not the first) resolvent identity: (2.28)

Tz (H0 ) − Tz (H1 ) = Tz (H1 )JTz (H0 ) = Tz (H0 )JTz (H1 )

The next proposition follows immediately from (2.2).

Proposition 2.26. Let (H, F ) be a rigged Hilbert space. Let H be a self-adjoint operator on H such that the rigging F is H -compact. For any bounded self-adjoint operator J on the auxiliary Hilbert space K and any nonreal z, the compact operators Tz (H)J and JTz (H) have the same nonzero eigenvalues (counting multiplicities). Moreover, if V = F ∗ JF, then these same eigenvalues are also shared by the compact operators Rz (H)V and V Rz (H). Using Proposition 2.26 and Lemma 2.7, from the sandwiched second resolvent identity we obtain (2.29)

Tz (H1 ) = Tz (H0 )(1 + JTz (H0 ))−1 = (1 + Tz (H0 )J)−1 Tz (H0 ),

where H1 − H0 = F ∗ JF. Proposition 2.27. Let (H, F ) be a rigged Hilbert space. Let a compatible affine space of self-adjoint operators A(F ) be (A) relatively compact, or (B) resolvent comparable. For any H ∈ A(F ) and any nonreal z, the following functions are continuous (i)

(ii) (iii)

A0 (F ) ∋ V 7→ Tz (H + V ) ∈ L∞ (K)

A0 (F ) ∋ V 7→ F Rz (H + V ) ∈ Lp (H, K)

A0 (F ) × A0 (F ) × A0 (F ) ∋ (V1 , V2 , V3 ) 7→ Rz (H + V1 )V2 Rz (H + V3 ) ∈ Lp (K)

where p = ∞ in case (A) and p = 1 in case (B).

Proof. Continuity of (i) follows from the sandwiched second resolvent identity: Since J 7→ JTz (H) is norm continuous, with V = F ∗ JF, it is clear that A0 (F ) ∋ V 7→ JTz (H) ∈ L∞ (K) is continuous. Hence by continuity of the inverse, V 7→ (1 + JTz (H))−1 is norm-continuous. Then the equality Tz (H + V ) = Tz (H)(1 + JTz (H))−1 shows that (i) is continuous. Continuity of (ii) then follows from the equality F Rz (H + V ) = (1 − Tz (H + V )J)F Rz (H),

since V 7→ 1 − Tz (H + V )J is norm-continuous and F Rz (H) ∈ Lp . Finally, continuity of (iii) follows from continuity of (ii) along with, in case (B), Hölder’s inequality.

24


Proposition 2.28. Let H0 be a self-adjoint operator on H and let V be a bounded selfadjoint operator which is Hp 0 -compact. Let Fk be an injective Hilbert-Schmidt operator on the kernel of V. Set F = |V | + Fk as a rigging operator and set A0 (F ) = F ∗ Bsa (H)F. Then A(F ) = H0 + A0 (F ) is a relatively compact affine space, such that A0 (F ) includes V as well as EV, where E is any spectral projection of V. Further, if the pair H0 , H0 + |V | is resolvent comparable, then the affine space A(F ) is resolvent comparable.

Proof. The operator V decomposes as F ∗ JF, where J is the partial isometry sgn(V ). Moreover, since any spectral projection E of V commutes with F, we have EV = F ∗ EJF. For nonreal z, since V is H0 -compact, the operator p p Rz¯(H0 )JV Rz (H0 ) = ( |V |Rz (H0 ))∗ ( |V |Rz (H0 )) p |V |Rz (H0 ) and so is F Rz (H0 ). Now suppose H and is compact, hence so is the operator H + |V | are resolvent comparable. Note that this holds if the pairs H0 , H0 + V± are resolvent comparable, where V± are the positive and negative parts of V. From resolvent comparability and the resolvent identities we get p p Rz¯(H0 )|V |Rz (H0 ) = ( |V |Rz (H0 ))∗ ( |V |Rz (H0 )) ∈ L1 (H). p |V |Rz (H0 ) is Hilbert-Schmidt. Thus F Rz (H0 ) is HilbertIt follows that the operator Schmidt. 2.4.1. Limiting Absorption Principle. Let H be a self-adjoint operator on a Hilbert space H and let F : H → K be a H -compact rigging operator. When the rigging operator is fixed we will use the notation Tλ±i0 (H) := lim F Rλ±iy (H)F ∗ y→0+

and

Im Tλ±i0 (H) := lim Im F Rλ±iy (H)F ∗ . y→0+

The set of regular points of H is defined by Λ(H, F ) := λ ∈ R Tλ+i0 (H) exists in B(K) .

If λ ∈ / Λ(H, F ), then H is said to be resonant at λ. Being the set of points of convergence of a family of continuous functions, Λ(H, F ) is a Borel set. We say that the Limiting Absorption Principle holds for H, if Λ(H, F ) is a full set. In this case, Corollary 2.30 below justifies interpreting points from the complement R \ Λ(H, F ) as singular points of H. Note that if Tλ+i0 (H) exists, then so does Tλ−i0 (H) by continuity of the adjoint operation, and hence so does Tλ+i0 (H) − Tλ−i0 (H) . Im Tλ+i0 (H) = 2i Thus, the set Λ(H, F ) simultaneously supports the absolutely continuous parts of the spectral measures µf,g of H, for any vectors f = F ∗ φ and g = F ∗ ψ from ran F ∗ (see Corollary 2.2). We have the following Theorem 2.29. Let E be the spectral measure of H. Then E(Λ(H, F )) is an orthogonal projection onto a linear subspace of the absolutely continuous subspace H(a) (H) of H with respect to H. Proof. Assume the contrary. Then there exists a null subset X of Λ(H, F ) such that E(X) 6= 0. Since the range of F ∗ is dense in H, it follows that there exists a vector F ∗ ψ such that (2.30)

hF ∗ ψ, E(X)F ∗ ψiH 6= 0.

On the other hand, by definition of the set Λ(H, F ), for all λ ∈ X ⊂ Λ(H, F ) there exists the finite limit 1 lim hψ, F Im Rλ+iy (H)F ∗ ψiK . π y→0+ By Theorem 2.1, the set of points λ where the limit above exists, is a full set whose complement is a support of the singular part of the measure µF ∗ ψ (∆) = hF ∗ ψ, E∆ F ∗ ψiH , and hence hF ∗ ψ, E(X)F ∗ ψiH = 0. This contradicts (2.30) and therefore completes the proof.


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Corollary 2.30. If Λ(H, F ) is a full set, then it supports the absolutely continuous spectral measure of H. Moreover, its complement R \ Λ(H, F ) is a minimal support of the singular spectral measure, in other words, a core of the singular spectrum of H. The set of regular points Λ(H, F ) cannot contain any eigenvalues of H. Since it obviously contains real points of the resolvent set R \ σ(H), this means that any extra regular points are points of the essential spectrum σess (H). A classical result, due to M. S. Birman and S. B. Entina, asserts that the Limiting Absorption Principle holds if F is Hilbert-Schmidt: Theorem 2.31. [BE] If H is a self-adjoint operator on a Hilbert space H and F is a HilbertSchmidt operator from H to another Hilbert space K, then for a.e. λ ∈ R the operator valued function F Rλ+iy (H)F ∗ has a limit in the Hilbert-Schmidt norm as y → 0. See also [Y, Theorem 6.1.9]. Note that for an arbitrary self-adjoint operator H, this result cannot be improved, in the sense that if F ∈ L2+ε with ǫ > 0, but F ∈ / L2 , then for some H the Limiting Absorption Principle fails, see [N]. Corollary 2.32. If H is a self-adjoint operator on a Hilbert space H and if F : H → K is a H -Hilbert-Schmidt rigging operator, then the Limiting Absorption Principle holds for H. Proof. Let E be the spectral measure of H. By assumption, F Rz (H) is Hilbert-Schmidt and hence F Rz (H)(H −z)E(∆) = F E(∆) is Hilbert-Schmidt for any bounded interval ∆. We have F Rλ+iy (H)F ∗ = F E(∆)Rλ+iy (H)F ∗ + F E(R \ ∆)Rλ+iy (H)F ∗ .

The first term (F E(∆))Rλ+iy (H)(F E(∆))∗ converges for a.e. λ by Theorem 2.31 and the second term F E(R \ ∆)Rλ+iy (E(R \ ∆)H)F ∗ converges for all λ in the interior of ∆ since λ ∈ ρ(E(R \ ∆)H). Hence, ∆ \ Λ(H, F ) is a null set for any bounded interval ∆ and hence Λ(H, F ) is a full set. In §3, the wave matrices w± (λ; H1 , H0 ) and the scattering matrix S(λ; H1 , H0 ) are constructed at any point which is regular for both H1 and H0 . However, when it comes to the SSF and its decomposition into the absolutely continuous and singular SSFs, more than the existence of the limit of the sandwiched resolvent Tλ+i0 (H) is required. Let H be self-adjoint and let F be a H -compact rigging operator. Let E be the spectral measure of H. We define the set Λ(H, F ; L1 ) of trace-regular points of H as follows. λ ∈ Λ(H, F ; L1 ) if λ ∈ Λ(H, F ) and if for some open neighbourhood ∆ ⊂ R of λ Im F E(∆)Rλ+iy (H)F ∗ → Im Tλ+i0 (H)

as y → 0+ in the trace class norm. A classical result in this direction: Theorem 2.33. [BE] If H is a self-adjoint operator on a Hilbert space H and F is a HilbertSchmidt operator from H to another Hilbert space K, then for a.e. λ ∈ R the operator valued function F Im Rλ+iy (H)F ∗ has a limit in the trace class norm as y → 0. See also [Y, Theorem 6.1.5]. Corollary 2.34. If H is self-adjoint and F is a H -Hilbert-Schmidt rigging operator, then the set of trace-regular points Λ(H, F ; L1 ) is a full set. Moreover, for any λ ∈ Λ(H, F ; L1 ) the limit Im Tλ+i0 (H) exists in the trace class. Proof. Let E be the spectral measure of H. As in the proof of Corollary 2.32, we write F Im Rλ+iy (H)F ∗ = F E(∆) Im Rλ+iy (H)F ∗ + F E(R \ ∆) Im Rλ+iy (H)F ∗ ,

for any bounded ∆. The first term, equal to (F E(∆)) Im Rλ+iy (H)(F E(∆))∗ , converges in the trace class norm for a.e. λ by Theorem 2.33. By Proposition 2.25, the second term converges to 0 in the trace class norm as ∆ → R. Hence, the set of real numbers where the imaginary part of the sandwiched resolvent has a trace class limit is a full set and hence so is Λ(H, F ; L1 ), by Corollary 2.32.

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For the purposes of this paper, the set Λ(H, F ; L1 ) could be replaced by (2.31) Λ(H, F ) ∩ λ ∈ R Im Tλ+i0 (H) exists in L1 (K) ,

because this set (2.31) coincides with Λ(H, F ; L1 ) under the assumption that F is H -HilbertSchmidt. In fact, for the set (2.31) to contain a single point λ the operator Im Tλ+iy (H) needs to be trace class for small enough y, in other words it requires F to be H -Hilbert-Schmidt (see Proposition 2.23). Therefore the set (2.31) must be a full set by Corollary 2.34. 2.4.2. Resonance points. Let H be a self-adjoint operator and let F be a H -compact rigging operator. Recall that H is regular or resonant at λ, by definition, if λ ∈ Λ(H, F ) or λ ∈ / Λ(H, F ) respectively. Since the set of points R \ Λ(H, F ) supports the singular spectrum of H, we can interpret the resonance of H at λ as an indication that the point λ may belong to the singular spectrum of H. Therefore it may be argued that considering the question of what happens to the inclusion λ ∈ Λ(H, F ) when the operator H is perturbed provides information about the movement of the singular spectrum. In the setting of a relatively compact affine space of self-adjoint operators A(F ), it is shown below that if λ ∈ Λ(H, F ) then along any analytic curve passing through H in A(F ) there is only a discrete subset of operators which are resonant at λ. If H + V is resonant at λ, then the compact operator V Rλ+iy (H) has a non-empty set of eigenvalues converging to −1 as y → 0+ . In [Az3 ] the number of these eigenvalues approaching −1 from above minus the number approaching from below is called the resonance index. The resonance index is studied in detail in [Az2 , Az3 , Az5 ]. It is a natural extension of the notion of (infinitesimal) spectral flow into the essential spectrum. The resonance index and its connection with the singular SSF will be discussed further in §4.3. The following convenient notation is borrowed from [Y, p. 115]. Let (H, F ) be a rigged Hilbert space and let H be an operator from a relatively compact affine space A(F ). We attach one copy of Λ(H, F ) to each complex half-plane forming the sets (2.32)

Π± (H, F ) := C± ∪ Λ(H, F )

and denote by Π(H, F ) the disjoint union of the sets Π± (H, F ) : (2.33)

Π(H, F ) := Π+ (H, F ) ⊔ Π− (H, F ).

