Mathematical Physics, Analysis and Geometry 7: 119–149, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands.
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Transformation Operators for Sturm–Liouville Operators with Singular Potentials Dedicated to Professor V. A. Marchenko on the occasion of his 80th birthday ROSTYSLAV O. HRYNIV1 and YAROSLAV V. MYKYTYUK2 1 Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv,
Ukraine and Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine. e-mail:
[email protected] 2 Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine. e-mail:
[email protected] (Received: 26 September 2002; in final form: 4 April 2003) Abstract. We construct transformation operators for Sturm–Liouville operators with singular potentials from the space W2−1 (0, 1) and show that these transformation operators naturally appear during factorisation of Fredholm operators of a special form. Some applications to the spectral analysis of Sturm–Liouville operators with singular potentials under consideration are also given. Mathematics Subject Classifications (2000): Primary: 34C20; secondary: 34B24, 34L05, 47A68. Key words: transformation operators, Sturm–Liouville operators, singular potentials.
1. Introduction In the present work we shall study transformation operators (TOs) for Sturm–Liouville (SL) operators generated in a Hilbert space H = L2 (0, 1) by the differential expressions (f ) := −f + qf
(1.1)
with complex-valued distributions q from the space W2−1 (0, 1). (The precise definitions of the SL operators considered and the TOs are given in the next section.) Starting from the works by Povzner [25], Marchenko [21], Gelfand and Levitan [9] TOs have been successfully used in the spectral analysis of SL operators and other classical operators of mathematical physics. In particular, TOs have proved to be an important tool for solution of inverse spectral problems for SL operators (see the original papers [9, 21] and the monographs [20, 22, 26] for extended reference lists) and have been thoroughly studied for the case of regular, i.e., locally integrable, potentials q. Recent development of the theory of Sturm–Liouville and The work was partially supported by the Ukrainian Foundation for Basic Research DFFD under
grant no. 01.07/00172.
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Schrödinger operators with singular (i.e., not locally integrable) potentials [4, 14– 16, 24, 28–30] (see also the books [1] and [2] for general theory and detailed bibliography) has made feasible a thorough spectral analysis for SL operators with potentials from W2−1 (0, 1) and, in turn, led to inverse spectral problems for such class of operators. That posed the problem of existence and properties of TOs for SL operators with singular potentials, which we study in detail in the present article. In the subsequent papers [17, 18] we use the TOs constructed to solve the inverse spectral problems for SL operators with potentials from the space W2−1 (0, 1). Note that inverse spectral problems for SL operators with nonsmooth coefficients (in particular, for SL operators in the impedance form), were treated in different manner in, e.g., [3, 5, 7, 13, 27, 31, 32]. The main aim of this article is two-fold. Firstly, we shall construct the TOs for SL operators in H with singular potentials q ∈ W2−1 (0, 1) and study their properties. Secondly, we shall point out connection between the TOs constructed and a factorisation problem for Fredholm operators of a special form. Although this connection is known in the regular case, in the singular case under consideration it takes a very explicit form and allows a complete description. The organisation of the paper is the following. In the next section we briefly introduce the related concepts and give formulation of the main results. Section 3 is devoted to construction of some special TOs, which are then used in Section 4 to construct TOs for SL operators from the class considered and then to study their dependence on the potential. In Section 5 we establish results on connection of TOs with factorisation of some Fredholm operators, and in Section 6 we give some applications of the TOs to the spectral analysis of singular SL operators. 2. Preliminaries and Formulation of Main Results Throughout the paper we denote by dom T and ker T respectively the domain and kernel of an operator T in a Banach space X, while Wps ([0, 1], X) and Lp ((0, 1), X) will stand for the Sobolev and Lebesgue spaces of X-valued strongly measurable functions on [0, 1]. We shall write Wps [0, 1] and Lp (0, 1) instead of Wps ([0, 1], R) and Lp ((0, 1), R) respectively; in particular, W2−1 (0, 1) is the dual space of W21 [0, 1] with respect to L2 (0, 1). Suppose that q ∈ W2−1 (0, 1); then there exists a function σ ∈ H such that q = σ in the distributional sense, and differential expression (1.1) can be recast in terms of σ as d d +σ − σ f − σ 2 f, σ (f ) := −(f − σf ) − σf = − dx dx where again the derivatives are understood in the sense of distributions. We denote by Tσ a differential operator in H given by Tσ f = σ (f ) on the domain dom Tσ := {f ∈ W21 [0, 1] | f [1] ∈ W11 [0, 1], σ (f ) ∈ H },
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where f [1] := f − σf is the quasi-derivative of a function f . It is shown in [29] that the operator Tσ is closed. We consider a family Tσ,h of restrictions of Tσ to the linear manifolds dom Tσ,h := {f ∈ dom Tσ | f [1] (0) = hf (0)}. Here h is an arbitrary number from the extended complex plane C = C ∪ {∞}, and for h = ∞ the above relation is interpreted as dom Tσ,∞ := {f ∈ dom Tσ | f (0) = 0}. It is easily seen that Tσ,h = Tσ +h,0 for any h ∈ C, so that {Tσ,h | σ ∈ H, h ∈ C} = {Tσ,0 | σ ∈ H } ∪ {Tσ,∞ | σ ∈ H }, and it suffices to consider only the cases h = 0 and h = ∞. We also observe that Tσ,∞ = Tσ +h,∞ for any h ∈ C, so that the operator Tσ,∞ depends on the equivalence class σˆ := σ + C of H /C rather than on σ itself. It turns out that within each of the orbits {Tσ,0 | σ ∈ H } and {Tσ,∞ | σ ∈ H } all the operators are similar to each other. We recall first some definitions. DEFINITION 2.1. We say that closed and densely defined operators A and B in a Banach space X are similar and write A ∼ B if there exists a bounded and boundedly invertible operator U = UA,B (called the transformation operator (TO) for the pair (A, B)) such that AU = U B. Remark 2.2. Our definition of TO slightly differs from the one given in [20]; despite some shortcomings, it is more convenient for our purposes. The set (A, B) of all TOs for a pair (A, B) has the form (A, B) = {U V | V ∈ MB }, where U is a fixed TO and MB denotes the group of all bounded and boundedly invertible operators that commute with B. A TO U for a pair (A, B) is said to be unique up to a scalar factor if (A, B) = {λU | λ ∈ C \ {0}}; according to the above remark this is equivalent to the equality MB = {λI | λ ∈ C \ {0}}. THEOREM 2.3. Suppose that σ ∈ H ; then the following statements are true: (i) Tσ,0 ∼ T0,0 and Tσ,∞ ∼ T0,∞ ; (ii) a transformation operator Uσ,h for the pair (Tσ,h , T0,h ), h = 0, ∞, is unique up to a scalar factor and can be chosen as Uσ,h = I + Kσ,h , where Kσ,h is an integral Volterra operator of Hilbert–Schmidt class of the form x kσ,h (x, t)f (t) dt; (2.1) Kσ,h f (x) = 0
(iii) the operators T0,0 and T0,∞ are not similar.
