arXiv:1608.00294v1 [math-ph] 1 Aug 2016
EIGENVALUES BEHAVIOURS FOR SELF-ADJOINT PAULI OPERATORS WITH UNSIGNED PERTURBATIONS AND ADMISSIBLE MAGNETIC FIELDS DIOMBA SAMBOU AND AMAL TAARABT Abstract. We investigate the behaviour of the eigenvalues of twodimensional Pauli operators with nonconstant magnetic fields perturbed by a sign-indefinite decaying electric potential V . We prove new eigenvalues asymptotics.
1. Introduction and results We consider the Pauli operator (1.1)
H(b, V ) :=
(−i∇ − A)2 − b 0
0 +V (−i∇ − A)2 + b
on L2 (R2 , C2 ),
where V = V (x), x ∈ R2 , is a 2 × 2 Hermitian matrix-valued potential, and A is a vector potential generating the magnetic field (1.2)
b = ∇ ∧ A.
We assume b to be an admissible magnetic field in the sense there exists a constant b0 > 0 such that b(x) = b0 + ˜b(x), and where the Poisson equation ∆ϕ˜ = ˜b
(1.3)
˜ < ∞ for α ∈ N, admits a solution ϕ˜ ∈ C2 (R2 ) satisfying supx∈R2 |D α ϕ(x)| |α| ≤ 2. We refer for instance to [3] for examples of admissible magnetic fields. In the unperturbed case where V = 0, the spectrum of H(b, 0) belongs to {0} ∪ [ζ, +∞) with (1.4)
˜ ζ = 2b0 e−2osc(ϕ) ,
osc(ϕ) ˜ := sup ϕ(x) ˜ − inf ϕ(x). ˜ x∈R2
x∈R2
Further, 0 is an eigenvalue of infinite multiplicity (see e.g. [3]). Notice that in the constant magnetic field case b = b0 , we have ζ = 2b0 the first Landau level of the shifted Schrödinger operator (−i∇ − A)2 + b. The case where 2010 Mathematics Subject Classification. Primary: 35B34; Secondary: 35P25. Key words and phrases. Pauli operators, eigenvalues, non-definite sign perturbations. This research has been partially supported by the Chilean Program Núcleo Milenio de Física Matemática RC120002. 1
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DIOMBA SAMBOU AND AMAL TAARABT
V is of definite sign has been already studied in [3]. In the present note, we are interested in the sign-indefinite potentials V of the form U (x) 0 , for x ∈ R2 , (1.5) V (x) := U (x) 0 where the function U (x) ∈ C satisfies (1.6)
|U (x)| = O(hxi−m ),
hxi :=
for some m > 0.
p
1 + |x|2 ,
Remark. The potentials V of the form (1.5) are sign-indefinite since their eigenvalues are given by ±|U (x)|. Under condition (1.6), V is relatively compact with respect to H(b, 0) so that σess H(b, V ) = σess H(b, 0) , where σess denotes the essential spectrum. However, H(b, V ) may have discrete spectrum σdisc H(b, V ) that can only accumulate at 0. The novelty of this work arise from sign-indefinite perturbations we consider and behaviours we obtain. This is probably one of the first works dealing with sign-indefinite perturbations in a magnetic setting, see also the recent work [4] where the case of Pauli operators in dimension 3 are studied. We denote (1.7)
H± := (−i∇ − A)2 ± b
on L2 (R2 , C),
the component operators of the Pauli operator (1.1). Let p := p(b) be the spectral projection of L2 (R2 ) onto the (infinite dimensional) kernel of H− . The corresponding projection in the constant magnetic field case will be 2 2 denoted p0 := p(b0 ). For a bounded operator B ∈ L L (R ) , we introduce the operator W(B) defined by (1.8) W(B)f (x) := U (x)B(U f )(x).