Elements of Π± (H, F ) in the disjoint union Π(H, F ) can be distinguished by their imaginary parts. That is, z ∈ Π(H, F ) can be written as λ ± iy, where λ = Re z and y = | Im z|, so that λ ± i0 distinguishes between the copies of Λ(H, F ) when λ is a regular point of H. The set Π(H, F ; L1 ) is similarly defined: just replace Λ(H, F ) by Λ(H, F ; L1 ) in the definition of Π(H, F ). The set Π(H, F ) indexes the operators Tz (H) : For any z ∈ Π(H, F ), ( F Rz (H)F ∗ for z = λ ± iy, with λ ∈ R and y > 0, Tz (H) := ∗ for z = λ ± i0, with λ ∈ Λ(H, F ). limy→0+ F Rλ±iy (H)F In fact there will be little need to distinguish λ + i0 from λ − i0 if the limits Tλ+i0 (H) and Tλ−i0 (H) are equal. In this regard, since z 7→ Tz (H) is continuous on the resolvent set ρ(H), in the definition (2.33) we could have replaced Λ(H, F ) by Λ(H, F ) \ ρ(H) = Λ(H, F ) ∩ σ(H). What is important is that for a regular point λ inside the essential spectrum σess the operators Tλ±i0 (H) may differ. Note that these operators may have no decomposition of the form F RF ∗ . Since A(F ) is relatively compact, the sandwiched resolvent Tz (H) is compact for any z ∈ Π(H, F ). For any H1 , H0 ∈ A(F ) with H1 − H0 = F ∗ JF, the sandwiched version of the second resolvent identity (2.28) holds for any z ∈ Π(H1 , F ) ∩ Π(H0 , F ). Indeed, since it holds for any nonreal z = λ + iy, where λ ∈ Λ(H1 , F ) ∩ Λ(H0 , F ), taking the limit as y → 0 shows that it holds for z = λ ± i0 as well.


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Proposition 2.35. Let H0 and H1 be two self-adjoint operators from a relatively compact affine space A(F ) with difference H1 − H0 = F ∗ JF. The operator 1 + JTz (H0 ) is invertible for any z ∈ Π(H0 , F ) ∩ Π(H1 , F ). Proof. Since JTz (H0 ) is compact, if 1 + JTz (H0 ) is not invertible, then by the Fredholm alternative there exists a nonzero φ ∈ K such that (2.34)

(1 + JTz (H0 ))φ = 0.

But then the sandwiched second resolvent identity Tz (H0 ) = Tz (H1 )(1 + JTz (H0 )) implies Tz (H0 )φ = 0, which by (2.34) gives the contradiction φ = 0. Suppose λ ∈ Λ(H0 , F ) and z = λ + iy with y > 0. Then from the second resolvent identity, we have (2.29). Letting y → 0 one infers that if (1 + Tλ+i0 (H0 )J)−1 exists then so does Tλ+i0 (H1 ). And by Proposition 2.35, existence of the operator Tλ+i0 (H1 ) implies invertibility of the operator 1 + Tλ+i0 (H0 )J. Let z ∈ Π(H0 , F ). Suppose Hr = H0 + Vr is an analytic path in A(F ). Then Vr = F ∗ Jr F, where Jr is an analytic path in B(K). In a neighbourhood of the real axis, consider the function r 7→ Tz (H0 )Jr for complex r. It is a holomorphic compact operator valued function. By the analytic Fredholm alternative (Theorem 2.4), the function r 7→ (1+Tz (H0 )Jr )−1 is meromorphic and its pole-set is the discrete set of points r for which −1 is an eigenvalue of the compact operator Tz (H0 )Jr . Note that the compact operator valued function r 7→ Tz (Hr ) = (1 + Tz (H0 )Jr )−1 Tz (H0 )

is also meromorphic with the same pole-set. Poles of either function are called resonance points of the path Hr corresponding to z. Resonance points for z¯ are called anti-resonance points for z. It can be checked that if rz is a resonance point for z, then r¯z is an anti-resonance point. It is the real resonance points of Hr corresponding to λ ∈ Λ(H0 , F ) that are of the most interest. The set of such points, denoted R(λ; {Hr }), is called the resonance set of the path Hr . Above, we have established the following Theorem 2.36. Let A(F ) be a relatively compact affine space. Let Hr = H0 + F ∗ Jr F be an analytic path in A(F ). The inclusion λ ∈ Λ(H0 , F ) implies the inclusion λ ∈ Λ(Hr , F ) for all r except those points of the resonance set R(λ; {Hr }). For a resolvent comparable affine space A(F ), the property of regular points expressed in Theorem 2.36 holds for trace-regular points as well: Theorem 2.37. Let A(F ) be a resolvent comparable affine space and let Hr be as in Theorem 2.36. Suppose λ ∈ Λ(H0 , F ; L1 ). Then λ ∈ Λ(Hr , F ; L1 ) if and only if r ∈ / R(λ; {Hr }). Proof. By Theorem 2.36, the inclusion λ ∈ Λ(Hr , F ; L1 ) ⊂ Λ(Hr , F ) means r ∈ / R(λ; {Hr }). We must show that if λ ∈ Λ(Hr , F ), then λ ∈ Λ(Hr , F ; L1 ). This follows from the well-known equality Im Tλ+i0 (Hr ) = (1 + Tλ−i0 (H0 )Jr )−1 Im Tλ+i0 (H0 )(1 + Jr Tλ+i0 (H0 ))−1 , which is also discussed below.

Let A(F ) be a relatively compact affine space containing the operators H0 , H1 , with H1 − H0 = F ∗ JF, and let z = λ + iy be nonreal. The first resolvent identity implies the equalities (2.35)

Im Tλ+iy (H1 ) = y F Rλ±iy (H1 )Rλ∓iy (H1 )F ∗ ,

and the second resolvent identity implies the equalities (2.36)

F Rz (H1 ) = (1 + Tz (H0 )J)−1 F Rz (H0 ), Rz (H1 )F ∗ = Rz (H0 )F ∗ (1 + JTz (H0 ))−1 .

Combining (2.35) and (2.36) gives the well-known identity (see e.g. [KK, p.144], [RS2 , (99)]) (2.37)

Im Tz (H1 ) = (1 + Tz¯(H0 )J)−1 Im Tz (H0 )(1 + JTz (H0 ))−1 .

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If λ ∈ Λ(H0 , F ) ∩ Λ(H1 , F ), then taking the limit as y → 0 shows that equality (2.37) holds for any z ∈ Π(H1 , F ) ∩ Π(H0 , F ). Let z ∈ Π(H0 , F ) and let Hr = H0 +F ∗ Jr F be an analytic path in A(F ). The equality (2.37) applied to Hr implies that, since the inverted factors of (2.37) are meromorphic, the compact operator valued function r 7→ Im Tz (Hr ) is also meromorphic. Hence so is the function (2.38) r 7→ Im Tλ+i0 (Hr )J˙r .

Note that if z is nonreal, then it follows from Proposition 2.35 that the function (2.38) is analytic in a neighbourhood of the real axis. However, if z = λ ± i0 then (2.38) has real poles at resonance points. The following lemma will be useful later. Lemma 2.38. Let A(F ) be a resolvent comparable affine space of self-adjoint operators and let Hr = H0 + F ∗ Jr F be an analytic path in A(F ). Suppose λ ∈ Λ(H0 , F ). Then although the trace class operator valued function (2.38) is meromorphic with real poles at resonance points, its trace r 7→ Tr Im Tλ+i0 (Hr )J˙r admits analytic continuation to the real axis.

Proof. This lemma will be proved in §4.2 as a consequence of Corollary 4.10. A modified version of that proof is sketched below. It uses properties of the operator M (r) := (1 + Tλ−i0 (H0 )Jr )(1 + Tλ+i0 (H0 )Jr )−1 = 1 − 2i Im Tλ+i0 (H0 )Jr (1 + Tλ+i0 (H0 )Jr )−1 .

In particular, when r is real this operator is unitary on K. This can be shown algebraically by applying its adjoint and using the equality (2.37). It follows that M (r) admits analytic continuation to the real axis. Its derivative at any nonresonant r can be calculated: M ′ (r) = −2i(1 + Tλ−i0 (H0 )Jr ) Im Tλ+i0 (Hr )J˙r (1 + Tλ+i0 (H0 )Jr )−1 . It follows that the function i r 7→ Tr(M ′ (r)M ∗ (r)) = Tr (1 + Tλ−i0 (H0 )Jr ) Im Tλ+i0 (Hr )J˙r (1 + Tλ−i0 (H0 )Jr )−1 2 = Tr Im Tλ+i0 (Hr )J˙r

admits analytic continuation to R.

3. Constructive approach to stationary scattering theory 3.1. Evaluation operator. Let (H, F ) be a rigged Hilbert space, and let H be a self-adjoint operator on H such that F is H -compact. Note that for any z ∈ C+ the imaginary part of the sandwiched resolvent Im Tz (H) is positive. For any λ ∈ Λ(H, F ) the norm-limit Im Tλ+i0 (H) is therefore also positive. The operator Im Tz (H) will often be accompanied by a scaling factor of π −1 p. Since it is positive for any z ∈ Π+ (H, F ) = C+ ∪ Λ(H, F ), we can consider its square root π −1 Im Tz (H). This operator will play an important role. For later convenience we will + let z denote the projection of z onto the upper half-plane Π+ (H, F ) (defined by (2.32)): z + := Re z + i | Im z| ∈ Π+ (H, F ) ∀z ∈ Π(H, F ), p so that the operator Im Tz + (H) is positive for any z ∈ Π(H, F ). Usually we write λ = Re z and y = | Im z|. p Proposition 3.1. The operator p Im Tz (H) is compact for any z ∈ Π+ (H, F ). Further, for λ ∈ Λ(H, Im Tλ+i0 (H) is Hilbert-Schmidt. If F is H -Hilbert-Schmidt, p p F ; L1 ), the operator Im Tλ+iy (H) → Im Tλ+i0 (H) in the Hilbert-Schmidt class L2 (K). then


29

Proof. Firstly, since Im Tz (H) is compact for any z ∈ Π+ (H, F ), so is its square root. Secondly, λ ∈ Λ(H, F ; L1 ) means that Im Tλ+i0 (H) is trace class, hence its square root is Hilbert-Schmidt. Lastly, if F is H -Hilbert-Schmidt, then Im Tλ+iy (H) converges to Im Tλ+i0p (H) in the trace p + class as y → 0 . It follows that its square root Im Tλ+iy (H) converges to Im Tλ+i0 (H) in the Hilbert-Schmidt class, see e.g. [Az, Lemma 1.6.5]. p Im Tz (H) has dense range: The range For nonreal z = λ + iy, y > 0, the operator of F ∗ , dense in H by definition, is mapped by the bounded operator Rz (H) to a dense set, which is in turn mapped to a dense set in K by the unitary operator Uz∗ (H) from the polar ∗ decomposition Rz (H)F ∗ = Uz (H)|Rp z (H)F |. This last dense subset of K, call it hz (H) = ran Uz∗ (H)Rz (H)F ∗ , is the range of Im Tz (H), since p p √ √ Im Tz (H) = yF Rz¯(H)Rz (H)F ∗ = y |Rz (H)F ∗ | = y Uz∗ (H)Rz (H)F ∗ . p However, for λ ∈ Λ(H, F ) the operator Im Tλ+i0 (H) may no longer have dense range. For each λ ∈ Λ(H, F ), we define the Hilbert space hλ ⊂ K as the closure of its range p (3.1) hλ (H, F ) := Im Tλ+i0 (H)K. We also write hλ (H) or hλ , if there is no danger of confusion.