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Remark 2.4. For the case of Sturm–Liouville operators on semiaxis with σ ∈ C (R+ ) claim (i) of the theorem is proved in [20, Theorem 1.2.1]. 1
The operators Kσ,h and Lσ,h := (I + Kσ,h )−1 − I , h = 0, ∞, have some other nice properties, to formulate which we have to introduce some functional spaces. Suppose that X is a Banach space with norm | · |; we shall also use the notation | · | for the operator norm in the algebra B(X) of all bounded operators in X. We denote by Gp (X), p 1, the set of all strongly measurable (classes of equivalent) functions k on [0, 1]2 with values in B(X) having the property that |k| ∈ Lp ((0, 1)2 ) and that the mappings x −→ k( · , x) ∈ Lp (0, 1), B(X) x −→ k(x, · ) ∈ Lp (0, 1), B(X) , are continuous on the interval [0, 1] (i.e., coincide a.e. with some continuous mappings of [0, 1] into Lp ((0, 1), B(X))). Also Gp (X) will denote the set of all integral operators with kernels from Gp (X). The set Gp (X) becomes a Banach space upon introducing the norm kGp (X) := max{ max k(x, · )Lp ((0,1),B(X)), max k( · , x)Lp ((0,1),B(X))}, x∈[0,1]
x∈[0,1]
while Gp (X) turns into a Banach algebra under the norm KGp (X) := kGp (X) with k being the kernel of K. It is easily seen that Gp (X) ⊂ G1 (X) for all p 1 and that the norm · G1 (X) coincides with the so-called Holmgren norm [8]; thus every operator K ∈ Gp (X), p 1, is continuous in all spaces Lq ((0, 1), X), q 1, see [6, Lemma XX.2.5]. Also every K ∈ G2 (X) acts continuously from L2 ((0, 1), X) into the space C([0, 1], X), and KL2 ((0,1),X)→C([0,1],X) KG2 (X) . Put + := {(x, t) ∈ (0, 1)2 | x > t}, − := {(x, t) ∈ (0, 1)2 | x < t} and − denote by G+ p (X) and Gp (X) the subspaces of Gp (X) consisting of all operators K whose kernels k satisfy the condition k(x, t) = 0 a.e. in − and k(x, t) = 0 − a.e. in + , respectively. G+ p (X) and Gp (X) form closed subalgebras of Gp (X) + − and, moreover, Gp (X) = Gp (X) Gp (X). Finally we observe that every element of G± p (X) is a Volterra operator in the space Lq ((0, 1), X) with any q 1. THEOREM 2.5. For any σ ∈ H and h = 0 or h = ∞ the operators Kσ,h and + Lσ,h = (I + Kσ,h )−1 − I belong to G+ 2 := G2 (C), and the mappings H σ −→ Kσ,h ∈ G+ 2,
H σ −→ Lσ,h ∈ G+ 2
are continuous from H into G+ 2. Consider now the sets of operators K0 := {Kσ,0 | σ ∈ H },
K∞ := {Kσ,∞ | σ ∈ H }.
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It turns out that these sets can be completely described in terms of the factorisation theory of Fredholm operators. We shall briefly recall its main notions, referring the reader to the books [10, 12] for further details. Let S2 denote the ideal of all Hilbert–Schmidt operators in H . Recall that every − operator in S2 is an integral operator; we denote by S+ 2 (S2 ) the set of those K ∈ − + S2 , whose kernels k vanish on (vanish on , respectively). It is obvious that − + − S2 = S+ 2 ⊕S2 and that the operators from S2 and S2 are Volterra ones. Also, the ± ± inclusions G2 (C) ⊂ S2 , G2 (C) ⊂ S2 hold and, moreover, KS2 KG2 (C) . DEFINITION 2.6. We say that an operator I + Q with Q ∈ S2 admits factori− ∈ S− sation (or is factorisable) if there exist operators K + ∈ S+ 2 and K 2 such that I + Q = (I + K + )−1 (I + K − )−1 . Note that an operator I + Q can admit at most one factorisation, so that the operators K ± = K ± (Q) are determined uniquely by Q. We denote by F2 the set of those Q ∈ S2 , for which I + Q is factorisable. The results of [23] imply the following statement. PROPOSITION 2.7. Put G2 := G2 (C); then (i) the set G2 ∩ F2 is open and everywhere dense in G2 ; (ii) for every Q ∈ G2 ∩ F2 the operators K ± (Q) belong to G± 2 and the operatorvalued mappings G2 ∩ F2 Q −→ K ± (Q) ∈ G± 2 are locally uniformly continuous. It turns out that the sets K0 and K∞ can be described as ranges of K + (·) when the argument Q runs through some special sets of operators in F2 . Namely, for φ ∈ L2 (0, 2) we denote by Fφ,0 and Fφ,∞ integral operators in G2 with kernels φ(x + t) + φ(|x − t|) and φ(x + t) − φ(|x − t|) respectively, i.e., 1 φ(x + t) + φ(|x − t|) f (t) dt, Fφ,0 f (x) := 0 1 φ(x + t) − φ(|x − t|) f (t) dt. Fφ,∞ f (x) := 0
We observe that, just like for Tσ,∞ , for any h ∈ C we have Fσ +h,∞ = Fσ,∞ , so that Fσ,∞ depends on the equivalence class φˆ := φ + C in L2 (0, 2)/C rather than on φ itself. Put for h = 0, ∞ Fh := {φ ∈ L2 (0, 2) | Fφ,h ∈ F2 }; then by Proposition 2.7 the sets F0 and F∞ are open in L2 (0, 2).
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THEOREM 2.8. The following equalities hold: K0 = {K + (Fφ,0 ) | φ ∈ F0 },
K∞ = {K + (Fφ,∞ ) | φ ∈ F∞ }.
In other words, for every σ ∈ H the operator Kσ,h can be obtained as a result of factorisation of I + Fφ,h for some φ ∈ L2 (0, 2) and, conversely, for every φ ∈ L2 (0, 2) such that I + Fφ,h is factorisable, the operator I + K + (Fφ,h ) is a TO for some SL operator Tσ,h . Thus Theorem 2.8 states that there exist bijections ∞ : H /C → F∞ /C, given by 0 : H → F0 and and H σ −→ φ0 =: 0 (σ ) ∈ F0 ˆ H /C σˆ −→ φ∞ =: ∞ (σˆ ) ∈ F∞ /C ∞ can be lifted to a mapping ∞ between H and F∞ and, respectively. In fact, moreover, the maps 0 and ∞ can be made more explicit. THEOREM 2.9. The mappings h , h = 0, ∞, are homeomorphic, and, moreover, with φσ,h := h (σ ) it holds x 1 lσ,h (x, t)2 dt, φσ,h (2x) = − σ (x) + 2 0 where lσ,h is the kernel of the integral operator Lσ,h = (I + Kσ,h )−1 − I .