Clearly, if I denotes the identity operator on L2 (R2 ), then W(I) is the multiplication operator by the function x 7−→ |U (x)|2 . This function will be denoted W(I) again. Our results are strongly related to the operator W(B) through the Toepliz operator (1.9)
pW(B)p,
−1 . B = I or H+
Since the spectrum of the invertible operator H+ belongs to [ζ, ∞) and U fulfills (1.6), then, it follows from [3, positive self-adjoint Lemma 3.5] that the −1 operators pW(I)p and pW H+ p are compact on L2 (R2 , C). Assumption (A). The real-valued function U satisfies −α x |x| (1 + o(1)), |x| → ∞, U (x) = U0 |x|
with α > 0, U0 ∈ C 0 (S1 ), U0 > 0, and |∇U (x)| = O(|x|−α−1 ) as |x| → ∞. Our first main result goes as follows:
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3
Theorem 1.1. Assume that V and U fulfil (1.5) and (1.6) respectively. Then, there exists a discrete set E ⊂ R∗ such that for any ε ∈ R∗ \ E and any r0 > 0 small enough, the following holds: (i) Localization: If z is a discrete o z ≤ 0. n eigenvalue of H(b, εV ), then, −1 (ii) Asymptotic: Suppose that # z ∈ σ(pW(H+ )p) : z > r → +∞ as r ց 0. Then, there exists a sequence (rℓ )ℓ tending to 0 such that (1.10) o o n n # z ∈ σdisc (H(b, εV )) : −r0 ≤ z < −rℓ
−1 ∼ # z ∈ σ(pW(H+ )p) : z > rℓ ,
as ℓ −→ ∞. (iii) Upper-bound: If W(I) satisfies Assumption (A), then,
(1.11)
n
# z ∈ σdisc (H(b, εV )) : −r0 ≤ z < −r
o
n
≤# z∈σ
o 1 pW(I)p ) : z > r , ζ
as r ց 0. Furthermore, if the magnetic field is constant (b = b0 ), we obtain the next lowers bounds: Theorem 1.2. Suppose that o n ∗ (1.12) # z ∈ σ (p0 U p0 p0 U p0 : z > 2rb0 = φ(r) 1 + o(1) , r ց 0, where φ r(1 ± ν) = φ(r) 1 + o(1) + O(ν) for any ν > 0 small enough. Then, n o # z ∈ σdisc (H(b, εV )) : −r0 ≤ z < −r o n (1.13) ∗ ≥ # z ∈ σ p0 U p0 p0 U p0 : z > 2rb0 , r ց 0. In particular, if U ≥ 0 and satisfies Assumption A), then, o n
(1.14)n
# z ∈ σdisc (H(b, εV )) : −r0 ≤ z < −r
as r ց 0.
≥ # z ∈ σ(p0 U p0 ) : z > 2rb0
12 o
,
Remarks. (i) Notice that the estimates (1.11) and (1.14) imply that in the constant magnetic field case, the number of negative eigenvalues of the operator H(b, εV ) near 0 is so that o n 1 1/m −1/m r , (1.15) # z ∈ σdisc H(b, εV ) : −r0 ≤ z < −r ∼ Cm 2b0 as r ց 0, for some constant Cm > 0. In particular, It holds from (1.15) that the negative eigenvalues of H(b0 , εV ) less than −r accumulate at zero with an accumulation rate of order r −1/m , whereas it was of order r −2/m for V of definite sign in [3].
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DIOMBA SAMBOU AND AMAL TAARABT
(ii) Otherwise, we can expect that this kind of accumulation also occurs near all the Laundau levels 2b0 q, q ∈ N, of the operator H(b0 , V ). 2. Strategy of the Proofs We explain the main ideas of the proofs and the relationship between the initial operator and the new quantities we are going to introduce. Let us introduce some useful notations. For H a separable Hilbert space, we denote S∞ (H ) (resp. GL(H )) the set of compact (resp. invertible) linear operators acting in H . Let D ⊆ C be a connected open set, Z ⊂ D be a discrete and closed subset, A : D\Z −→ GL(H ) be a finite meromorphic operator-valued function see e.g. [2, Definition 2.1] and Fredholm at each point of Z. For an operator A that does not vanish on γ a positive oriented contour, the index of A with respect to γ is defined by Z Z 1 1 ′ −1 tr A (z)A(z) dz = tr A(z)−1 A′ (z)dz. (2.1) Indγ A := 2iπ 2iπ γ γ 2.1. Reduction of the problem. We reduce our problem to a semi-effective Hamiltonian. Let us consider the punctured disk (2.2) D(0, ǫ)∗ := z ∈ C, 0 < |z| < ǫ for 0 < ǫ < ζ.