Lemma 3.2. The field of fibre Hilbert spaces (3.2)

{hλ : λ ∈ Λ(H, F )}

is measurable in the following sense: if Pλ is the orthogonal projection onto hλ , then for all φ, ψ ∈ K the function Λ(H, F ) ∋ λ 7→ hφ, Pλ ψi is measurable. Proof. Let φ and ψ be two vectors from K. The function E Dp p Im Tλ+i0 (H)φ, Im Tλ+i0 (H)ψ = lim hφ, Im F Rλ+iy (H)F ∗ ψi , λ 7→ y→0+

being a pointwise limit of measurable functions, is measurable. Since hφ, Pλ ψi = hPλ φ, Pλ ψi and p each vector in the range of Pλ is the limit of some sequence of vectors from the range of Im Tλ+i0 (H), we have Dp E p hφ, Pλ ψi = lim Im Tλ+i0 (H)φn , Im Tλ+i0 (H)ψn , n→∞ p p where Pλ φ = limn→∞ Im Tλ+i0 (H)φn and Pλ ψ = limn→∞ Im Tλ+i0 (H)ψn . Hence the function λ 7→ hφ, Pλ ψi is a pointwise limit of measurable functions and is therefore measurable. It follows that {hλ : λ ∈ Λ(H, F )} defines a direct integral of Hilbert spaces (see §2.1.5) Z ⊕ hλ (H, F ) dλ. (3.3) H(H, F ) = Λ(H,F )

This direct integral can be seen as the closed subspace of L2 (Λ(H, F ), K) consisting of those square-integrable K -valued functions on Λ(H, F ) such that f (λ) ∈ hλ (H, F ) for a.e. λ ∈ Λ(H, F ). p For λ ∈ Λ(H, F ), we can consider the bounded operator Im Tλ+i0 (H) as an operator from ran F ∗ ⊂ H to K by first applying the inverse of F ∗ . We denote this operator, scaled by π −1/2 , by p (3.4) Eλ (H, F ) := π −1 Im Tλ+i0 (H)(F ∗ )−1 .

It will be called the evaluation operator. We may write Eλ (H) if there is no danger of confusion. Since its range is contained in hλ by definition, the evaluation operator can be considered as an operator from ran F ∗ to hλ (H, F ). One can see that for any f = F ∗ φ ∈ ran F ∗ the function Λ(H, F ) ∋ λ 7→ Eλ (H, F )f

30


is a measurable section of the field (3.2). In fact, this section is square integrable, so that it is an element of the Hilbert space (3.3). Indeed, Z 1 ∗ 2 (3.5) kE• (H)F φ kH = lim hF ∗ φ, Im Rλ+iy (H)F ∗ φi dλ = kEΛ(H,F ) F ∗ φk2 , π Λ(H,F ) y→0+

where the first equality is an immediate consequence of the definition (3.4) and the second equality follows from Corollary 2.2 and Theorem 2.29. From this equality (3.5) and the density of ran F ∗ in H it follows that the collection of mappings E(H, F ) = {Eλ (H, F ) : λ ∈ Λ(H, F )} defines a bounded operator (3.6)

E(H, F ) : H → H(H, F )

with norm 6 1. Moreover, the equality (3.5) implies that this bounded operator is isometric on the subspace EΛ(H,F ) H ⊂ H(a) (H) and vanishes on ER\Λ(H,F ) H ⊃ H(s) (H). Note that for arbitrary f ∈ H, f (·) := E(H, F )f ∈ H(H, F ) is an equivalence class of functions equal a.e., so we cannot ask the value of the ‘function’ f (·) at a point from Λ(H, F ). ∗ ∗ However, p for those f = F φ in the dense linear subspace ran F , there is an obvious choice: f (λ) = π −1 Im Tλ+i0 (H) φ. The following theorem, which has a value of its own, is the key to the approach to scattering theory given here. It gives an explicit and constructive diagonalisation of the absolutely continuous part of a self-adjoint operator. Versions of this theorem were given in [Az] and [Az4 ]. The present version has simpler formulation, is more general in one aspect and has a simpler proof. Theorem 3.3. The operator (3.6), defined in (3.4), is a partial isometry with initial space H H EΛ(H,F ) H and final space H(H, F ). Moreover, E(H, F ) diagonalises the operator EΛ(H,F ) H in the sense that for all f ∈ dom H the equality

(3.7)

(E(H)Hf )(λ) = λ(E(H)f )(λ)

holds for a.e. λ ∈ Λ(H, F ). Further, if h is a bounded measurable function whose minimal Borel support is a subset of Λ(H, F ), then for all f ∈ H and for a.e. λ ∈ Λ(H, F ) (3.8)

(E(H)h(H)f )(λ) = h(λ)(E(H)f )(λ).

Proof. It has already been remarked after (3.5) that E(H, F ) is a partial isometry with initial H space EΛ(H,F ) H. It follows from the equality (3.5) that for any Borel ∆ ⊂ Λ(H, F ) and any f ∈ H we have kE(H)E∆ f kH = kE∆ f k.

Hence, E∆ f = E∆ g if and only if (E(H)f )(λ) = (E(H)g)(λ) for a.e. λ ∈ ∆, and in particular for any Borel subset ∆ of Λ(H, F ) and for any f ∈ H the equality (3.9)

(E(H)E∆ f )(λ) = χ∆ (λ)(E(H)f )(λ).

holds for a.e. λ ∈ Λ(H, F ). The equality (3.9) implies that (3.8) holds for step functions h(λ). Hence, by continuity of E(H), this equality holds for all bounded measurable functions h. This plainly implies that (3.7) also holds. Now we show that the range of E(H) coincides with H. Since E(H) is a partial isometry, for this it is enough to show that the range of E(H) is dense in H. Let g(·) be an element of H which is orthogonal to the range of E(H) and let ψ1 , ψ2 , . . . be a basis of the auxiliary Hilbert space K. Since the range of F ∗ is dense in H, the vectors f1 = F ∗ ψ1 , f2 = F ∗ ψ2 , . . . form a total set in H. The equality (3.8) implies that if the range of E(H) contains a function f (·), then it also contains all functions of the form h(·)f (·), where h is a bounded measurable (scalar-valued) function. Hence, g(·) is orthogonal to all vectors h(·)fj (·) = h(·)(E(H)fj )(·). One infers that for all j, g(λ) ⊥ Eλ (H)fj in hλ for a.e. λ ∈ Λ(H, F ). Since the range of Eλ F ∗ is dense in hλ , the set of vectors {Eλ (H)fj : j = 1, 2, . . .} is total in hλ . Hence, g(λ) = 0 for a.e. λ ∈ Λ(H, F ). This completes the proof.


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Corollary 3.4. If H is a self-adjoint operator and F is a rigging operator such that the Limiting Absorption Principle holds, then the operator E(H, F ) : H → H(H, F ) diagonalises the absolutely continuous part of H. This corollary is the spectral theorem for the absolutely continuous part of a self-adjoint operator. The conclusion of this corollary is strong due to its strong premise, the Limiting Absorption Principle. 3.2. Wave matrices. For the rest of the current section, we fix a rigged Hilbert space (H, F ) and a relatively compact affine space A(F ) of self-adjoint operators which is compatible with F.

Proposition 3.5. Let H1 and H0 be operators from A(F ) such that H1 − H0 = F ∗ JF. Let z be a nonreal complex number and let y = | Im z |. Then (3.10)

yF Rz¯(H1 )Rz (H0 )F ∗ = Im Tz + (H1 ) (1 + JTz (H0 ))

= (1 − Tz¯(H1 )J) Im Tz + (H0 ). Proof. These equalities can be easily obtained from the equalities (2.35) and (2.36). For example, rewriting Rz (H0 )F using (2.35), we get yF Rz¯(H1 )Rz (H0 )F ∗ = yF Rz¯(H1 )Rz (H1 )F (1 + JTz (H0 )). Then by (2.36), the right hand side is equal to Im Tz + (H1 ) (1 + JTz (H0 )) . The second equality is proved similarly. If λ = Re z ∈ Λ(H1 , F ) ∩ Λ(H0 , F ), then the limit y → 0+ can be taken in (3.10). The result is a compact operator, whose scaling by π −1 will be denoted a(λ ± i0; H1 , H0 ), depending on the sign of Im z. That is, for any z ∈ Π(H1 , F ) ∩ Π(H0 , F ), we define (compare with [Y, (2.7.4)]) 1 (3.11) a(z; H1 , H0 ) := Im Tz + (H1 ) (1 + JTz (H0 )) π 1 = (1 − Tz¯(H1 )J) Im Tz + (H0 ). π Let H ∈ A(F ) and let z be a complex number with y = | Im z| > 0. Note the equalities p √ √ (3.12) Im Tz + (H) = y |Rz (H)F ∗ | = y |Rz¯(H)F ∗ |. In §3.1, the unitary operator Uz (H) : K → H was mentioned in passing. It is the unitary operator from the polar decomposition of the compact operator Rz (H)F ∗ :

(3.13)

Uz (H)|Rz (H)F ∗ | = Rz (H)F ∗ .

For any H1 , H0 ∈ A(F ) and any nonreal z, the unitary operator w(z; H1 , H0 ) on the auxiliary Hilbert space K defined by (3.14)

w(z; H1 , H0 ) := Uz∗ (H1 )Uz (H0 )

will be called an off-axis wave matrix. The wave matrices themselves emerge in the limit | Im z| → 0. That is, if λ ∈ Λ(H1 , F ) ∩ Λ(H0 , F ), then as operators acting from the fibre Hilbert space hλ (H0 ) to the fibre Hilbert space hλ (H1 ), the wave matrices are equal to the limits of the off-axis wave matrices w(λ ± iy; H1 , H0 ) as y → 0+ . This will be shown below. It follows from Proposition 3.5, (3.11), (3.12), and (3.13) that the off-axis wave matrices satisfy the equality p p Im Tz + (H1 )w(z; H1 , H0 ) Im Tz + (H0 ) = πa(z, H1 , H0 ). p Since for any H ∈ A(F ), the range of Im Tz + (H) is dense in K, the off-axis wave matrices are determined by the numbers E Dp p Im Tz + (H1 )φ, w(z; H1 , H0 ) Im Tz + (H0 )ψ = π hφ, a(z, H1 , H0 )ψi for φ, ψ ∈ K.

32


The formula (3.14) defines the wave matrices w(z; H1 , H0 ) for non-real values of z. Now we proceed to a definition of w(z; H1 , H0 ) which holds for regular real values z too. Let z ∈ Π(H0 , F ) ∩ Π(H1 , F ). Define the sesquilinear form p p w(z; H1 , H0 ) : Im Tz + (H1 )K × Im Tz + (H0 )K → C

by the formula

p p Im Tz + (H1 )φ, Im Tz + (H0 )ψ = π hφ, a(z; H1 , H0 )ψiK . w(z; H1 , H0 ) p p Note that if Im Tz + (H1 )φ = 0 or Im Tz + (H0 )ψ = 0, then hφ, a(z; H1 , H0 )ψi = 0. From this it follows that the form (3.15) is well-defined. As in [Az, Proposition 5.2.2] (see also [Y]) we show that the form (3.15) is bounded operator from p with bound 6 1, and hence defines a bounded p the closure of the range of Im Tz + (H0 ) to the closure of the range of Im Tz + (H1 ). If z is nonreal, then the resulting bounded operator is of course the off-axis wave matrix w(z; H1 , H0 ). If z = λ ± i0, where λ ∈ Λ(H1 , F ) ∩ Λ(H0 , F ), then the resulting bounded operators (3.15)

(3.16)

w(λ ± i0; H1 , H0 ) : hλ (H0 ) → hλ (H1 )

define the wave matrices. For the wave matrices we also use the notation w± (λ; H1 , H0 ) := w(λ ± i0; H1 , H0 ). Proposition 3.6. For any z ∈ Π(H1 , F ) ∩ Π(H0 , F ) the form (3.15) is bounded with norm 6 1. Hence the wave matrices are bounded operators with norm less or equal to 1. Proof. We already know this to be true for the off-axis wave matrices. Let Re z = λ ∈ Λ(H1 , F )∩ Λ(H0 , F ) and y = | Im z| > 0. Using the Schwartz inequality, for any φ, ψ ∈ K π |hφ, a(z; H1 , H0 )ψiK | = y |hRz (H1 )F ∗ φ, Rz (H0 )F ∗ ψiH |

6 y kRz (H1 )F ∗ φkH kRz (H0 )F ∗ ψkH

p

p

= Im Tz + (H1 )φ Im Tz + (H0 )ψ . K

Taking the limit y →

0+

we obtain,

p

π |hφ, a(λ ± i0; H1 , H0 )ψi| 6 Im Tλ+i0 (H1 )φ

hλ (H1 )

completing the proof.