3. Some Special Transformation Operators The aim of this section is to construct some special TOs for first order systems of differential equations on an interval. In the next section these TOs will be used to find TOs for SL operators with singular potentials from the space W2−1 (0, 1). The systems we shall consider here arise in a very natural way. Namely, according to our definition of σ the equality σ (u) = v is to be interpreted as −(u[1] ) − σ u[1] − σ 2 u = v with u[1] = u − σ u, or, in other words, as the first order system d u1 u1 0 −σ (x) −1 . + V (x) = , V (x) := u2 −v σ 2 (x) σ (x) dx u2 Observe that for σ ∈ H the entries of V (x) are integrable functions so that any solution to the above system is absolutely continuous and enjoys the standard uniqueness properties. It is reasonable to expect that TOs for the SL operators with singular potentials could be constructed through the analogous TOs for the d + V in the space H × H . More precisely, we shall seek for bounded operator dx ± operators A such that d ± V A± f = A∓ f dx
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for any function f ∈ W21 [0, 1] × W21 [0, 1] with f(0) = 0. Since V 2 = 0, the upperd d left components of ( dx ∓ V )( dx ± V ) coincide with −±σ , and hence the upper-left components of A± are strongly connected with the TOs for T±σ . In fact, without any technical complication we can consider a more abstract setting. Suppose that X is a Hilbert space with norm | · | and that v(·) is a function from L1 ((0, 1), B(X)). We denote by V an operator from L∞ ((0, 1), X) into L1 ((0, 1), X) given by ‘pointwise multiplication’, (Vf )(x) = v(x)f (x), and by · p the norm in the space Lp ((0, 1), B(X)), p 1. THEOREM 3.1. There exist operators A± ∈ B(L2 ((0, 1), X)) such that the equalities d + V A+ f = A− f , dx (3.1) d − + −V A f = A f dx hold for all f ∈ W21 ([0, 1], X) with f (0) = 0. The operators A± have the form x ± a ± (x, t)f (t) dt, f ∈ H , (3.2) A f (x) = f (x) + 0
and (A± − I ) ∈ G+ 1 (X). The proof of the theorem will rely on several lemmata. Consider first the case where v is smooth, i.e., v ∈ C ∞ ([0, 1], B(X)). Putting B + := 12 (A+ + A− ),
B − := 12 (A+ − A− ),
(3.3)
we rewrite system (3.1) as d (B + f ) + V B − f = B + f , dx d (B − f ) + V B + f = −B − f . dx
(3.4)
In terms of the kernels b± := (a + ± a − )/2 of the operators B ± (assuming that they are continuously differentiable in + ) these equations read (recall that f (0) = 0) x ∂ ∂ + − + b (x, t) + v(x)b (x, t) f (t) dt = 0, ∂x ∂t 0 (3.5) x ∂ ∂ − + − − b (x, t) + v(x)b (x, t) f (t) dt = − v(x) + 2b (x, x) f (x). ∂x ∂t 0
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ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
It is easily seen that relations (3.5) hold for all f ∈ W21 ([0, 1], X) with f (0) = 0 if the kernels b± satisfy the system x + v(ξ )b− (ξ, ξ − x + t) dξ, b (x, t) = − x−t (3.6) x 1 x +t − + b (x, t) = − . v(η)b (η, x + t − η) dη − v x+t 2 2 2 For convenience we assume the kernels a ± and b± to be extended by zero outside the domain + . We find a solution of system (3.6) by successive approximation method in the form b± =
∞
bn± ,
(3.7)
n=1
where the kernels bn± , n ∈ N, satisfy the following recurrent relations 1 x +t − , b1 (x, t) = − v 2 2 x + v(ξ )bn− (ξ, ξ − x + t) dξ, bn (x, t) = − x−t x − (x, t) = − v(η)bn+ (η, x + t − η) dη bn+1
(3.8)
x+t 2
for (x, t) ∈ + and equal zero otherwise. Remark 3.2. Suppose that v ∈ C ∞ ([0, 1], B(X)) and denote C := maxx∈(0,1) |v(x)| + maxx∈(0,1) |v (x)|. The standard induction arguments applied to (3.8) show that the functions bn± are continuously differentiable in + and also yield the following inequalities for all n ∈ N and all (x, t) ∈ + : 1 C 2n−1 2n−2 x , 2 (2n − 2)! 1 C 2n x 2n−1 , |bn+ (x, t)| 2 (2n − 1)!
2n−1
∂ −
b (x, t) , ∂ b− (x, t) (2n − 1)C (1 + x)2n−2 , n n
∂t
∂x (2n − 2)!
2n
∂ +
b (x, t) , ∂ b+ (x, t) (2n)C (1 + x)2n−1 .
∂t n
(2n − 1)!
∂x n
|bn− (x, t)|
This implies that series (3.7) as well as the series obtained after term-by-term differentiation of (3.7) in x or t converge uniformly in the domain + . Henceforth the functions b± and a ± are continuously differentiable in + and the operators A± given by formulae (3.2) satisfy equalities (3.1).
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Our ultimate goal is to construct the operators A± satisfying equalities (3.1) for an arbitrary v ∈ L1 ((0, 1), B(X)), and to this end we shall study convergence of (3.7) in a more suitable norm. LEMMA 3.3. Suppose that v ∈ C ∞ ([0, 1], B(X)). Then the functions bn± , n ∈ N, verify the inequalities x 2n−1 1 bn− (x, ·)1 |v(ξ )| dξ , (2n − 1)! 0 (3.9) x 2n 1 |v(ξ )| dξ . bn+ (x, ·)1 (2n)! 0 Proof. Using recurrent formulae (3.8), we find that x 1 x |v(x/2 + t/2)| dt |v(ξ )| dξ, b1− (x, ·)1 = 2 0 0 x x + dt |v(ξ )| |bn− (ξ, ξ − x + t)| dξ bn (x, ·)1 0 x−t x x dξ |v(ξ )| |bn− (ξ, ξ − x + t)| dt 0 x−ξ x |v(ξ )| bn− (ξ, ·)1 dξ, 0 x x − dt |v(η)| |bn+ (η, x + t − η)| dη bn+1 (x, ·)1
x+t 2
0
x x 2
0
dη |v(η)| 0
x
2η−x
|bn+ (η, x + t − η)| dt
|v(η)| bn+ (η, ·)1 dη.
Inequalities (3.9) follow now by induction if we use the identity x ξ n n+1 1 1 x |v(ξ )| |v(τ )| dτ dξ = |v(τ )| dτ . n! 0 (n + 1)! 0 0
(3.10) 2
The lemma is proved.
LEMMA 3.4. Suppose that v ∈ C ∞ ([0, 1], B(X)); then the functions b+ , b− belong to G1 (X) and b− G1 (X) sinh(v1 ),
b+ G1 (X) cosh(v1 ) − 1.
(3.11)
Remark 3.5. Observe that since the functions b+ and b− are continuous, they belong to the spaces Gp (X) for all p 1; however, inequalities analogous to (3.11) with p > 1 in general do not hold.
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Proof of Lemma 3.4. Integrating relations (3.8) in x and using Fubini’s theorem, we get 1 − b1 (·, t)1 |v(ξ )| dξ, 0 1 x + dx |v(ξ )| |bn− (ξ, ξ − x + t)| dξ bn (·, t)1
t
x−t 1
ξ +t
dξ |v(ξ )|
0
|bn− (ξ, ξ − x + t)| dx
ξ 1
|v(ξ )| bn− (ξ, ·)1 dξ, 0 1 x − (·, t)1 dx |v(η)| |bn+ (η, x + t − η)| dη bn+1
1
x+t 2
t
t
0
|bn+ (η, x + t − η)| dx
η 1
2η−t
dη |v(η)|
|v(η)| bn+ (η, ·)1 dη.
Recalling (3.9) and using (3.10), we easily establish the inequalities bn+ G1 (X)
1 v2n 1 , (2n)!
bn− G1 (X)
1 v2n−1 , 1 (2n − 1)! 2
and relations (3.11) follow.
Now we show that the functions b± ∈ G1 (X) depend continuously on the function v in the L1 ((0, 1), B(X))-norm. ± and LEMMA 3.6. Assume that v, v ∈ C ∞ ([0, 1], B(X)) and let bn± = bn,v ± ± bn = bn, v be the corresponding kernels constructed as above. Then for every n ∈ N and all x ∈ [0, 1] the following inequalities are satisfied: x 2n−2 v − v 1 − − , |v(ξ )| + | v (ξ )| dξ bn (x, ·) − bn (x, ·)1 (2n − 2)! 0 x 2n−1 (3.12) v − v 1 + + bn (x, ·)1 . |v(ξ )| + | v (ξ )| dξ bn (x, ·) − (2n − 1)! 0
bn± results Proof. Applying the arguments of the proof of Lemma 3.3 to bn± and in b1− (x, ·)
− b1− (x, ·)1
x
|v(ξ ) − v (ξ )| dξ, 0
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bn+ (x, ·)1 bn+ (x, ·) −
0
x
|v(ξ ) − v (ξ )|bn− (ξ, ·)1 dξ + x
+ 0
− − (x, ·) − bn+1 (x, ·)1 bn+1
| v (ξ )|bn− (ξ, ·) − bn− (ξ, ·)1 dξ, x
0
|v(η) − v (η)|bn+ (η, ·)1 dη + x
+ 0
| v (η)|bn+ (η, ·) − bn+ (η, ·)1 dη.