For z ∈ D(0, ǫ)∗ small enough, we have 1 0 H(b, V )−z −(H+ − z)−1 U (H+ − z)−1 (2.3) H− − z − U (H+ − z)−1 U U (H+ − z)−1 = , 0 1 so that the following characterisation holds: (2.4) H(b, V ) − z
is invertible ⇔ H− − z − U (H+ − z)−1 U
is invertible.
Thus we reduce the study of the discrete eigenvalues of H(b, V ) near z = 0 to the analysis of the non-invertibility of the operator H− −z−U (H+ −z)−1 U . It is not difficult to prove the following lemma which gives a new representation of the operator U (H+ − z)−1 U that turns to be useful. Lemma 2.1. Let z be small enough. Then, the operator U (H+ − z)−1 U admits the representation 1/2 (2.5) U (H+ − z)−1 U = w∗ 1 + M (z)H+ w, X 1/2 −1/2 −k−1 is analytic z k H+ where w := H+ U , and z 7−→ M (z)H+ := z k≥0
near z = 0. Let R− (z) denotes the resolvent of the operator H− . Under the notations of Lemma 2.1, we have the following:
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Lemma 2.2. For z small enough, the operator-valued function 1/2 D(0, ǫ)∗ ∋ z 7−→ TV (z) := 1 + M (z)H+ wR− (z)w∗ , is analytic with values in S∞ L2 (R2 ) .
Proof. Since M (z) and R− (z) are well defined and are analytic for z in D(0, ǫ)∗ , the analyticity of TV (z) follows. The compactness holds from that of U R− (z)U , combining the diamagnetic inequality and [5, Theorem 2.13]. We have the following Birman-Schwinger principle:
Proposition 2.1. For z near zero, the following assertions are equivalent: (i) z is a discrete eigenvalue of H(b, V ), (ii) I − TV (z) is not invertible. −1 Proof. Let R(z) := H− − z − U (H+ − z)−1 U . Then, the proof follows directly from the identity 1/2 1/2 (2.6) I − 1+M (z)H+ wR− (z)w∗ I + 1+M (z)H+ wR(z)w ∗ = I,
together with (2.4).
In Proposition 2.1,(ii), z is said to be a characteristic value of the operatorvalued function I − TV (·). The multiplicity of a discrete eigenvalue z is defined by (2.7) mult(z) := Indγ I − TV (·) ,
where γ is a small positively oriented contour containing z as the unique discrete eigenvalue of H(b, V ) see (2.1) . We will denote the set of characteristic values of I − TV (·) by char(•).
2.2. Sketch of proof of Theorem 1.1. First, we split the resolvent R− as R− (z) = (H− − z)−1 p + (H− − z)−1 p⊥ = −z −1 p + (H− − z)−1 p⊥ , since p is the spectral projection onto ker H− and p⊥ := 1−p is its orthogonal. In particular, we have (2.8) 1 1/2 1/2 TV (z) = − wpw∗ −z −1 M (z)H+ wpw∗ + 1 + M (z)H+ wR− (z)p⊥ w∗ . z We first deal with the first term of the r.h.s. of (2.8) such that we write wpw∗ = (pw∗ )∗ (pw∗ ). From the definition of w in Lemma 2.1, it follows −1 U and since σ(H+ ) ⊂ [ζ, ∞), we obtain that w∗ w = U H+ (2.9)
−1 )p ≤ ζ −1 pW(I)p, pw∗ wp = pW(H+
where W(•) is the operator defined by (1.8).