K

p

Im Tλ+i0 (H0 )ψ

hλ (H0 )

,

It directly follows from their definition (3.14) that the off-axis wave matrices are unitary, satisfy the equality w(z; H0 , H0 ) = 1, have the multiplicative property w(z; H2 , H1 )w(z; H1 , H0 ) = w(z; H2 , H0 ), and satisfy the equality w∗ (z; H1 , H0 ) = w(z; H0 , H1 ). Each of these properties can be alternatively proved from the form definition (3.15). These alternate proofs work for the wave matrices too. In the rest of this subsection we present these proofs. Note that it follows from the definition, see (3.15) and (3.11), that for any z ∈ Π(H0 , F ), (3.17)

w(z; H0 , H0 ) = 1.

Lemma 3.7. Let H0 , H1 , H2 be three self-adjoint operators from A with F ∗ J1 F = H1 − H0 and F ∗ J2 F = H2 − H1 . For any z ∈ Π(H0 , F ) ∩ Π(H1 , F ) ∩ Π(H2 , F ), the equality (3.18)

π a(z; H2 , H0 ) = (1 − Tz¯(H2 )J2 ) Im Tz + (H1 )(1 + J1 Tz (H0 ))

holds, where both sides are bounded operators on K. Proof. Let y > 0. Using the resolvent identities (2.35) and (2.36), (1−Tλ∓iy (H2 )J2 ) Im Tλ+iy (H1 )(1 + J1 Tλ±iy (H0 )) = y(1 − Tλ∓iy (H2 )J2 )F Rλ∓iy (H1 )Rλ±iy (H1 )F ∗ (1 + J1 Tλ±iy (H0 ))

= yF Rλ∓iy (H2 )Rλ±iy (H0 )F ∗ .


33

This completes the proof for nonreal z, with λ = Re z and y = | Im z| . When λ ∈ Λ(H0 , F ) ∩ Λ(H1 , F ) ∩ Λ(H2 , F ), taking the limit as y → 0+ completes the proof in the case that z = λ ± i0. Theorem 3.8. Let H0 , H1 , H2 ∈ A. If z ∈ Π(H0 , F ) ∩ Π(H1 , F ) ∩ Π(H2 , F ), then w(z; H2 , H0 ) = w(z; H2 , H1 )w(z; H1 , H0 ).

p Proof. Let φ, ψ ∈ K and let y = | Im z|. Since the linear subspace Im Tz + (H)K is dense in either K if y > 0, or hλ (H) if y = 0, it is enough to show that the number E Dp p Im Tz + (H2 )φ, w(z; H2 , H1 )w(z; H1 , H0 ) Im Tz + (H0 )ψ (E) := Dp E p Im Tz + (H2 )φ, w(z; H2 , H0 ) Im Tz + (H0 )ψ . is equal to p process, there exBy the density of Im Tz + (H1 )K and the Gram-Schmidt orthogonalisation p ist vectors b1 , b2 , . . . ∈ K (depending on z + ) such that the vectors Im Tz + (H1 )bj , j = 1, 2, . . . form an orthonormal basis of the Hilbert space K if y > 0, or of the Hilbert space hλ (H1 ) if y = 0, in which case it is possible that there are finitely many bj ’s. It follows that (E) =

∞ Dp X j=1

E p Im Tz + (H2 )φ, w(z; H2 , H1 ) Im Tz + (H1 )bj ×

Dp

E p Im Tz + (H1 )bj , w(z; H1 , H0 ) Im Tz + (H0 )ψ ,

where the sum may be finite if y = 0. Let H1 −H0 = F ∗ J1 F and H2 −H1 = F ∗ J2 F. Combining this equality with definition (3.15) of w(z; H1 , H0 ) gives ∞ X hφ, π a(z; H2 , H1 )bj iK hbj , π a(z; H1 , H0 )ψiK (E) = =

j=1 ∞ X j=1

hφ, [1 − Tz¯(H2 )J2 ] Im Tz + (H1 )bj i hbj , Im Tz + (H1 )[1 + J1 Tz (H0 )]ψi ,

where in the second equality we have used (3.11). Using again the fact that the vectors p Im Tz + (H1 )bj , j = 1, 2, . . . form an orthonormal basis, we obtain the equality ∞ Dp E X p (E) = Im Tz + (H1 )[1 − Tz¯(H2 )J2 ]∗ φ, Im Tz + (H1 )bj j=1

Dp

Im Tz + (H1 )bj ,

= hφ, [1 − Tz¯(H2 )J2 ] Im Tz + (H1 )[1 + J1 Tz (H0 )]ψi .

p

Im Tz + (H1 )[1 + J1 Tz (H0 )]ψ

E

Now, Lemma 3.7 implies that

(E) = hφ, π a(z; H2 , H0 )ψi = and the proof is complete.

Dp

E p Im Tz + (H2 )φ, w(z; H2 , H0 ) Im Tz + (H0 )ψ

Corollary 3.9. For any z ∈ Π(H0 , F ) ∩ Π(H1 , F ), the operator w(z; H1 , H0 ) is unitary and w∗ (z; H1 , H0 ) = w(z; H0 , H1 ). Proof. This follows verbatim the proof of [Az, Corollary 5.3.8]. By the multiplicative property of Theorem 3.8 and the equality (3.17), we have w(z; H0 , H1 )w(z, H1 , H0 ) = w(z; H0 , H0 ) = 1, w(z; H1 , H0 )w(z, H0 , H1 ) = w(z; H1 , H1 ) = 1. Since kw(z; H1 , H0 )k 6 1 by Proposition 3.6, it follows that w(z; H1 , H0 ) is unitary and w∗ (z; H1 , H0 ) = w(z; H0 , H1 ).

34


3.3. Wave operators. Recall that throughout the remainder of this section, A(F ) denotes a relatively compact affine space of self-adjoint operators. Let H0 , H1 ∈ A(F ) and put Λ := Λ(H0 , F ) ∩ Λ(H1 , F ), which may or may not be a full set. The operator valued functions Λ ∋ λ 7→ w± (λ; H1 , H0 ) are measurable and their values are unitary operators from the fibre Hilbert space hλ (H0 ) to the fibre Hilbert space hλ (H1 ). Therefore, we can define the (partial) wave operators Z ⊕ Z ⊕ hλ (H1 ) dλ. hλ (H0 ) dλ → W± (H1 , H0 ; Λ) : Λ

Λ

by the formulas (3.19)

W± (H1 , H0 ; Λ) =

Z

⊕

w± (λ; H1 , H0 ) dλ.

Λ

R⊕ Since the Hilbert space Λ hλ (H0 , F ) dλ is naturally isomorphic to the subspace EΛH0 H ⊂ H(a) (H0 ), it follows that the wave operators W± (H1 , H0 ; Λ), thus defined, can also be considered as operators (3.20)

W± (H1 , H0 ; Λ) : EΛH0 H → EΛH1 H.

As a consequence of the definition (3.19) and the corresponding properties of the wave matrices, we obtain the following theorem. Theorem 3.10. The wave operators (3.20) defined by (3.19) have the following properties: (i) (ii) (iii) (iv)

The operators (3.20) are unitary. If Λ = Λ(H0 , F )∩Λ(H1 , F )∩Λ(H2 , F ), then W± (H2 , H0 ; Λ) = W± (H2 , H1 ; Λ)W± (H1 , H0 ; Λ). W±∗ (H1 , H0 ; Λ) = W± (H0 , H1 ; Λ). The operator W± (H0 , H0 ; Λ) is the identity operator on EΛH0 H.

Theorem 3.11. For any bounded measurable function h on R such that the support of h is a subset of Λ(H0 , F ) ∩ Λ(H1 , F ), we have h(H1 )W± (H1 , H0 ) = W± (H1 , H0 )h(H0 ). H0 Also, if we set the operator (3.20) to be zero on ER\Λ H then

H1 W± (H1 , H0 ) = W± (H1 , H0 )H0 . Proof. This follows from the definition (3.19) of the wave operators and Theorem 3.3.

Now we assume that the sets Λ(Hj , F ) have full Lebesgue measure, that is, we assume that the Limiting Absorption Principle holds for each operator Hj . If we define the operator (3.20) to be zero on the singular subspace H(s) (H0 ), then part (iv) of Theorem 3.10 becomes W± (H0 , H0 ) = P (a) (H0 ). It follows from (ii) with H2 = H0 , (iii) and (iv) that W± (H1 , H0 ) = W± (H1 , H0 )P (a) (H0 ) = P (a) (H1 )W± (H1 , H0 ). Hence, we have the following corollary, which is a generalised Kato-Rosenblum theorem. Corollary 3.12. Assume that the sets Λ(Hj , F ), j = 0, 1, have full Lebesgue measure. Then the restrictions of the operators H0 and H1 to their absolutely continuous subspaces H(a) (H0 ) and H(a) (H1 ) are unitarily equivalent. Corollary 3.13. Assume that the sets Λ(Hj , F ) have full Lebesgue measure. Then the wave operators W± (H1 , H0 ) exist and are complete.


35

3.4. Connection with the time-dependent approach. Theorems 3.10 and 3.11 are wellknown properties of the wave operators. The next theorem states that definition (3.19) of the wave operators coincides with their classical time-dependent definition. Theorem 3.14. Let the sets Λ(Hj , F ), j = 0, 1, have full Lebesgue measure and suppose H that the operators F E∆ j , j = 0, 1, are compact for any bounded Borel set ∆. Then the strong operator limits (a)

lim eitH1 e−itH0 P0

t→±∞

exist and coincide with E∗ (H1 )W± (H1 , H0 )E(H0 ). Since proof of this theorem uses a standard method and since it will not be used further, we will give only a brief sketch of a proof. The Limiting Absorption Principle implies that the condition of [Y, Lemma 5.3.1] holds for a dense set F ∗ K of vectors. Therefore, the conclusion of [Y, Lemma 5.3.1] holds. This lemma allows to prove by standard methods (see e.g. [Y, §5.3]) that the weak wave operators exist and coincide with the wave operators defined by formula (3.19). Combined with the multiplicative property of the wave operators (Theorem 3.8), this implies the existence of the strong wave operators and that they coincide with (3.19). Details are omitted. 3.5. Scattering matrix. It turns out that once the wave matrices w± (λ; H1 , H0 ) have been defined and their properties have been established, the definition of the scattering matrix S(λ; H1 , H0 ) and proof of its properties, such as the stationary formula, follow the same lines as in [Az, §7] and [Az4 ]. Still, though the formulations of these theorems and their proofs are almost the same, the content of theorems is different given the new definitions of the set Λ(H, F ), the evaluation operator Eλ (H, F ), etc. For this reason, we shall repeat this material here. As in [Az, §7] and [Az4 ], given a number λ ∈ Λ(H0 , F ) ∩ Λ(H1 , F ), we define the scattering matrix S(λ; H1 , H0 ) as an operator on the fibre Hilbert space hλ (H0 ) by the formula ∗ S(λ; H1 , H0 ) = w+ (λ; H1 , H0 )w− (λ; H1 , H0 ).

For any z ∈ Π(H0 , F ) ∩ Π(H1 , F ) we define the operator S(z; H1 , H0 ) by the formula (3.21)

S(z; H1 , H0 ) = w∗ (z; H1 , H0 )w(¯ z ; H1 , H0 ) = w(z; H0 , H1 )w(¯ z ; H1 , H0 ).