Using Lemma 3.3 and the induction assumption for bn− (x, ·) − bn− (x, ·)1 , we get bn+ (x, ·)1 bn+ (x, ·) − x 2n−1 v − v 1 |v(ξ )| dξ + (2n − 1)! 0 ξ 2n−2 v − v 1 x | v (ξ )| dξ |v(u)| + | v (u)| du + (2n − 2)! 0 0 ξ 2n−2 v − v 1 x dξ |v(ξ )| + | v (ξ )| |v(u)| + | v(u)| du (2n − 2)! 0 0 x 2n−1 v − v1 . |v(u)| + | v (u)| du = (2n − 1)! 0 − − (x, ·) − bn+1 (x, ·)1 In the same manner we obtain the required estimate for bn+1 based on that for bn+ (x, ·) − bn+ (x, ·)1 , and the proof by induction is complete. 2
Modifying similarly the arguments of Lemma 3.4 and using Lemma 3.6, we arrive at the following conclusion. b± are the LEMMA 3.7. Suppose that v, v ∈ C ∞ ([0, 1], B(X)) and that b± , corresponding kernels. Then b+ G1 (X) v − v 1 sinh(v1 + v 1 ), b+ − − − v 1 cosh(v1 + v 1 ). b − b G1 (X) v − We now return to the kernels a ± and the corresponding operators A± . Recall that by definition a + = b+ + b− ,
a − = b+ − b− .
Denote by av± (respectively A± v ) the kernels (respectively operators) that correspond to v ∈ C ∞ ([0, 1], B(X)). Since the space G1 (X) is complete, Lemma 3.7 and extension by continuity yield the following statement.
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COROLLARY 3.8. The mappings v → av± extend uniquely to continuous functions L1 (0, 1), B(X) v −→ av± ∈ G1 (X), and for arbitrary v, v ∈ L1 ((0, 1), B(X)) it holds v1 exp(v1 + v 1 ). av± − a v± G1 (X) v − It turns out that the functions av± even for an arbitrary v ∈ L1 ((0, 1), B(X)) still are the kernels of the corresponding transformation operators as stated in Theorem 3.1. Proof of Theorem 3.1. Suppose that v is an arbitrary function from L1 ((0, 1), ∞ B(X)). We fix a sequence (vn )∞ n=1 ⊂ C ([0, 1], B(X)) that converges to v in ± the L1 -norm and denote by Vn and An the corresponding operators of ‘pointwise multiplication’ by vn and transformation operators, respectively. It follows from Corollary 3.8 that there exist operators A± = A± v that are continuous in the space ± Lp ((0, 1), X) for every p ∈ [1, ∞] and such that A± n f → A f in Lp ((0, 1), X) for any f ∈ Lp ((0, 1), X). Take an arbitrary f ∈ W21 ([0, 1], X) with f (0) = 0; then (recall Remark 3.2) we have d ∓ ± Vn A± (3.13) n f = An f . dx Since f ∈ L∞ ((0, 1), X) ∩ L1 ((0, 1), X) and f ∈ L2 ((0, 1), X) ⊂ L1 ((0, 1), X), ± ± we have that A± n f → A f in L∞ ((0, 1), X) and L1 ((0, 1), X) and that An f → ± A f in L1 ((0, 1), X) as n → ∞. Convergence of vn to v in the L1 -norm now ± implies that Vn A± n f → V A f in L1 ((0, 1), X), so that by (3.13) we get ∓ ± (A± n f ) −→ A f ∓ V A f
as n → ∞ in the norm of the space L1 ((0, 1), X). It follows that A± f ∈ W11 ([0, 1], X) and (A± f ) = A∓ f ∓ V A± f. Thus operators A± verify equalities (3.1) and the theorem is proved.
2
The next theorem shows how equalities (3.1) should be modified for an arbitrary function f ∈ W21 ([0, 1], X). THEOREM 3.9. Suppose that v ∈ L1 ((0, 1), B(X)) and f ∈ W21 ([0, 1], X). Then A± f ∈ W11 ([0, 1], X) and d ± V A± f = A∓ f + a ∓ (x, 0)f (0). dx
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Proof. Consider the sequence fn = φn f , where φn (x) = min{nx , 1} for x ∈ [0, 1] and n ∈ N. Then fn ∈ W21 ([0, 1], X) and fn (0) = 0, so that by virtue of equalities (3.1) we get (A± f ) ± V A± fn = A∓ fn , or
(A± − I )fn
= (A∓ − I )(φn f ) + + (A∓ − I )(φn f ) ∓ V A± fn ,
n ∈ N.
(3.14)
Observe that fn → f in Lp ((0, 1), X) as n → ∞ for all p ∈ [1, ∞] and thus A± fn → A± f in L1 ((0, 1), X) ∩ L∞ ((0, 1), X) and V A± fn → V A± f in L1 ((0, 1), X). Convergence φn f → f in L1 ((0, 1), X) as n → ∞ implies that A∓ (φn f ) → A∓ f in L1 ((0, 1), X). Since f is continuous and a ∓ ∈ G1 (X), we have that 1/n ∓ (A − I )(φn f )(x) = n a ∓ (x, t)f (t) dt −→ a ∓ (x, 0)f (0) 0
in L1 ((0, 1), X). The above reasonings show that the right-hand side of equality (3.14) converges in L1 ((0, 1), X) to (A∓ − I )f + a ∓ (x, 0)f (0) ∓ V A± f as n → ∞. Since the space W11 ([0, 1], X) is complete, the statements of the theorem follow. 2 Now we establish some additional property of the kernels a ± constructed that will be essentially used for Sturm–Liouville operators. a ± = a v± LEMMA 3.10. Suppose that v, v ∈ C ∞ ([0, 1], B(X)) and a ± = av± , denote the corresponding kernels constructed for v and v respectively. Let P be an arbitrary bounded operator in X; then with g(x) = (1 + x) exp(2x) the following inequalities hold: a ± 2G2 (X) P a ± − P 12(P v − P v 22 + v − v 21 P v 22 )g(v1 + v 1 ).
(3.15)
b± , where b± := Proof. It suffices to prove analogous inequalities for P b± − P − a )/2. It follows from (3.6) that ( a ± +
b+ (x, t)| |P b+ (x, t) − P x |P v(ξ ) − P v (ξ )| |b− (ξ, ξ − x + t)| dξ + x−t x |P v (ξ )| |b− (ξ, ξ − x + t) − b− (ξ, ξ − x + t)| dξ. + x−t
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The Cauchy–Schwarz inequality yields the estimate |P b+ (x, t) − P b+ (x, t)|2 x 2C1 |P v(ξ ) − P v (ξ )|2 |b− (ξ, ξ − x + t)| dξ + x−t x |P v (ξ )|2 |b− (ξ, ξ − x + t) − b− (ξ, ξ − x + t)| dξ, + 2C2 x−t
where
C1 := max
1
y∈[0,1] y
C2 := max
y∈[0,1] y
1
|b− (ξ, ξ − y)| dξ, |b− (ξ, ξ − x + t) − b− (ξ, ξ − x + t)| dξ.
Integration in x and t now produces the inequality b+ 2G2 (X) 2C1 P (v − v )22 b− G1 (X) + P b+ − P v 22 b− − b− G1 (X) , + 2C2 P and it remains to estimate C1 and C2 in a suitable way. Using again (3.6) and Fubini’s theorem, we find that 1 ξ 1 − |b (ξ, ξ − y)| dξ dξ |v(η)| |b+ (η, 2ξ − y − η)| dη + y
y
ξ− y
2
y
1 1
dξ v ξ− + 2 y
2
1 v1 b+ G1 (X) 2
+ 12 v1 ,
so that 2C1 v1 b+ G1 (X) + v1 . Analogously we estimate 2C2 as follows: v 1 b+ G1 (X) + v 1 b+ − b+ G1 (X) + v − v 1 . 2C2 v − Combining these inequalities with the estimates of Lemmata 3.4 and 3.7 we arrive b+ 2G2 (X) with the constant 6 instead of 12. The estimate for at (3.15) for P b+ − P b− 2G2 (X) is derived analogously, and the result follows. 2 P b− − P Passing to the limit and completeness of the corresponding spaces justify now the following statement. COROLLARY 3.11. Suppose that v, v ∈ L1 ((0, 1), B(X)) and P ∈ B(X) are such that P v, P v belong to L2 ((0, 1), B(X)). Then for the corresponding kernels a ± = a v± the inclusions P a ± , P a ± ∈ G2 (X) take place, and, moreover, a ± = av± , inequality (3.15) holds.