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DIOMBA SAMBOU AND AMAL TAARABT
According to Proposition 2.1, the discrete eigenvalues z of H(b, εV ) near 0 correspond to the characteristic values of the operator I − TεV (z). Setting KV (z) = −wpw∗ + zA(z),
(2.10) where
1/2 1/2 A(z) = −z −1 M (z)H+ wpw∗ + 1 + M (z)H+ wR− (z)p⊥ w∗ , we have ε2 KV (z), z so that KV (0) = −wpw∗ . Let Π0 be the orthogonal projection onto ker KV (0). The compactness of the operator KV′ (0)Π0 implies the existence of a sequence (εn )n such that the operator I − εKV′ (0)Π0 is invertible for each ε ∈ R\{εn , n ∈ N}. For L ∈ S∞ L2 (R2 ) , we let
(2.11)
(2.12)
I − TεV (z) = I −
n+ (r, L) := rank P(r,∞) (L),
r > 0,
where P(r,∞) (L) is the orthogonal projection of L on the interval (r, ∞). We have −1 ∗ ∗ (2.13)
n+ r, wpw
= n+ r, pw wp = n+ r, pW H+
p ,
Then, (2.9) together with the mini-max principle imply that pW(I)p −1 ∗ (2.14)
n+ r, wpw
= n+ r, pW H+
p ≤ n+ r,
ζ
r > 0.
,
r > 0.
We return to the proof of Theorem 1.1. The point (i) follows immediately from [2, Corollary 3.4] with z replaced by −z/ε2 . To deal with the aim (ii), let us introduce the sector Cα (a, b) := {x + iy ∈ C : a ≤ x ≤ b, −αx ≤ y ≤ αx}, with a > 0 tending to 0 and b > 0. The property (i) of Theorem 1.1 asserts that for z small enough, the discrete eigenvalues are concentrated in the sector z ∈ −ε2 Cα (r, r0 ) ∩ R, for any α > 0. Hence, one has n o # z ∈ σdisc H(b, εV ) : −r0 ≤ z < −r n o (2.15) = # z ∈ char(•) : −z/ε2 ∈ Cα (r, r0 ) ∩ R + O(1), r ց 0 In view of (2.14), we have Tr 1[r,r0 ] K(0) ≤ n+ r, pW(I)p + O(1). Thus, ζ the point (ii) follows from (2.15) together with [2, Corollary 3.9] and the equality in (2.14). Concerning (iii), if the function W(I) satisfies Assumption (A), then pW(I)p = Cm (ζr)−1/m 1+o(1) , Tr 1[r,r0] T (0) ≤ n ˜ [r, r0 ] := Tr 1[r,r0 ] ζ
DISTRIBUTION OF MAGNETIC EIGENVALUES
7
as r ց 0. Now, if φ(r) = r −γ , γ > 0, then φ r(1 ± ν) = r −γ (1 ± ν)−γ = φ(r) 1 + O(ν) . If n ˜ [r, 1] = φ(r) 1 + o(1) with φ r(1 ± δ) = φ(r) 1 + o(1) + O(δ) , then (2.16) n ˜ r(1 − ν), r(1 + ν) = n ˜ [r, 1] o(1) + O(ν) . Therefore, the result follows from (2.15) together with [2, Theorem 3.7] and (2.16).
2.3. Sketch of Proof of Theorem 1.2. If the magnetic field is constant, then −1 −1 p0 = (2b0 )−1 p0 . ≥ H+ H+ ∗ −1 This implies that p0 W H+ p0 ≥ (2b0 )−1 p0 U p0 p0 U p0 . Then the minimax principle holds ∗ −1 (2.17) Tr 1[r,r0 ] p0 W H+ p0 ≥ Tr 1[r,r0 ] (2b0 )−1 p0 U p0 p0 U p0 .
Thus, in the same spirit of the proof of (iii) in Theorem 1.1, (1.13) follows. This concludes the proof of Theorem 1.2. References [1] J. Avron, I. Herbst, B. Simon, Schrödinger operators with magnetic fields. I. General interactions, Duke Math. J. 45 (1978), 847-883. [2] J.-F. Bony, V. Bruneau, G. Raikov, Counting function of characteristic values and magnetic resonances, Commun. PDE. 39 (2014), 274-305. [3] G. Raikov, Spectral asymptotics for the perturbed 2D Pauli Operator with oscillating magnetic Fields. I. Non-zero mean value of the magnetic field, Markov Process. Related Fields 9 (2003), 775-794. [4] D. Sambou, Counting function of magnetic eigenvalues for non-definite sign perturbations, Operator Theory: Advances and Applications, 254 (2016), 205-221. [5] B. Simon, Trace ideals and their applications, Lond. Math. Soc. Lect. Not. Series, 35 (1979), Cambridge University Press. Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile E-mail address:
[email protected] Instituto de Física, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile E-mail address:
[email protected]