When z is nonreal, we call S(z; H1 , H0 ) the off-axis scattering matrix. Note that S ∗ (z; H1 , H0 ) = S(¯ z ; H1 , H0 ). Of course when z = λ + i0, this is the scattering matrix, which we write as S(λ; H1 , H0 ) = S(λ + i0; H1 , H0 ). Its adjoint is S ∗ (λ; H1 , H0 ) = S(λ − i0; H1 , H0 ). Now we list some properties of the scattering matrix, which immediately follow from the definition and the properties of the wave matrices already established. Theorem 3.15. Let z ∈ Π(H0 , F )∩Π(H1 , F )∩Π(H2 , F ). The scattering matrix (3.21) possesses the following properties. (i) The operator S(z; H1 , H0 ) is a unitary operator. (ii) S(z; H2 , H0 ) = w∗ (z; H1 , H0 )S(z; H2 , H1 )w(¯ z ; H1 , H0 ). (iii) S(z; H2 , H0 ) = w∗ (z; H1 , H0 )S(z; H2 , H1 )w(z; H1 , H0 )S(z; H1 , H0 ). Another important property of the scattering matrix is the stationary formula: Theorem 3.16. Let H1 , H0 be operators from the affine space A(F ), with H1 − H0 = F ∗ JF. If z ∈ Π± (H0 , F ) ∩ Π± (H1 , F ), then p p S(z; H1 , H0 ) = 1 ∓ 2i Im Tz + (H0 )J(1 + Tz (H0 )J)−1 Im Tz + (H0 ).

The proof of this formula given in [Az, Theorem 7.2.2] applies in the current setting verbatim. Below we give a simplified version of that proof.

36


Proposition 3.17. Let H1 , H0 ∈ A(F ) with H1 − H0 = F ∗ JF. If z ∈ Π(H0 , F ) ∩ Π(H1 , F ), then we have the equalities p p w(z; H1 , H0 ) Im Tz + (H0 ) = Im Tz + (H1 )[1 + JTz (H0 )], (3.22) p p Im Tz + (H1 )w(z; H1 , H0 ) = [1 − Tz¯(H1 )J] Im Tz + (H0 ). Proof. It is enough to see that for any φ, ψ ∈ K Dp E p Im Tz + (H1 )φ, w(z; H1 , H0 ) Im Tz + (H0 )ψ E Dp p Im Tz + (H1 )φ, Im Tz + (H1 )[1 + JTz (H0 )]ψ = E D p p = φ, [1 − Tz¯(H1 )J] Im Tz + (H0 ) Im Tz + (H0 )ψ , which follows from the definition (3.15) of the wave matrix and (3.11).

When z = λ ± i0 is on the boundary ∂Π(H1 , F ) ∩ ∂Π(H0 , F ), the equalities in the second line of (3.22) can be identified with equations for the stationary scattering states from potential scattering theory, see for instance [T, (10.8)]. (There are two sign disagreements with [T, (10.8)]. One comes from the traditional difference between physicists and mathematicians in their choice of sign convention for the definition of the wave operators. The other comes from apdifference in the sign convention for the resolvent.) Here we are interpreting the operator are identifying K with Im Tλ+i0 (H) as the evaluation operator Eλ (H), or in other words wep ∗ the dense subspace F K of H (see (2.27)). Under this interpretation, Im Tλ+i0 (H) projects a state (from F ∗ K) onto the energy shell of H with energy λ. From (3.22) and the second resolvent identity we can easily obtain the Lippman-Schwinger equation (see [T, (10.12)]): p Im Tλ+i0 (H1 )w(λ ± i0; H1 , H0 ) p p = Im Tλ+i0 (H0 ) − Tλ∓i0 (H0 )J Im Tλ+i0 (H1 )w(λ ± i0; H1 , H0 ). The following lemma, whose proof is a simple algebraic manipulation based on the second resolvent identity, is taken from [Az, Lemma 7.2.1]. Lemma 3.18. Let H1 , H0 ∈ A(F ) with H1 − H0 = F ∗ JF. If z ∈ Π(H0 , F ) ∩ Π(H1 , F ), then (1 + Tz¯(H0 )J) Im Tz + (H1 )(1 + JTz¯(H0 ))

= Im Tz + (H0 ) [1 − 2iJ(1 − Tz (H1 )J) Im Tz (H0 )] . Proof. Consider the following equalities. 1 − 2iJ(1 − Tz (H1 )J) Im Tz (H0 ) = 1 − J(1 − Tz (H1 )J)(Tz (H0 ) − Tz¯(H0 ))

= 1 − J(1 − Tz (H1 )J)Tz (H0 ) + J(1 − Tz (H1 )J)Tz¯(H0 )

= 1 − JTz (H1 ) + J(1 − Tz (H1 )J)Tz¯(H0 )

= (1 − JTz (H1 ))(1 + JTz¯(H0 )). Now the required equality follows from (3.11).

We will now prove the stationary formula for the scattering matrix. Proof of Theorem 3.16. Since the second resolvent identity implies the equality (1 + Tz (H0 )J)−1 = 1 − Tz (H1 )J,

it suffices to show that for any φ, ψ ∈ K E Dp p Im Tz + (H0 )φ, S(z; H1 , H0 ) Im Tz + (H0 )ψ h E Dp ip p p Im Tz + (H0 )φ, 1 ∓ 2πi Im Tz + (H0 )J(1 − Tz (H1 )J) Im Tz + (H0 ) Im Tz + (H0 )ψ . =


37

By the definition of the scattering matrix (3.21) the left hand side is equal to D E p p LHS = w(z; H1 , H0 ) Im Tz + (H0 )φ, w(¯ z ; H1 , H0 ) Im Tz + (H0 )ψ

Then using (3.22), we obtain E Dp p Im Tz + (H1 )[1 + JTz (H0 )]φ, Im Tz + (H1 )[1 + JTz¯(H0 )]ψ LHS = = hφ, [1 + Tz¯(H0 )J] Im Tz + (H1 )[1 + JTz¯(H0 )]ψi .

Now the required equality follows by Lemma 3.18.

Note that given the stationary formula, we can consider the scattering matrix S(λ; H1 , H0 ) as an operator on K by defining it as the identity on hλ (H0 )⊥ . In this case, the off-axis scattering matrix S(λ + iy; H1 , H0 ) converges in B(K) to the scattering matrix as y → 0+ . 3.6. Scattering operator. Let H0 , H1 ∈ A(F ). The scattering operator Z ⊕ Z ⊕ hλ (H1 , F ) dλ, hλ (H0 , F ) dλ → S(H1 , H0 ) : Λ(H1 ,F )

Λ(H0 ,F )

is defined by the equality (3.23)

S(H1 , H0 ) =

Z

⊕

S(λ; H1 , H0 ) dλ.

Λ(H0 ,F )∩Λ(H1 ,F )

Theorem 3.19. The scattering operator satisfies S(H1 , H0 ) = W+∗ (H1 , H0 )W− (H1 , H0 ), that is, the definition (3.23) agrees with the usual definition. This theorem follows from the definitions of the wave operators (3.19) and the scattering matrix (3.21), and their properties. Obviously, the scattering operator admits properties analogous to those of the scattering matrix given in Theorem 3.15 (similar to those in [Az, Theorem 7.1.3]). In particular, the scattering operator S(H1 , H0 ) is unitary and commutes with H0 . 4. Singular spectral shift function 4.1. Absolutely continuous and singular spectral shift functions. Let A be a resolvent comparable affine space of self-adjoint operators such that any direction V ∈ A0 admits a decomposition V = V1 −V2 into positive directions 0 6 V1 , V2 ∈ A0 . As shown in §2.3 (Theorem 2.20 and Corollary 2.18), for a pair of operators H0 and H1 from A, the SSF is the density of the spectral shift measure, which in turn is the integral of the infinitesimal spectral shift measure Φ over a continuously differentiable path connecting H0 and H1 . In this subsection it is shown that if the affine space A is compatible with a rigging operator F, then the Lebesgue decomposition of the infinitesimal spectral shift measure ΦH (V ) into its absolutely continuous and singular parts results in a decomposition of the spectral shift measure along a particular path into the sum of two absolutely continuous measures. These two measures, or functions, which depend on the path of integration, are called the absolutely continuous and singular SSFs. Throughout the remainder of this section A(F ) will denote a resolvent comparable affine space of self-adjoint operators, which is compatible with a rigging operator F. It will be assumed that any direction may be decomposed as the difference of positive directions. This is not a restrictive assumption; if it doesn’t hold then A(F ) can be enlarged so that it does. Indeed, we can just put A0 (F ) = F ∗ Bsa (K)F. Due to the compatibility of the affine space A(F ) with the rigging F, the infinitesimal spectral shift measure of a pair (H, V ), as a functional defined by (2.19), may be written as (4.1)

ΦH (V )(φ) = Tr(JF φ(H)F ∗ ),

for any compactly supported bounded Borel function φ, where V = F ∗ JF.

38


Recall that if A, B are bounded operators and BA is trace class, then (outside of any proof) by Tr(AB) we denote Tr(BA). (s)

(a)

Proposition 4.1. The absolutely continuous ΦH (V ) and singular ΦH (V ) parts of the infinitesimal spectral shift measure (4.1) are given, respectively, by the functionals Cc (R) ∋ φ 7→ Tr(V φ(H)P (a) (H))

and

Cc (R) ∋ φ 7→ Tr(V φ(H)P (s) (H)).

Proof. Let E be the spectral measure of H. If Z is a bounded Borel set of zero measure, (a) then E (a) (Z) = 0, so that Tr(JF E (a) (Z)F ∗ ) = 0. Hence, the measure ΦH (V ) is absolutely continuous. On the other hand, if Z is a support of E (s) with |Z| = 0, then for any bounded (s) Borel X which does not intersect Z, we have EX = 0, hence Tr(JF E (s) (X)F ∗ ) = 0. It (s) (a) (s) follows that Z is a null support of ΦH (V ). Finally, since ΦH (V ) + ΦH (V ) = ΦH (V ), the proof is complete. Let Hr be a piecewise analytic path in A(F ) from H0 to H1 with derivative H˙ r = F ∗ J˙r F. For any λ ∈ Λ(H1 , F ; L1 ) ∩ Λ(H0 , F ; L1 ), the absolutely continuous SSF is defined by Z 1 1 (a) (4.2) ξ (λ; {Hr }) := Tr Im Tλ+i0 (Hr )J˙r dr, π 0

where the integrand is analytically continued through resonance points (see Lemma 2.38). If Hr is a straight line, we write ξ (a) (λ; H1 , H0 ) := ξ (a) (λ; {Hr }). The absolutely continuous SSF is zero outside of the common essential spectrum σess , since Im Tλ+i0 (Hr ) = 0 if λ is outside the spectrum of Hr and any λ from (Λ(H1 , F ; L1 ) ∩ Λ(H0 , F ; L1 )) \ σess is outside of the spectrum of Hr for all r ∈ [0, 1] except a discrete set. We now work towards Theorem 4.5, which shows that the absolutely continuous spectral shift function ξ (a) ( · ; {Hr }) when considered as a functional, coincides with the integral of Φ(a) along the path Hr . Lemma 4.2. For any H ∈ A(F ), the positive function λ 7→ Tr(Im Tλ+i0 (H)) is integrable over any bounded Borel subset ∆ of Λ(H, F ; L1 ), and its integral over ∆ is equal to π Tr(F E(∆)F ∗ ), where E is the spectral measure of H. Proof. Let {ψj } be an orthonormal basis in K. Then Z X Z ∞ hψj , Im Tλ+i0 (H)ψj i dλ. Tr(Im Tλ+i0 (H)) dλ = ∆ j=1

∆

Using monotone convergence we can interchange the sum and the integral. Then using Corollary 2.2 we obtain Z ∞ Z X lim hF ∗ ψj , Im Rλ+iy F ∗ ψj i dλ Tr(Im Tλ+i0 (H)) dλ = +

∆

=

y→0 j=1 ∆ ∞ X ∗ j=1

π hF ψj , E(∆)F ∗ ψj i

= π Tr(F E(∆)F ∗ ). Theorem 4.3. If H ∈ A(F ) and V = F ∗ JF ∈ A0 (F ), then for any bounded Borel function φ with compact support, Z 1 (a) φ(λ) Tr (J Im Tλ+i0 (H)) dλ. (4.3) Tr V φ(H)P (H) = π R


39

Proof. Without loss of generality we can assume that J is positive. Let {ψj } be an orthonormal basis of K. Then the right hand side of (4.3) can be written as Z X ∞ 1 φ(λ) hψj , J Im Tλ+i0 (H)ψj i dλ, π R j=1

where the sum is absolutely convergent. We have X N φ(λ) hψj , J Im Tλ+i0 (H)ψj i 6 |φ(λ)| kJk Tr(Im Tλ+i0 (H)). j=1

Since φ is bounded, by Lemma 4.2 and dominated convergence we may interchange the sum and integral. Consider the terms of the resulting sum: Z 1 aj := φ(λ) hψj , J Im Tλ+i0 (H)ψj i dλ. π R

The integral is unchanged by removing a null set: Z 1 aj = φ(λ) hψj , J Im Tλ+i0 (H)ψj i dλ π Λ(H,F ;L1) Z 1 φ(λ) lim hF ∗ Jψj , Im Rλ+iy (H)F ∗ ψj i dλ. = π Λ(H,F ;L1) y→0+

By Corollary 2.2 and Corollary 2.30, E E D D aj = F ∗ Jψj , φ(H)P (a) (H)F ∗ ψj = ψj , JF φ(H)P (a) (H)F ∗ ψj .