TRANSFORMATION OPERATORS
133
4. Transformation Operators for Sturm–Liouville Operators In this section we use the results of the previous section to construct the TOs for the pairs of SL operators (Tσ,h , T0,h ), where σ ∈ H and h = 0, ∞. Denote by M2 := B(C2 ) the Banach space of all 2 × 2 matrices with complex entries and put H := L2 ((0, 1), C2 ). For an arbitrary but fixed σ ∈ H the function −σ (x) −1 v(x) := σ 2 (x) σ (x) belongs to L1 ((0, 1), M2 ). Henceforth by Theorems 3.1 and 3.9 there exist operators A± ∈ B(H) of the form x A± f(x) = f(x) + a ± (x, t)f(t) dt, 0 ±
such that a ∈ G1 (C ) and that for any f ∈ W21 ([0, 1], C2 ) the following relation holds: d ± V A± f = A∓ f + a ∓ (·, 0)f(0). (4.1) dx 2
With respect to the natural decomposition H = L2 (0, 1) × L2 (0, 1) the operators A± and the kernels a ± can be represented in the matrix form ± ± ± A11 A± a11 a12 ± ± 12 , a = . (4.2) A = ± ± ± A± a21 a22 21 A22 It follows that aij± ∈ G1 (C); also with 1 0 P = 0 0 ± ∈ G2 (C) depends continwe have P v(·) ∈ L2 ((0, 1), M2 ), which implies that a11 uously on σ ∈ H in view of Corollary 3.11. Taking f = (f, 0)T with f ∈ W21 [0, 1] in (4.1), we get the following equalities: d − + − − σ (A+ 11 f ) = A11 f + A21 f + a11 (·, 0)f (0), dx d + − + + σ (A− (4.3) 11 f ) = A11 f − A21 f + a11 (·, 0)f (0), dx d − − 2 + + σ (A+ 21 f ) = A21 f − σ A11 f + a21 (·, 0)f (0). dx
In particular, for f ∈ W22 [0, 1] with f (0) = f (0) = 0 this gives d d d + + +σ − σ A11 f = + σ (A− 11 f + A21 f ) dx dx dx 2 + = A+ 11 f − σ A11 f,
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+ i.e., σ (A+ 11 f ) = A11 (0 (f )). To have this equality satisfied for all f ∈ dom Tσ,h with h = 0 or h = ∞, we should modify the operator A+ 11 in a suitable way. For the sake of brevity we put − (·, 0), α1 = a11
+ α2 = a11 (·, 0),
− α3 = a21 (·, 0).
Take m = mh ∈ H , denote by M = Mh an integral operator in H given by x f (x − t)m(t) dt, Mf (x) = 0
and put ± A˜ ± ij = Aij (I + M).
Since (Mf ) = Mf + f (0)m,
(Mf )(0) = 0
for all f ∈ W21 [0, 1], equalities (4.3) result in d − ˜− ˜+ − σ (A˜ + 11 f ) = A11 f + A21 f + (A11 m + α1 )f (0), dx d + ˜+ ˜− + σ (A˜ − 11 f ) = A11 f − A21 f + (A11 m + α2 )f (0), dx d − 2 ˜+ ˜− + σ (A˜ + 21 f ) = A21 f − σ A11 f + (A21 m + α3 )f (0). dx
(4.4)
We are now in a position to prove similarity of the operators Tσ,h and T0,h , h = 0, ∞. We start with the simpler case h = ∞. LEMMA 4.1. The operators Tσ,∞ and T0,∞ are similar. −1 Proof. Take m∞ := −(A+ 11 ) α2 in equalities (4.4); then for all f ∈ dom Tσ,0 we get (recall that f (0) = 0): d d d ˜+ +σ − σ A˜ + + σ (A˜ − f = 11 11 f + A21 f ) dx dx dx 2 ˜+ = A˜ + 11 f − σ A11 f. ˜+ ˜+ It follows that A˜ + 11 f ∈ dom Tσ,∞ and Tσ,∞ A11 f = A11 T0,∞ f , i.e., that ˜+ A˜ + 11 T0,∞ ⊂ Tσ,∞ A11 . ˜+ ˜ + −1 Since A˜ + 11 is a bounded and boundedly invertible operator, for S := A11 T0,∞ (A11 ) we get S ⊂ Tσ,∞ .
(4.5)
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Fix an arbitrary nonzero λ ∈ C; it is clear that ran(T0,∞ − λI ) = H,
dim ker(T0,∞ − λI ) = 1,
so that also ran(S − λI ) = H,
dim ker(S − λI ) = 1.
(4.6)
Assuming that S = Tσ,∞ , we deduce from (4.5) and (4.6) that dim ker(Tσ,∞ − λI ) > 1. It follows then that there exists a nonzero function f ∈ W21 [0, 1] such that f [1] ∈ W11 [0, 1], f (0) = f [1] (0) = 0, and d f f 0 +V = −λ . f [1] f dx f [1] This is impossible by uniqueness arguments, so that S = Tσ,0 and the lemma is proved. 2 LEMMA 4.2. The operators Tσ,0 and T0,0 are similar. Proof. Since any f ∈ dom T0,0 satisfies f (0) = 0, we find that d ˜− ˜+ − σ A˜ + 11 f = A11 f + A21 f + pf (0) dx with p := A− 11 m + α1 and − d d d +σ − σ A˜ + + σ A˜ 11 f + A˜ + f = 11 21 f + pf (0) dx dx dx + = A˜ 11 f − σ 2 A˜ + 11 f + qf (0) with
q :=
d + σ p + A− 21 m + α3 . dx
We shall prove below that for a suitable choice of m ∈ W21 [0, 1] we get p ∈ ˜+ W21 [0, 1], p(0) = 0, and q ≡ 0. Then A˜ + 11 T0,0 ⊂ Tσ,0 A11 and the arguments similar to those used in the proof of Lemma 4.1 do the rest. To produce the required m, we denote by J the integration operator, x f (t) dt, Jf (x) := 0 −1 and take m in the form m = (A− 11 ) (J r − α1 ) for some r ∈ H . Then p = J r ∈ W21 [0, 1], p(0) = 0, and the equation q = 0 can be recast as − −1 − − −1 r + σ J r + A− 21 (A11 ) J r − A21 (A11 ) α1 + α3 = 0
(4.7)
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in terms of r. It is clear that − −1 R := σ J + A− 21 (A11 ) J
is a Hilbert–Schmidt operator with lower triangular kernel; therefore R is a Volterra operator in H and (4.7) has a unique solution − −1 r = (I + R)−1 (A− 21 (A11 ) α1 + α3 ) ∈ H.
2
The proof is complete.
The previous lemmata also show that the TO for the pair Tσ,h and T0,h can be + + chosen as A˜ + 11 = A11 + A11 M, which is of the form (2.1) with x + + a11 (x, s)mh (s − t) ds. kσ,h (x, t) = a11 (x, t) + t
Remark 4.3. Observe that although the kernel kσ,h need not be smooth enough for the function Kσ,h f to belong to W21 [0, 1] even if f ∈ C ∞ [0, 1], the function g := (I + Kσ,h )f has quasi-derivative g [1] ∈ W11 [0, 1] for any f ∈ dom T0,h and x x − + [1] a˜ 11 (x, t)f (t) dt + a˜ 21 (x, t)f (t) dt + ph (x)f (0) g (x) = f (x) + 0
with p∞ ≡ 0 and p0 ∈
0
W21 [0, 1]
constructed in the proof of Lemma 4.2.