Therefore the sum of these terms aj is the trace of the operator JF φ(H)P (a) (H)F ∗ , which is the left hand side of (4.3). Lemma 4.4. Let Hr , r ∈ R, be a piecewise analytic path in resolvent comparable affine space of self-adjoint operators A, with derivative H˙ r = F ∗ J˙r F. Then the set of points (λ, r) in the plane for which λ is a regular point of the operator Hr , Γ({Hr }, F ) := (λ, r) ∈ R2 : λ ∈ Λ(Hr , F ; L1 ) , has full measure in R2 and the function

is measurable.

Γ({Hr }, F ) ∋ (λ, r) 7→ Tr J˙r Im Tλ+i0 (Hr )

Proof. Without loss of generality we can assume that the path Hr is an analytic path. First, the set Γ({Hr }, F ) is measurable as the set of convergence points as y → 0+ of the two families of continuous functions Tλ+iy (Hr ) and Im Tλ+iy (Hr ) of the variables (λ, r). As a limit of measurable functions, the function (λ, r) 7→ Tr J˙r Im Tλ+i0 (Hr ) = lim Tr J˙r Im Tλ+iy (Hr ) , y→0+

is measurable. Finally, to show that Γ({Hr }, F ) is a full set, let χ denote the set R2 \Γ({Hr }, F ) R and also its indicator function. The Lebesgue measure of χ is R2 χ(λ, r) dλdr. Since the function χ is measurable and positive, by Fubini’s theorem we may replace the double integral by an iterated one. Now we just need to note that the cross sections of χ are null sets. The cross section at a fixed r is the set R \ Λ(Hr , F ; L1 ) of points λ at which the operator Hr is resonant, while the cross section at almost every fixed λ is the discrete set R(λ, {Hr }) of real resonance points of the path Hr .

40


Theorem 4.5. Let Hr be a piecewise analytic path in A(F ) from H0 to H1 . Then the functional Z 1 Z 1 (a) Tr H˙ r φ(Hr )P (a) (Hr ) dr ΦHr (H˙ r )(φ) dr = (4.4) Cc (R) ∋ φ 7→ 0

0

defines an absolutely continuous measure whose density is the absolutely continuous SSF ξ (a) (λ; {Hr }) defined by (4.2). Proof. Let φ be a bounded Borel function with compact support. Consider the function 1 (4.5) (λ, r) 7→ φ(λ) Tr J˙r Im Tλ+i0 (Hr ) π over the strip R × [0, 1]. It is measurable by Lemma 4.4. Next we show that it is integrable. For this it suffices to show that (4.5) as a function of λ has locally bounded L1 -norm with respect to r. Let χ be a smoothed indicator of a bounded interval containing the support of φ. Then Z Z 1 1 ˙ χ(λ) Tr |J˙r | Im Tλ+i0 (Hr ) dλ, φ(λ) Tr Jr Im Tλ+i0 (Hr ) dλ 6 kφk∞ π R π R where we have used the fact that if J is a bounded self-adjoint operator and T is a positive trace class operator, then | Tr(JT )| 6 Tr(|J|T ). Using Theorem 4.3, the right hand side of this inequality is equal to the left hand side of the following inequality. √ kφk∞ Tr |J˙r |F χ(Hr )P (a) (Hr )F ∗ 6 kφk∞ kJ˙r kkF χ(Hr )k22 . √ Since J˙r is norm continuous by assumption and F χ(Hr ) is continuous in L2 (H, K) by Proposition 2.27, the right hand side of the last inequality is locally bounded with respect to r. The integrability of (4.5) follows. Therefore by Fubini’s theorem, the integral of (4.5) over the strip R × [0, 1] is equal to each of the iterated integrals. On one hand, from the definition of ξ (a) (λ; {Hr }), Z Z 1 Z 1 φ(λ) Tr J˙r Im Tλ+i0 (Hr ) dr dλ. φ(λ)ξ (a) (λ; {Hr }) dλ = π R 0 R On the other hand, by Theorem 4.3, Z 1 Z 1 Z 1 (a) ∗ ˙ ˙ φ(λ) Tr Jr Im Tλ+i0 (Hr ) dλ dr. Tr(Jr F φ(Hr )P (Hr )F ) dr = π R 0 0 Hence, combining these equalities, Z (a) φ(λ) ξ (a) (λ; {Hr }) dλ, ξ (φ; {Hr }) = R

for any compactly supported bounded Borel function φ. This completes the proof.

We denote the absolutely continuous spectral shift measure (4.4) by ξ (a) (φ; {Hr }) or simply ξ (a) (φ), if there is no ambiguity. Corollary 4.6. If Hr is a piecewise analytic path in A(F ) and φ is a compactly supported bounded Borel function, then the function (s)

(a)

r 7→ ΦHr (H˙ r )(φ) = ΦHr (H˙ r )(φ) − ΦHr (H˙ r )(φ)

is locally integrable, as the difference of locally integrable functions.

Let Hr , r ∈ [0, 1], be a piecewise analytic path in A(F ). The singular spectral shift measure ξ (s) ( · ; {Hr }) is defined by Z 1 Z 1 (s) Tr H˙ r φ(Hr )P (s) (Hr ) dr ∀φ ∈ Cc (R). ΦHr (H˙ r )(φ) dr = ξ (s) (φ; {Hr }) := 0

0

Therefore, along the path Hr , the spectral shift admits the decomposition ξ(φ; H1 , H0 ) = ξ (a) (φ; {Hr }) + ξ (s) (φ; {Hr })

∀φ ∈ Cc (R).


41

Since both the spectral shift measure and the absolutely continuous spectral shift measure are absolutely continuous (Theorem 2.20 and Theorem 4.5), so is the singular spectral shift measure. The locally integrable density of the singular spectral shift measure, denoted by the same symbol ξ (s) , is the singular SSF. Note that the absolutely continuous and singular spectral shifts are additive along a given curve {Hr }, in the sense that if γ1 ⊔ γ2 is the concatenation of two paths γ1 and γ2 then (4.6)

ξ (a) (φ; γ1 ⊔ γ2 ) = ξ (a) (φ; γ2 ) + ξ (a) (φ; γ2 ).

However, unlike the SSF itself, they depend on the curve {Hr }, as shown in [Az, §8.3]. 4.2. Exponential representation of the scattering matrix. In this subsection, the proof of theorem from the abstract is completed. That is, the singular SSF is shown to be almost everywhere integer-valued. This can be seen as a consequence of Theorem 4.12 below. Throughout this subsection, Hr = H0 + F ∗ Jr F will denote a piecewise analytic path in A(F ). Recall that in this section A(F ) denotes a resolvent comparable affine space of selfadjoint operators, which is assumed to be such that any direction decomposes into positive directions. By U1 (H) we denote the set of unitary operators of the form 1 + T where T is trace class, in other words, U1 (H) = {U ∈ 1 + L1 (H) : U is unitary}. This is a complete metric space with metric (U, V ) 7→ kU − V k1 . Consider the stationary formula (Theorem 3.16) for S(z; Hr , H0 ) : p p (4.7) S(z; Hr , H0 ) = 1 − ∓2i Im Tz + (H0 )Jr (1 + Tz (H0 )Jr )−1 Im Tz + (H0 ),

which holds for any z ∈ Π± (H0 , F ) ∩ Π± (Hr , F ). Then Proposition 3.1 implies

Proposition 4.7. For any z ∈ Π(Hr , F ) ∩ Π(H0 , F ; L1 ), the operator S(z; Hr , H0 ) is an element of U1 (K). Moreover, the off-axis scattering matrix S(λ + iy; Hr , H0 ) converges to the scattering matrix S(λ; Hr , H0 ) in U1 (K) as y → 0+ . The stationary formula (4.7) holds for all r ∈ R if z is nonreal. If z = λ + i0 for λ ∈ Λ(H0 , F ; L1 ), then Theorem 2.37 implies that it holds for all r ∈ / R(λ; {Hr }). (4.7) allows us to consider the operator S(z; Hr , H0 ) as a function of r. By the analytic Fredholm alternative, the factor (1 + Tz (H0 )Jr )−1 is meromorphic and hence, in a neighbourhood of the real axis, (4.8)

r 7→ S(z; Hr , H0 ) − 1 ∈ L1 (K)

is a meromorphic function. If z is nonreal, (4.8) must be holomorphic in a neighbourhood of R. Indeed, since 1 + Tz (H0 )Jr is invertible for real r, the factor (1 + Tz (H0 )Jr )−1 is holomorphic in a neighbourhood of R. Now suppose z = λ + i0 for λ ∈ Λ(H0 , F ; L1 ). Then although the factor (1 + Tλ+i0 (H0 )Jr )−1 has poles at real resonance points r ∈ R(λ; {Hr }), the scattering matrix S(λ; Hr , H0 ) is unitary for all non-resonant real r, which means that (4.8) must be bounded on the real axis. Thus it cannot have poles there and must admit analytic continuation to the real axis. We formulate these observations as a proposition. Proposition 4.8. Let λ ∈ Λ(H0 , F ; L1 ). The scattering matrix S(λ; Hr , H0 ) as a function of r is a meromorphic function with values in U1 (K), which admits analytic continuation to the real axis. Similarly for nonreal z, the off-axis scattering matrix r 7→ S(z; Hr , H0 ) ∈ U1 (K) as a function of r is analytic in a neighbourhood of R. Theorem 4.9. Let z ∈ Π± (H0 , F ; L1 ). For any r ∈ R such that z ∈ Π(Hr , F ; L1 ), the derivative of S(z; Hr , H0 ) with respect to r is given by (4.9)

d S(z; Hr , H0 ) dr p p = ∓ 2i w+ (z; H0 , Hr ) Im Tz (Hr )J˙r Im Tz + (Hr )w+ (z; Hr , H0 )S(z; Hr , H0 ),

where the derivative is taken in the norm of L1 (K).

42


Proof. We consider real z = λ + i0. For nonreal z, the formula holds for any r and the calculation of the derivative is identical. Consider the meromorphic function r 7→ Jr (1+ Tz (H0 )Jr )−1 , which appears in the stationary formula (4.7). Its derivative can be calculated to be (4.10)

d Jr (1 + Tz (H0 )Jr )−1 dr = J˙r (1 + Tz (H0 )Jr )−1 − Jr (1 + Tz (H0 )Jr )−1 Tz (H0 )J˙r (1 + Tz (H0 )Jr )−1 .