We show next that the TO I + Kσ,h is unique up to a scalar factor. To this end it suffices to prove that the set of bounded and boundedly invertible operators commuting with T0,h in H consists of nonzero multiples of the identity operator. Denote sλ (x) :=
sin λx , λ
cλ (x) := cos λx,
with λ an arbitrary complex number and s0 (x) ≡ x. Observe that the mappings λ → sλ and λ → cλ are analytic H -valued functions of the argument λ. LEMMA 4.4. Suppose that a bounded and boundedly invertible operator U in H satisfies one of the following conditions: (a) for all λ ∈ C the function sλ is an eigenvector of U ; (b) for all λ ∈ C the function cλ is an eigenvector of U . Then U = cI for some nonzero c ∈ C. Proof. Assume first that condition (a) is satisfied. Then there exists a function g such that U sλ = g(λ)sλ
(4.8)
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for all λ ∈ C. For a given λ0 ∈ C the relation g(λ) =
(U sλ , sλ0 ) (sλ , sλ0 )
shows that g analytic in a neighbourhood of λ0 consisting of those λ, for which (sλ , sλ0 ) = 0; since λ0 is arbitrary, g is entire. By (4.8) the function g satisfies the inequality U −1 −1 |g(λ)| U for all λ ∈ C; the Liouville theorem now proves that g ≡ c for some nonzero c. Since the system {sλ | λ ∈ C} is complete in H (e.g., the sequence (sπn )n∈N forms a basis of H ), this implies that U = cI as claimed. The case (b) is considered analogously. 2 LEMMA 4.5. The operators T0,0 and T0,∞ are not similar. Proof. Assume that the claim of the lemma is false. Then there exists a bounded and boundedly invertible operator U such that T0,∞ U = U T0,0 . Since for an arbitrary λ ∈ C we have ker(T0,0 − λ2 I ) = lin{sλ },
ker(T0,∞ − λ2 I ) = lin{cλ },
there exists a function f : C → C such that U sλ = f (λ)cλ
(4.9)
for all λ ∈ C. The relation f (λ) =
(U sλ , cλ0 ) , (cλ , cλ0 )
shows that f is analytic in a neighbourhood of any point λ0 ∈ C and hence is entire. Equality (4.9) implies that |f (λ)|
sλ U , cλ
λ ∈ C.
Writing λ = ξ + iη with ξ, η ∈ R, we find that 1 1 sinh 2η sin 2ξ 2 − , | sin λx| dx = 2 2η 2ξ 0 1 1 sinh 2η sin 2ξ + . | cos λx|2 dx = 2 2η 2ξ 0
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ROSTYSLAV O. HRYNIV AND YAROSLAV V. MYKYTYUK
Therefore lim
λ→∞
sλ =0 cλ
and hence f (λ) = o(1),
λ → ∞.
The Liouville theorem now implies f ≡ 0, which contradicts invertibility of U . 2 The contradiction derived shows that T0,0 and T0,∞ are not similar. With all these results in hand, we can prove Theorems 2.3 and 2.5. Proof of Theorem 2.3. Lemmata 4.1 and 4.2 establish similarity of claim (i) and existence of the TO of required form of claim (ii). Uniqueness of claim (ii) follows from Lemma 4.4, and part (iii) is proved in Lemma 4.5. 2 Proof of Theorem 2.5. We consider first the operators Kσ,h , h = 0, ∞. It follows from the proof of Lemmata 4.1 and 4.2 that + I + Kσ,h = A+ 11,σ + A11,σ Mσ,h ,
(4.10)
where the operator Mσ,h acts according to x mσ,h (x − t)f (t) dt, Mσ,h f (x) := 0
with mσ,0 and mσ,∞ defined in the proofs of Lemmata 4.2 and 4.1 respectively. It is easily seen that Mσ,h − Mσ˜ ,h G+2 mσ,h − mσ˜ ,h L2 (0,1), so that the mapping L2 (0, 1) σ −→ Mσ,h ∈ G+ 2 is continuous as soon as such is the mapping L2 (0, 1) σ −→ mσ,h ∈ L2 (0, 1). Assuming this already established and recalling that G+ 2 is a Banach algebra and + depends continuously on σ ∈ H in G , we conclude from the above that A+ 11,σ 2 arguments and representation (4.10) that the mapping L2 (0, 1) σ −→ Kσ,h ∈ G+ 2 is continuous. Thus it remains to show that mσ,0 and mσ,∞ depend continuously on σ .
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For h = ∞ we have −1 + mσ,∞ := (A+ 11,σ ) a11,σ (·, 0),
and the required continuity follows from Corollary 3.8. For h = 0 the function mσ,0 is given by − −1 mσ,0 := (A− 11,σ ) (J rσ − a11,σ (·, 0)),
where rσ := (I + Rσ )−1 pσ ,
− − −1 − pσ := A− 21,σ (A11,σ ) a11,σ (·, 0) + a21,σ (·, 0)
and − −1 Rσ := σ J + A− 21,σ (A11,σ ) J
with J being the integration operator. It follows from Corollary 3.8 that the mappings → pσ ∈ L1 (0, 1), L2 (0, 1) σ − → Rσ ∈ B L1 (0, 1), L2 (0, 1) L2 (0, 1) σ − are continuous. This implies continuity of the mapping L2 (0, 1) σ −→ rσ ∈ L2 (0, 1) as well as the one L2 (0, 1) σ −→ mσ,0 ∈ L2 (0, 1), and the proof for the operators Kσ,h , h = 0, ∞, is complete. + To treat the operators Lσ,h , we observe first that the mapping L: G+ 2 → G2 given by L(K) = (I + K)−1 − I =
∞
(−K)n
(4.11)
n=1
is continuous. In fact, for the kernel k of an operator K ∈ G+ 2 we find that
1
k(x, s)k(s, t) ds
k(x, ·)L2 (0,1)k(·, t)L2 (0,1) k2G2
0
if x > t, so that by induction K 2n G2 K2n G2 /(n − 1)! and thus series (4.11) converges locally uniformly in K ∈ G2 . It follows now that Lσ,h = (I + Kσ,h )−1 − I = L(Kσ,h ) ∈ G+ 2 depends continuously on σ ∈ H , and the theorem is proved. 2
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5. Proof of Theorems 2.8 and 2.9 In this section, we shall study the question how the TOs Kσ,h enter factorisation of some special operators. Note that the relations we establish below are known for regular potentials (e.g., formula (5.2) can be found in [22]); we derive them here not only for the sake of completeness but also to give a precise characterisation of the TOs for the SL operators with singular potentials from W2−1 (0, 1). Recall that for an integral operator K in H with kernel k its associated operator K is defined as the integral operator with kernel k (x, t) = k(t, x). Claims of Theorems 2.8 and 2.9 are basically contained in the following two lemmata. LEMMA 5.1. Suppose that σ ∈ H . Then for h = 0, ∞ the following equality holds: −1 ) = I + Fφ,h , (I + Kσ,h )−1 (I + Kσ,h
in which the function φ = φσ,h ∈ L2 (0, 2) is given by x 1 lσ,h (x, t)2 dt, x ∈ [0, 1], φσ,h (2x) = − σ (x) + 2 0
(5.1)
(5.2)
and lσ,h is the kernel of the operator Lσ,h = (I + Kσ,h )−1 − I . Proof. First we shall prove equality (5.1) for the case h = 0 and under the assumption that σ ∈ C 2 [0, 1]. Put x kσ,0 (x, t) cos λt dt, λ ∈ C, x ∈ [0, 1]. (5.3) y(λ, x) := cos λx + 0
By the definition of the TO I + Kσ,0 the function y satisfies the equation −y (λ, x) + q(x)y(λ, x) = λ2 y(λ, x) and the initial conditions y(λ, 0) = 1,
y (λ, 0) = ay(λ, 0),
where q = σ and a := σ (0). Since I +Lσ,0 is the inverse of I +Kσ,0 , relation (5.3) can be recast as x lσ,0 (x, t)y(λ, t) dt. cos λx = y(λ, x) + 0
It is shown in [22, Ch. I.2] that the kernel lσ,0 of Lσ,0 is twice continuously differentiable in the closure of the domain + = {(x, t) ∈ (0, 1)2 | x > t} and is a unique solution of the partial differential equation = −ltt + q(t)l, −lxx
(x, t) ∈ + ,
subject to the boundary conditions 1 x q(t) dt, l(x, x) = −a − 2 0
lt (x, t)|t =0 − al(x, 0) = 0.