Since J0 = 0, the derivative (4.10) evaluated at r = 0 is simply J˙0 . It follows from the stationary formula that p p d (4.11) S(λ; Hr , H0 ) = −2i Im Tλ+i0 (H0 )J˙0 Im Tλ+i0 (H0 ). dr r=0

Using the property of the scattering matrix expressed in Theorem 3.15(iii), the derivative at 0 is all we need: By Theorem 2.37, λ ∈ Λ(Hr , F ; L1 ) if and only if r ∈ / R(λ; {Hr }). Let λ ∈ Λ(Hr , F ; L1 ). Since the resonance set R(λ; {Hr }) is discrete, λ ∈ Λ(Hr+h , F ; L1 ) for any small enough h. Theorem 3.15(iii) gives d d S(λ; Hr , H0 ) = S(λ; Hr+h , H0 ) dr dh h=0 d = w+ (λ; H0 , Hr ) S(λ; Hr+h , Hr )w+ (λ; Hr , H0 )S(λ; Hr , H0 ) dh h=0 and the proof is completed by substituting (4.11).

Corollary 4.10. Let z ∈ Π(H0 , F ; L1 ). The function p p (4.12) r 7→ w(z; H0 , Hr ) Im Tz + (Hr )J˙r Im Tz + (Hr )w(z; Hr , H0 ) ∈ L1 (K)

although only defined for nonresonant values of r, admits analytic continuation to the real axis.

A second proof of Lemma 2.38 follows from this corollary by taking the trace of the function (4.12) and using Corollary 3.9. Proof. At any nonresonant r, it can be seen from Theorem 4.9 and the unitarity of S(z; Hr , H0 ), that the function (4.12) is equal to 1 d S(z; Hr , H0 ) S(¯ z ; Hr , H0 ). ∓ 2i dr The result now follows from Proposition 4.8.

The next two theorems follow directly from Theorem 4.9 and properties of the ordered exponential (see §2.1.4). Theorem 4.11. Let z ∈ Π± (H0 , F ) ∩ Π± (H1 , F ). The operator S(z; H1 , H0 ), in particular the scattering matrix, has the ordered exponential representation Z 1 p p w(z; H0 , Hr ) Im Tz + (Hr )J Im Tz + (Hr )w(z; Hr , H0 ) dr . S(z; H1 , H0 ) = Texp ∓2i 0

Theorem 4.12. Let H1 and H0 be two self-adjoint operators from a resolvent comparable affine space, which is compatible with a rigging of the Hilbert space, and let z ∈ Π± (H0 , F ; L1 ) ∩ Π± (H1 , F ; L1 ). Then Z 1 ˙ Tr(Im Tz (Hr )Jr ) dr . (4.13) det S(z; H1 , H0 ) = exp ∓2i 0


43

This theorem can be interpreted as a variant of the Birman-Krein formula. The classical BirmanKrein formula can be recovered as follows. For nonreal z = λ + iy, y > 0, the formula (4.13) can be rewritten as (4.14)

det S(z; H1 , H0 ) = e−2πi ξ(z;H1 ,H0 ) ,

where ξ(z; H1 , H0 ) is the smoothed SSF (see p.21). The limit of ξ(z; H1 , H0 ) as y → 0+ is a.e. equal to the SSF ξ(λ; H1 , H0 ). On the other hand, for any λ ∈ Λ(H0 , F ; L1 ) ∩ Λ(H1 , F ; L1 ), the off-axis scattering matrix S(λ + iy; H1 , H0 ) converges in U1 (K) to the scattering matrix S(λ; H1 , H0 ) as y → 0+ (see Proposition 4.7). Therefore, taking the limit of (4.14) as y → 0+ results in Corollary 4.13. (Birman-Krein formula) For any two self-adjoint operators H0 and H1 from a resolvent comparable affine space, which is compatible with a rigging of the Hilbert space, the spectral shift function ξ(λ; H1 , H0 ) satisfies (4.15)

det S(λ; H1 , H0 ) = e−2πi ξ(λ;H1 ,H0 ) ,

for a.e. λ ∈ Λ(H0 , F ; L1 ) ∩ Λ(H1 , F ; L1 ). In addition to the classical formula (4.15), Theorem 4.12 explicitly gives a subtly different formula for the determinant of the scattering matrix: For any λ ∈ Λ(H0 , F ; L1 ) ∩ Λ(H1 , F ; L1 ), Z 1 Tr(Im Tλ+i0 (Hr )J) dr . det S(λ; H1 , H0 ) = exp −2i 0

which may be rewritten, in view of the definition (4.2) of the absolutely continuous SSF, as (4.16)

det S(λ; H1 , H0 ) = e−2πiξ

(a) (λ;{H

r })

.

Combining (4.15) and (4.16) gives the equality e−2πi ξ which holds for a.e. theorem.

(s) (λ;{H

r })

= 1,

λ ∈ Λ(H1 , F ; L1 ) ∩ Λ(H0 , F ; L1 ). Thus, we have proved the following

Theorem 4.14. Along any piecewise analytic path in a resolvent comparable affine space, which is compatible with a rigging of the Hilbert space, the singular SSF ξ (s) is a.e. integer-valued. 4.3. Singular spectral shift function and resonance index. The aim of this subsection is to show the connection between the singular SSF and the resonance index. As mentioned already, the resonance index is studied in detail in [Az3 ]. A brief review of the definition is given below. In this subsection, we consider only straight paths in the resolvent comparable affine space A(F ). Previous notation involving an arbitrary analytic path is modified for a straight path Hr = H0 + rV as follows: the resonance set is denoted R(λ; H0 , V ) instead of R(λ, {Hr }) and the absolutely continuous and singular SSFs are denoted ξ (a) (λ; H1 , H0 ) and ξ (s) (λ; H1 , H0 ) instead of ξ (a) (λ; {Hr }) and ξ (s) (λ; {Hr }). Let H0 ∈ A(F ). Let λ ∈ Λ(H0 , F ) and let V = F ∗ JF ∈ A0 (F ). We will consider the straight line Hr = H0 + rV. Let rλ be a fixed real resonance point from the resonance set R(λ; H0 , V ). As discussed in §2.4, this means that rλ is a real pole of the meromorphic function r 7→ Tλ+i0 (Hr )J, or of the meromorphic function r 7→ (1 + rTλ+i0 (H0 )J)−1 , or equivalently that the real number σλ := −rλ−1 is an eigenvalue of the compact operator Tλ+i0 (H0 )J. Consider shifting the point λ + i0 ∈ Π(H0 , F ) slightly to λ + iy for small positive (or negative) y. The function y 7→ Tλ+iy (H0 )J is analytic and by a well known result on the perturbation of isolated eigenvalues, the eigenvalue σλ may in general split into finitely many 1 N eigenvalues σλ+iy , . . . , σλ+iy . These N eigenvalues are stable for small y and are collectively known as the σλ -group. Or the rλ -group, since they correspond via j j σλ+iy = −(rλ+iy )−1 ,

j = 1, . . . , N,

44


j to poles rλ+iy of the meromorphic function r 7→ Tλ+iy (Hr )J. By Lemma 2.7 and Proposition 2.26, none of the eigenvalues of the σλ -group are real. Note that for any j = 1, . . . , N, the j j pole rλ+iy and the eigenvalue σλ+iy lie in the same complex half-plane. When y is positive, the number of eigenvalues (or poles) counting multiplicities which lie in the upper complex halfplane C+ is denoted N+ . The number lying in the lower half-plane C− is denoted N− . The difference N+ − N− is the resonance index, which is denoted

indres (λ; Hrλ , V ) = N+ − N− . The resonance index depends only on the triple (λ; Hrλ , V ) appearing in the notation. The elements of this triple are: a point λ which is essentially regular, meaning that it is a regular point of at least one operator from A(F ); an operator Hrλ which is resonant at λ, that is, λ∈ / Λ(Hrλ , F ); and a regular direction V, meaning that the line through Hrλ in the direction V contains an operator which is regular at λ (hence all but a discrete set of operators on this line are regular at λ). For a point λ outside of the essential spectrum, the resonance index indres (λ; Hrλ , V ) counts the net number of eigenvalues of the path Hr = H0 +rV which cross the point λ (in the positive direction) as r crosses rλ . The resonance index extends this notion of infinitesimal spectral flow into the essential spectrum. To make sense of the resonance index, we merely need to require relative compactness of the rigging operator F. The resonance index has other descriptions. Two of these are the signature of the resonance matrix and the singular µ -invariant. That the resonance index is equal to these numbers is proved, in full generality, in [Az3 ] and [Az5 ] respectively. The resonance index is also known to be equal to the singular SSF, at least in the case of trace class perturbations V, that is, in the case that the rigging operator F is Hilbert-Schmidt. This was established in [Az2 ]. Here we extend this result to the case of a relatively Hilbert-Schmidt rigging operator F. This will be proved following the same argument as that of [Az2 ]. For this purpose we briefly review another characterisation of the resonance index. Let z ∈ Π(H0 , F ) and let rz be a resonance point corresponding to z. By Pz (rz ) we denote the Riesz idempotent I 1 (σ − Tz (H0 )J)−1 dσ, Pz (rz ) = 2πi C(σz ) where C(σz ) is a small positively oriented circle enclosing the point σz = −rz−1 . The finite-rank operator Pz (rz ) is a projection (not necessarily orthogonal) onto the generalised eigenspace of the compact operator Tz (H0 )J at the isolated eigenvalue −rz−1 . The trace of Pz (rz ) is the multiplicity of the eigenvalue −rz−1 . A simple argument [Az3 , Proposition 3.2.3] shows that Pz (rz ) is also the residue of the meromorphic function r 7→ Tz (Hr )J = (1 + rTz (H0 )J)−1 Tz (H0 )J at the pole rz : (4.17)

1 Pz (rz ) = 2πi

I

Tz (Hr )J dr,

C(rz )

where C(rz ) is a small circle enclosing the point rz . For λ ∈ Λ(H0 , F ) and small positive or negative y, we define ↑ Pλ+iy (rλ ) =

X

j Pλ+iy (rλ+iy ),

j rλ+iy ∈C+

j where the sum is taken over those resonance points rλ+iy of the rλ -group which belong to ↓ the upper complex half-plane C+ . Similarly, the idempotent Pλ+iy (rλ ) is defined as the sum


45

j j of idempotents Pλ+iy (rλ+iy ) for resonance points rλ+iy of the rλ -group in the lower halfplane C− . For small positive y, we have ↑ ↓ (4.18) indres (λ; Hrλ , V ) = Tr Pλ+iy (rλ ) − Tr Pλ+iy (rλ ) ↑ ↑ (4.19) = Tr Pλ+iy (rλ ) − Tr Pλ−iy (rλ ) ,

where the second equality follows from the symmetry of the spectra of Tz (H0 )J and Tz¯(H0 )J. If C(rλ ) is a small circle enclosing the resonance point rλ , then for small positive y, it also j encloses all resonance points rλ+iy of the rλ -group. Note that by symmetry of the spectra of Tz (H0 )J and Tz¯(H0 )J, the circle C(rλ ) also encloses all anti-resonance points, that is, j resonance points rλ−iy of the rλ -group. Let C+ (rλ ) denote the upper closed semicircle formed by cutting C(rλ ) at the real axis (see the figure below). Then from (4.17), for small positive y we get I I I 1 1 1 Im Tλ+iy (Hr )J dr = Tλ+iy (Hr )J dr − Tλ−iy (Hr )J dr 2πi C+ (rz ) 2πi C+ (rz ) C+ (rz ) π ↑ ↑ = Pλ+iy (rλ ) − Pλ−iy (rλ ).