(5.4)
(5.5)
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Denote by I + F the operator of the left-hand side of (5.1). Then I + F := (I + Lσ,0 )(I + L σ,0 ), so that F is an integral operator with kernel 1 lσ,0 (x, s)lσ,0 (t, s) ds. f (x, t) = lσ,0 (x, t) + lσ,0 (t, x) + 0
It follows that the function f is twice continuously differentiable in [0, 1]2 and is symmetric with respect to x and t. We shall use (5.4) and (5.5) to show that f = −ftt so that f (x, t) = φ1 (x + t) + satisfies in + the wave equation −fxx φ2 (x − t) for some φ1 , φ2 ∈ C 2 [0, 1], and then derive from (5.5) that φ1 = φ2 = φσ,0 with φσ,0 of (5.2). The details are as follows. We write l instead of lσ,0 for brevity and observe that for (x, t) ∈ + it holds t l(x, s)l(t, s) ds. (5.6) f (x, t) = l(x, t) + 0
Differentiating (5.6) in x twice and using (5.4)–(5.5) at various points, we obtain t lxx (x, s)l(t, s) ds fxx (x, t) = lxx (x, t) + 0 t t (x, t) + lss (x, s)l(t, s) ds − l(x, s)q(s)l(t, s) ds = lxx 0 0 t lss (x, s)l(t, s) ds + = lxx (x, t) + 0 t l(x, s) ltt (t, s) − lss (t, s) ds + 0 t l(x, s)ltt (t, s) ds + = lxx (x, t) + 0
t + ls (x, s)l(t, s) − l(x, s)ls (t, s) 0 t = lxx (x, t) + l(x, s)ltt (t, s) ds + 0 + lt (x, t)l(t, t)
Differentiation in t gives ftt (x, t)
=
ltt (x, t)
t
+ 0
− l(x, t)ls (t, s)|s=t .
l(x, s)ltt (t, s) ds +
+ [l(x, t)l(t, t)]t + l(x, t)lt (t, s)|s=t , so that in view of (5.4) and (5.5) (x, t) − ftt (x, t) = −q(t)l(x, t) − 2[l(t, t)]t l(x, t) = 0 fxx
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as stated. Therefore f (x, t) = φ1 (x + t) + φ2 (x − t) for some φ1 , φ2 ∈ C 2 [0, 1]. It follows that ft (x, t)|t =0 = φ1 (x) − φ2 (x), while (5.6) and (5.5) give ft (x, t)|t =0 = lt (x, t)|t =0 + l(x, 0)l(0, 0) = lt (x, t)|t =0 − al(x, 0) = 0. Thus φ1 − φ2 ≡ c for some constant c, and with φ0 := φ1 − c/2 = φ2 + c/2 we conclude that f (x, t) = φ0 (x + t) + φ0 (x − t), i.e., that F = Fφ0 ,0 . To find φ0 , we observe that (5.6) and (5.5) for x = t give x 1 x φ0 (2x) + φ0 (0) = −a − q(s) ds + l(x, s)2 ds, 2 0 0 so that φ0 (0) = −a/2. Recalling that σ is a primitive of q with σ (0) = a, we arrive at the equality x l(x, s)2 ds. φ0 (2x) = − 12 σ (x) + 0
Summarizing, we have shown that F = Fφ0 ,0 with φ0 = φσ,0 given by (5.2) (i.e., proved the theorem) under the additional assumption that σ ∈ C 2 [0, 1]. Suppose now that σ is an arbitrary function in H and take a sequence (σn ) ⊂ C 2 [0, 1] that converges to σ in H . Denoting by φn ∈ L2 (0, 2) the functions as in (5.2) but corresponding to σn , we conclude from Theorem 2.5 that φn converge in L2 (0, 2) to φσ,0 of (5.2). Therefore lim Fφn ,0 − Fφσ,0 ,0 G2 = 0,
n→∞
and, passing to the limit in the equality Lσn ,0 + L σn ,0 + Lσn ,0 Lσn ,0 = Fφn ,0
results in (5.1) for h = 0 and an arbitrary σ ∈ H . The case h = ∞ is treated analogously; the only reservations are that the function cos λx should be replaced with (sin λx)/λ and boundary conditions (5.5) with the ones 1 x q(t) dt, lσ,∞ (x, 0) = 0. lσ,∞ (x, x) = − 2 0 The lemma is proved.
2
LEMMA 5.2. Suppose that φ ∈ F0 . Then there exists a unique σ0 ∈ H such that −1 ) = I + Fφ,0 ; (I + Kσ,0 )−1 (I + Kσ,0
(5.7)
moreover, σ0 depends continuously on φ ∈ F0 . Similarly, for φ ∈ F∞ there exists a unique σˆ ∞ ∈ H /C such that, for σ ∈ σˆ , )−1 = I + Fφ,∞ ; (I + Kσ,∞ )−1 (I + Kσ,∞
moreover, σˆ ∞ depends continuously on φˆ := φ + C ∈ F∞ /C.
(5.8)
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Proof. We shall consider only the case h = 0 as the other one is completely analogous. Denote by F = Fφ,0 an integral operator with kernel f (x, t) = φ(x + t) + φ(|x − t|); then by assumption the operator I + F is factorisable as explained in Section 2. Put K := K+ (F ); since the integral operator F has a symmetric kernel, it is easily seen that K− (F ) = K , i.e., that I + Fφ,0 = (I + K)−1 (I + K )−1 . Multiplying both sides of the above relation by I + K and equating the corresponding kernels, we arrive at the so-called Gelfand–Levitan–Marchenko (GLM) equation x k(x, s)f (s, t) ds = 0, (x, t) ∈ + , (5.9) k(x, t) + f (x, t) + 0
in which k is the kernel of K. Recall that the operator I + F is factorisable if and only if the Gelfand–Levitan–Marchenko equation is soluble for k. Suppose now that the function φ is smooth (e.g., infinitely differentiable). Then it is shown in [22, Ch. II.3] and [20, Ch. II.4] that the solution k of the GLM equation (5.9) is smooth as well and satisfies the partial differential equation (x, t) + q(x)k(x, t), −ktt (x, t) = −kxx
q(x) := 2[k(x, x)]x
and the boundary condition kt (x, t)|t =0 = 0. Moreover, k is the kernel of the TO I + Kσ,0 with σ (x) := 2[k(x, x) − k(0, 0)]. Denote by l the kernel of the integral operator L := (I + K)−1 − I . Then l = lσ,0 , so that by equality (5.2) x l(x, s)2 ds, x ∈ [0, 1]. (5.10) σ (x) = −2φ(2x) + 2 0
In view of Proposition 2.7 the operator K ∈ G2 (C) (and thus L ∈ G2 (C)) depends continuously on φ ∈ F0 , whence the function σ ∈ H of (5.10) is continuous in φ ∈ F0 ∩ C ∞ [0, 2] in the L2 (0, 2)-norm. Take now an arbitrary φ ∈ F0 and put ck (x) := cos( π2 kx);since the sys∞ = tem (ck )k0 is an orthonormal k=0 αk ck with m basis of L2 (0, 2), we have φ ∞ αk := (φ, ck ). Put φm := k=0 αk ck ; then the sequence (φm )m=1 converges to φ in L2 (0, 2) and therefore by Proposition 2.7 the infinitely differentiable functions φm belong to F0 for all m large enough, say for m m0 . Let (σm)∞ m=m0 be the sequence of functions constructed for φm through formula (5.10); in particular, I + Fφm ,0 = (I + Kσm ,0 )−1 (I + Kσm ,0 )−1 .
(5.11)
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From the above said it follows that (σm )∞ m=m0 is a Cauchy sequence in H and hence converges to some function σ ∈ H . By virtue of Theorem 2.5 we can pass to the limit in equation (5.11) to obtain −1 I + Fφ,0 = (I + Kσ,0 )−1 (I + Kσ,0 ) .