Therefore we obtain the following Theorem 4.15. [Az3 , Proposition 5.3.2] Let H0 be a self-adjoint operator from a relatively compact affine space A(F ), compatible with the rigging F, and let λ ∈ Λ(H0 , F ). Let V ∈ A0 (F ) and suppose rλ ∈ R(λ; H0 , V ). Let C(rλ ) be a small positively oriented circle enclosing rλ and no other points from R(λ; H0 , V ). Let C+ (rλ ) be the positively oriented closed semicircle forming the boundary of the region of C+ enclosed by C(rλ ). Then for small enough y, ! Z 1 Im Tλ+iy (Hr )J dr . indres (λ; Hrλ , V ) = Tr C+ (rλ ) π We are now in a position to make a connection to the singular SSF. Let rλ be a resonance point of the triple (λ, H0 , V ). Let [a, b] be an interval of R containing rλ and no other resonance points of (λ, H0 , V ). Consider z = λ + iy, 0 < y ≪ 1. Let L be a contour in C from a to b, which circumvents all resonance and anti-resonance points of the group of rλ in the upper half-plane as shown in the figure below. The closed contour C+ (rλ ) encloses all resonance and anti-resonance points of the group of rλ lying in the upper half-plane. C+ (rλ )

L a

rλ

b

rλ

For all small y, we have Z Z Z 1 b 1 1 Im Tλ+iy (Hr )J dr + Im Tλ+iy (Hr )J dr Im Tλ+iy (Hr )J dr = π a π L π C+ (rλ ) Z 1 ↑ ↑ Im Tλ+iy (Hr )J dr + Pλ+iy (rλ ) − Pλ−iy (rλ ). = π L Taking the trace of this equality and using (4.18) gives Z Z 1 b 1 (4.20) Tr(Im Tλ+iy (Hr )J) dr + indres (λ; Hrλ , V ). Tr(Im Tλ+iy (Hr )J) dr = π a π L The left hand side of (4.20) is the smoothed SSF, which by Theorem 2.1 and Proposition 2.21 converges to the spectral shift function ξ(λ; Hb , Ha ) as y → 0 for a.e. λ. Now we show that the first term on the right hand side of (4.20) converges to the absolutely continuous SSF

46


ξ (a) (λ; Hb , Ha ). Firstly, since on the contour L the function r 7→ Tr(Im Tλ+iy (Hr )J) uniformly converges to the function r 7→ Tr(Im Tλ+i0 (Hr )J) as y → 0+ , we have Z Z 1 1 Tr(Im Tλ+iy (Hr )J) dr = Tr(Im Tλ+i0 (Hr )J) dr. lim y→0 π L π L

Further, since by Lemma 2.38 the function Tr(Im Tλ+i0 (Hr )J) is analytic within the appropriately chosen region C+ (rλ ) for small enough y, in the last integral we can replace the contour L by [a, b]. The resulting integral is equal to ξ (a) (λ; Hb , Ha ), by the definition (4.2) of the latter. Therefore, taking the limit of both sides of (4.20) as y → 0 (and also using additivity along a curve as in (4.6)) proves the following theorem.

Theorem 4.16. Let H0 and H1 be self-adjoint operators from a resolvent comparable affine space A(F ), which is compatible with the rigging operator F. Let H1 − H0 = V. Then for a.e. λ from the full set Λ(H0 , F ; L1 ) ∩ Λ(H1 , F ; L1 ), X ξ(λ; H1 , H0 ) = ξ (a) (λ; H1 , H0 ) + indres (λ; Hrλ , V ), rλ ∈[0,1]

where the sum is taken over resonance points rλ from the finite set R(λ; H0 , V ) ∩ [0, 1]. Theorem 4.16 says that the singular SSF is equal almost everywhere to the total resonance index: X (4.21) ξ (s) (λ; H1 , H0 ) = indres (λ; Hrλ , V ). rλ ∈[0,1]

Since the total resonance index is explicitly defined on the full set Λ(H0 , F ; L1 ) ∩ Λ(H1 , F ; L1 ), Theorem 4.16 allows (4.21) to be taken as the definition of the singular SSF. Since the absolutely continuous SSF ξ (a) (λ; H1 , H0 ) is defined for every λ from this full set, this also allows the SSF ξ(λ; H1 , H0 ) to be explicitly defined on the same full set. Since the total resonance index is integer-valued by definition, this gives another proof of Theorem 4.14 along straight paths. References [Ag] [ASch] [Az] [Az2 ] [Az3 ] [Az4 ] [Az5 ] [ACDS] [AzD] [AzS] [BW] [BE] [BK] [BS] [BS2 ]

Sh. Agmon, Spectral properties of Schr¨ odinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (1975), 151–218. P. Alsholm, G. Schmidt, Spectral and scattering theory for Schr¨ odinger operators, Arch. Rational Mech. Anal. 40 (1971), 281–311. N. A. Azamov, Absolutely continuous and singular spectral shift functions, Dissertationes Math. 480 (2011), 1–102. N. A. Azamov, Resonance index and singular spectral shift function, arXiv:1104.1903. N. A. Azamov, Spectral flow inside essential spectrum, arXiv 1407.3551v2. N. A. Azamov, A constructive approach to stationary scattering theory, arXiv math-ph 1302.4142. N. A. Azamov, Resonance index and singular µ-invariant, arXiv 1501.07673. N. A. Azamov, A. L. Carey, P. G. Dodds, F. A. Sukochev, Operator integrals, spectral shift and spectral flow, Canad. J. Math. 61 (2009), 241–263. N. A. Azamov, Th. Daniels, Singular spectral shift function for Schr¨ odinger operators, II, work in progress. N. A. Azamov, F. A. Sukochev, Spectral averaging for trace compatible operators, Proc. Amer. Math. Soc. 136 (2008), 1769–1778. H. Baumg¨ artel, M. Wollenberg, Mathematical scattering theory, Basel; Boston, Birkhauser, 1983. ` M. Sh. Birman, S. B. Entina, The stationary approach in abstract scattering theory, Izv. Akad. Nauk SSSR, Ser. Mat. 31 (1967), 401–430; English translation in Math. USSR Izv. 1 (1967). M. Sh. Birman, M. G. Kre˘ın, On the theory of wave operators and scattering operators, Dokl. Akad. Nauk SSSR 144 (1962), 475–478. M. Sh. Birman, M. Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert space, D. Reidel Publishing Co., Dordrecht, 1987. M. Sh. Birman, M. Z. Solomyak, Remarks on the spectral shift function, J. Soviet math. 3 (1975), 408– 419.


[BY] [DS] [D] [Dy] [GK] [I]

[Ka] [KK] [Kr] [Kr2 ] [Ku] [Ku2 ] [Ku3 ] [L] [N] [RS] [RS2 ] [RS3 ] [Ru] [Sch] [S] [S2 ] [T] [Y]

47

M. Sh. Birman, D. R. Yafaev, Spectral properties of the scattering matrix (in Russian), Algebra i Analiz 4 (1992), no. 5, 1–27; English translation in St. Petersburg Math. J. 4 (1993), no. 6. M. Dimassi, J. Sj¨ ostrand, Spectral asymptotics in the semi-classical limit, Cambridge University Press, London Mathematical Society Lecture Notes Series, vol. 268. W. F. Donoghue, Jr, A theorem of the Fatou type, Monatshefte f¨ ur Mathematik, 67/3. E. M. Dyn’kin, An operator calculus based upon the Cauchy-Green formula, Zapiski Nauchn. semin. LOMI, 30 (1972), 33–40 and J. Soviet Math., 4 4 (1975), 329-334. I. C. Gohberg, M. G. Kre˘ın, Introduction to the theory of non-selfadjoint operators, Providence, R. I., AMS, Trans. Math. Monographs, AMS, 18, 1969. T. Ikebe, Eigenfunction expansions associated with the Schr¨ odinger operators and their application to scattering theory, Arch. Rational Mech. Anal. 5 (1960), 1–34; Erratum, Remarks on the orthogonality of eigenfunctions for the Schr¨ odinger operator on Rn , J. Fac. Sci. Univ. Tokyo, 17 (1970), 355-361. T. Kato, Some results on potential scattering, Proc. Intrn. Conf. on Functional Anal. and Related Topics, Tokyo, 1969, Tokyo Univ. Press 1970, 206–215. T. Kato, S. T. Kuroda, The abstract theory of scattering, Rocky Mountain J. Math. 1 (1971), 127–171. M. G. Kre˘ın, On the trace formula in perturbation theory, Mat. Sb., 33 75 (1953), 597–626. M. G. Kre˘ın, On perturbation determinants and the trace formula for unitary and self-adjoint operators, (Russian) Dokl. Akad. Nauk SSSR 144 (1962), 268–271. S. T. Kuroda, An introduction to scattering theory, Aarhus Universitet, Lecture Notes Series 51, 1976. S. T. Kuroda, Scattering theory for differential operators, I, operator theory, J. Math. Soc. Japan 25 (1973), 75–104. S. T. Kuroda, Scattering theory for differential operators, II, self-adjoint elliptic operators, J. Math. Soc. Japan 25 (1973), 222–234. I. M. Lifshitz, On a problem in perturbation theory connected with quantum statistics, Uspekhi Mat. Nauk 7 (1952), 171–180 (Russian). S. N. Naboko, Boundary values of analytic operator functions with a positive imaginary part, J. Soviet math. 44 (1989), 786–795. M. Reed, B. Simon, Methods of modern mathematical physics: 1. Functional analysis, Academic Press, New York, 1972. M. Reed, B. Simon, Methods of modern mathematical physics: 3. Scattering theory, Academic Press, New York, 1979. M. Reed, B. Simon, Methods of modern mathematical physics: 4. Analysis of operators, Academic Press, New York. W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1987. K. Schm¨ udgen, Unbounded self-adjoint operators on Hilbert space, Springer, GTM 265, 2012. B. Simon, Schr¨ odinger semigroups, Bull. Amer. Math. Soc. 7 (1982), 447–526. B. Simon, Trace ideals and their applications: Second Edition, Providence, AMS, 2005, Mathematical Surveys and Monographs, 120. J. R. Taylor, Scattering theory, John Wiley & Sons, Inc. New York. D. R. Yafaev, Mathematical scattering theory: general theory, Providence, R. I., AMS, 1992.

Index A0 , a real vector space of self-adjoint operators V, associated with A, pp. 12,23 A, a real affine space of self-adjoint operators H, associated with A0 , pp. 12,23 Cc (R), the class of infinitely differentiable functions on R with compact support C, the field of complex numbers C+ , the open upper half-plane C− , the open lower half-plane Eλ , evaluation operator, p. 29 F, a rigging operator H → K, satisfies the equality V = F ∗ JF, p. 21 H, Hilbert space H0 , a self-adjoint operator Hs , a path of operators H0 + sV hλ , a fiber Hilbert space, p. 5,10

48


J, a bounded self-adjoint operator on K, such that V = F ∗ JF, p. 23 K, auxiliary Hilbert space Lp (H), Schatten ideals rλ , resonance point, pp. 27,43 Rz (H) = (H − z)−1 , resolvent of H R, the field of real numbers Tz (H), sandwiched resolvent F Rz (H)F ∗ Uz (H), pp. 29,31 V, self-adjoint perturbation operator y, the imaginary part of the spectral parameter z z = λ + iy, complex number, spectral parameter z + , projection into the upper half plane: z + = Re z + i| Im z| λ, the real part of the spectral parameter z Λ(H, F ), the Borel set of λ ∈ R such that Tλ+i0 exists in uniform norm, p. 24 Λ(H, F ; L1 ), the set of λ ∈ Λ(H, F ) such that Im Tλ+i0 exists in trace class norm, p. 25 ξ(λ), spectral shift function, p. 20 ξ (a) (λ), absolutely continuous spectral shift function, p. 38 ξ (s) (λ), singular spectral shift function, p. 41 ξ(φ), spectral shift measure, p. 17 ξ (a) (φ), absolutely continuous spectral shift measure, p. 40 ξ (s) (φ), singular spectral shift measure, p. 40 Π, domain of the spectral parameter z Π(H, F ), union of C+ , C− and two copies of Λ(H, F ), p. 26 Π(H, F ; L1 ), union of C+ , C− and two copies of Λ(H, F ; L1 ), p. 25 σess , common essential spectrum of operators from an affine space A, p. 12 ΦH (V ), infinitesimal spectral shift measure, pp. 16,37 (a) ΦH (V ), absolutely continuous part of the infinitesimal spectral shift measure ΦH (V ), p. 38 (s) ΦH (V ), singular part of the infinitesimal spectral shift measure ΦH (V ), p. 38 Affine space, — relatively compact, p. 12 — resolvent comparable, p. 12 — compatible with F, p. 23 Resonance — point, p. 27, 43 — index, p. 44 Spectral shift function, 20 — absolutely continuous, p. 38 — singular, p. 41 — smoothed, p. 21 School of Computer Science, Engineering and Mathematics, Flinders University, Tonsley, 5042 Clovelly Park, SA, Australia E-mail address: [email protected] E-mail address: [email protected]

Singular spectral shift function for Schr\" odinger operators

Singular spectral shift function for Schr\" odinger operators

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