The above arguments also imply that the function σ so constructed depends con2 tinuously on φ ∈ F0 . The proof (for the case h = 0) is complete. Proof of Theorem 2.8. The statements of the theorem are corollaries of Lemmata 5.1 and 5.2. 2 Proof of Theorem 2.9. Continuity of the mappings h : σ → φσ,h , h = 0, ∞, follows from Theorem 2.5 and formula (5.2), and that of the inverse mappings from Lemma 5.2. 2
6. Some Applications In this section we demonstrate usefulness of the TOs constructed for the spectral analysis of Sturm–Liouville operators with singular potentials q from the space W2−1 (0, 1). Namely, we shall establish eigenvalue asymptotics, completeness properties of eigenfunctions, and similarity of related operators. Note that some of these results were established by different methods in [28–30]. Assume that q = σ for some complex-valued function σ ∈ H and denote by Tσ,DD the restriction of the operator Tσ,∞ defined in Section 2 by the Dirichlet boundary condition at the point x = 1. Then Tσ,DD has a discrete spectrum [29] that accumulates at +∞; we denote by λ2k , k ∈ N, the eigenvalues of Tσ,DD counted with multiplicities and ordered so that Re λk+1 Re λk and |Im λk+1 | |Im λk | in the case of equality above. Here and in the following, λk are taken from the closed right half-plane. THEOREM 6.1. For the eigenvalues λ2k ordered as explained above we have λk = λk ) belongs to 2 . πk + λk , where the sequence ( Proof. Observe first that for any λ ∈ C the solution to the equation σ y = λ2 y satisfying the boundary condition y(0) = 0 equals x kσ,∞ (x, t)sλ (t) dt; y(x, λ) = sλ (x) + 0
here sλ (x) = (sin λx)/λ for λ = 0 and s0 (x) ≡ x, and kσ,∞ is the kernel of the TO I + Kσ,∞ . Therefore λ2 is an eigenvalue of the operator Tσ,DD if and only if y(1, λ) = 0, and in that case y(x, λ) is a corresponding eigenfunction. In
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145
other words, the spectrum of Tσ,DD coincides with the squared zeros of the entire function 1 sin λt sin λ kσ,∞ (1, t) (λ) := + dt. λ λ 0 Observe also that λ is a zero of the function of multiplicity k 1 if and only ∂ ∂ k−1 y(x, λ), . . . , ( ∂λ ) y(x, λ) form a chain of eigen- and if the functions y(x, λ), ∂λ associated functions of the operator Tσ,DD corresponding to the eigenvalue λ2 , i.e., the multiplicity of a zero λ of coincides with the algebraic multiplicity of the eigenvalue λ2 of Tσ,DD . Thus it suffices to study the asymptotics of zeros of the function (λ) in the half-plane Re λ 0. Since is an odd function, its zeros are symmetric with respect to the origin and we can study the zeros in the whole plane C. Put 1 kσ,∞ (1, t) sin µt dt; Q(µ) := 0
then Q is an entire function of exponential type and lim e−|Im µ| Q(µ) = 0
|µ|→∞
(6.1)
by [22, Lemma 1.3.1]. For any n ∈ Z, denote by n the boundary of the rectangular Rn := {µ = ν + iτ | ν, τ ∈ R, |ν − π n| < π/2, |τ | < 1}. Then inf |sin µ| = c > 0
µ∈n
is independent of n, while sup |Q(µ)| −→ 0 µ∈n
as |n| → ∞ by virtue of (6.1). By Rouche’s theorem each Rn contains exactly one zero of the function for all n large enough. Representing this solution as µn , we see that µn = π n + µn ) −→ 0 sin µn = (−1)n+1 Q(π n + and whence µn → 0 as n → ∞. It follows now that 1 n+1 kσ,∞ (1, t) sin π nt dt + o(| µn |); µn = (−1) √
since { implies
0
2 sin π nt}∞ n=1 that ( µn ) ∈ 2 .
is an orthonormal basis of H and kσ,∞ (1, t) ∈ H , this
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Finally we observe that for all n large enough sin λ/λ and have the same number of zeros (counting multiplicities) inside the regions Pn := {µ ∈ C | |Re µ| π n + π/2, |Im µ| < n}; this easily follows by Rouche’s theorem (see details in [22, Ch. 1.3]). Therefore λn with λn = µn for all n large enough; in particular, we may write λn = π n + 2 2 (λn ) ∈ . The theorem is proved. It follows from Theorem 6.1 that all eigenvalues λ2n but maybe finitely many are simple. We denote by φn , n ∈ N, the system of eigen- and associated functions of the operator Tσ,DD corresponding to the eigenvalues λ2n . For all n large enough of Tσ,DD ) we normalize the eigenfunctions (e.g., such that λ2n is a simple eigenvalue √ [1] φ (0) = 2λ ; observe that for such n we have φn (t) = φn by the condition n n √ (I + Kσ,∞ ) 2 sin λn t. THEOREM 6.2. The system (φn )∞ n=1 forms a Bari basis of H (i.e., a basis that is quadratically close to√an orthonormal one). 2 Proof. Put ψn := √2 sin λn x if n ∈ N is of √such that λn is a simple eigenvalue √ k Tσ,DD and put ψn := 2 sin λn x, ψn+1 = 2x sin λn x, . . . , ψn+k = 2x sin λn x if λ2n = λ2n+1 = · · · = λ2n+k is an eigenvalue of Tσ,DD of algebraic multiplicity k + 1. Then due to the asymptotics of λn established in the previous theorem the system (ψn )∞ n=1 is complete in H , see [19, App. III]. For all n large enough we have φn = (I + Kσ,∞ )ψn ; we can also choose the remaining functions φn accordingly so that the above equality will hold for all n ∈ N. Since I +Kσ,∞ is a homeomorphism, we conclude that the√system (φn ) is complete in H . Put now ψn,0 := 2 sin π nx; then (ψn,0 ) is an orthonormal basis of H and φk − ψk,0 = (I + Kσ,∞ )(ψk − ψk,0 ) + Kσ,∞ ψk,0 . Observe that √ λk cos(π k + λk /2) = O(| λk |) ψk (x) − ψk,0 (x) = 2 2 sin 2 for all k large enough, so that ψk − ψk,0 2 < ∞ on account of the inclusion ( λk ) ∈ 2 . Since Kσ,∞ is a Hilbert–Schmidt operator and (ψk,0 ) is an orthonormal basis of H , we also have Kσ,∞ ψk,0 2 < ∞. Therefore φk − ψk,0 2 < ∞,
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and thus (φn )∞ n=1 is a Bari basis of H (see [11, Ch. VI]).
2
Our final result shows that existence of TOs implies that the operators Tσ,DD with σ ∈ H are similar to rank one perturbations of the potential-free operator T0,DD . THEOREM 6.3. Suppose that σ ∈ H . Then there exists a function p = pσ ∈ H such that Tσ,DD is similar to the operator S given by (Sf )(x) = −f on the domain
2 dom S = f ∈ W2 [0, 1] | f (0) = f (1) +
1
p(t)f (t) dt = 0 .
0
The similarity is performed by the operator I + Kσ,∞ . Proof. Since Tσ,∞ (I +Kσ,∞ ) = (I +Kσ,∞ )T0,∞ and Tσ,DD is a one-dimensional restriction of Tσ,∞ , the operator Tσ,DD is similar to a one-dimensional restriction S of the operator T0,∞ . The domain of S is found from the requirement that (I + Kσ,∞ ) dom S = dom Tσ,DD . Thus every f ∈ dom S belongs to W22 [0, 1] and satisfies the conditions 1 kσ,∞ (1, t)f (t) dt = 0, f (0) = f (1) + 0
and it remains to observe that p(t) := kσ,∞ (1, t) belongs to H .
2
Similar results also hold for the restriction Tσ,DN of the operator Tσ,∞ by the Neumann boundary condition f [1] (1) = hf (1), h ∈ C, at the point x = 1 and for the restrictions Tσ,ND and Tσ,NN of the operator Tσ,0 by the Dirichlet and Neumann boundary conditions respectively at the point x = 1. References 1. 2. 3. 4. 5.
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