1 Introduction
Abstract

We consider the Schrödinger operator with constant magnetic field defined on the half-plane with a Dirichlet boundary condition, , and a decaying electric perturbation . We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of by studying the Spectral Shift Function (SSF) associated to the pair . For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.

Keywords: Magnetic Schrödinger operators; Half-plane; Dirichlet and Neumann boundary conditions; Spectral Shift Function; Pseudodifferential Calculus.

Mathematics Subject Classification 2010: 35P20, 35J10, 47F05, 81Q10.

Threshold Singularities of the Spectral Shift Function for a Half-Plane Magnetic Hamiltonian


Vincent Bruneau111 Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux, 351 cours de la Libération, 33405 Talence Cedex, France. e-mail: vbruneau@math.u-bordeaux.fr and Pablo Miranda222 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173. Santiago, Chile. e-mail: pablo.miranda.r@usach.cl






1 Introduction

Motivations

We consider (resp., ), the Dirichlet (resp., Neumann) self-adjoint realization of the magnetic Schrödinger operator

(1.1)

in the half-plane ().

Our goal is to analyze the effects on the spectrum when a relatively compact perturbation of or is introduced. The perturbations under consideration will be real electric potentials that decays to zero at infinity in .

Such effect is now well understood for perturbations of the so-called Landau Hamiltonian i.e., the magnetic Schrödinger operator (1.1) but defined for the whole plane . The Landau Hamiltonian admits pure point spectrum with eigenvalues of infinite multiplicity (the so called Landau levels , ). It is established that perturbations by a decaying electric potential of a definite sign, even if it is compactly supported, produce an accumulation of discrete eigenvalues around the Landau levels (see [32, 21, 35, 25, 13, 37]). Using variational methods, it can be seen that the distribution of these eigenvalues is governed by the counting function of the eigenvalues of the compact Toeplitz operators , where is the spectral projection on Ker. Then, depending on the decay rate of , it is possible to obtain the asymptotic behavior of the counting functions of the eigenvalues of near each Landau Level. Tools from pseudodifferential analysis, together with variational and Tauberian methods have been used to obtain this behavior for V: power-like decaying [32, 21], exponentially decaying [35], compactly supported [35, 25, 13, 31, 36]. Also magnetic [37] and geometric perturbations [30, 27, 16] have been considered.

In our case, on the half-plane, the spectrum of (resp. ) is rather different from that of . It is purely absolutely continuous, given by (resp. , ). From the dynamical point of view, this difference is related to the fact that in the classical trajectories are circles, while in there exist propagation phenomena along the boundary . The accumulation of the discrete spectrum of and of for compactly supported potentials was studied in [7]. However, to our best knowledge, there are no results concerning the continuous spectrum.

A natural tool to extend this notion of spectral density, from the discrete spectrum into the continuous spectrum, is the spectral shift function (SSF) (see (1.6) and (2.11) below). For example, this function is studied for the Schrödinger operator with constant magnetic field in , which has purely absolutely continuous spectrum, and it is proved that it admits singularities at the Landau levels (see [12, 4] and other magnetic examples in [5, 34, 39]). The operators considered in these works admit an analytically fibered decomposition, and the singularities of the SSF are present at the minima of the corresponding band functions. Is is important to notice that in all these models, the minima of the band functions are non-degenerate. These points are called thresholds, because they are points where the Lebesgue multiplicity of the a.c. spectrum changes.

From a general point of view, it is known that the extrema of the band functions play a significant role in the description of the spectral properties of fibered operators(see [15]). In the particular case where these extrema are reached and non-degenerate, there is a well known procedure to obtain effective Hamiltonians that allows to describe the distribution of eigenvalues (as in [33, 7]) and the singularities of the SSF (see [5]).

Our operator also admits an analytically fibered decomposition (see (1.2) below), and as we will see, the singularities of the spectral shift function are present at the infima of the band functions as well. However, the nature of these infima is completely different from the ones mentioned above. They are not reached, they are the limits of the band functions at infinity. The derivative is zero only at infinity. This is the source of one of the main technical difficulties that we have to overcome in order to describe the properties of the SSF.

Since our extremal points are only the limits on the band functions, it is necessary to modify the analysis of previous works. The phenomenon of thresholds given by limits of the band functions is also present for some quantum Hall effect models (see [6]) and for some Iwatsuka models (see [26]). In these works, the SSF was studied only in the region where it counts the number of discrete eigenvalues. Similar results are expected to hold also inside the continuous spectrum, but it requires a more precise description of some Birman-Schwinger operators, especially when the considered energy levels cross the corresponding band functions.

Dirichlet magnetic Schrödinger operator on the half-plane

In the paper, we will focus on the Dirichlet realization of magnetic Schrödinger operator (1.1) and write instead of . We will discuss the Neumann boundary condition at the end of Section 2.

The operator is generated in by the closure of the quadratic form

defined originally on . This is the Hamiltonian of a 2D spinless nonrelativistic quantum particle in a half-plane, subject to a constant magnetic field of scalar intensity .

Let be the partial Fourier transform with respect to , i.e.

Then, we have the identity

(1.2)

where the operator is the Dirichlet realization in of

The domain of the operator is:

(1.3)

Note that it does not depends on . Also, the family , is real analytic in the sense of Kato [24], and for each the operator has a discrete and simple spectrum. Let be the increasing sequence of the eigenvalues of . For , the function is called the -th band function. By the Kato analytic perturbation theory, is a real analytic functions of . Further, it is proved in [11] (see also [21, Chapter 15.A]), that for any , the band function is strictly decreasing, and

(1.4)

In consequence, the spectrum of is purely absolutely continuous and is given by

Using decomposition (1.2) and the monotonicity of , it is possible to see that the Lebesgue multiplicity of the a.c. spectrum of changes at any point in the set . Such points are called thresholds in .

Perturbation and Spectral Shift Function

Suppose that the electric potential is a Lebesgue measurable function that satisfies

(1.5)

for some positive constant , , and . On the domain of introduce the operator

self-adjoint in . Estimate (1.5) combined with the diamagnetic inequality in the half-plane (see [20]) imply that for any real the operator is Hilbert–Schmidt, and hence the resolvent difference is a trace-class operator. In particular, the absolutely continuous spectrum of coincides with . Furthermore, there exists a unique function called the spectral shift function for the operator pair , that satisfies the Lifshits-Kreĭn trace formula

for each , and vanishes identically in [40].

In scattering theory, the SSF can be seen as the scattering phase of the operator pair , namely we have the Birman-Kreĭn formula

where is the scattering matrix of the operator pair . In addition, for almost every the SSF coincides with the eigenvalue counting function of the operator , i.e.

(1.6)

where denotes the characteristic function of the Borel set .

In this article, for of a definite sign, we investigate the properties of the SSF , particularly its behavior near the Landau levels. Using the Pushnitski representation formula of the SSF, we will reduce our analysis to the study of counting function of some parametrized compact operators (see Theorem 2.1). To prove these results, in Section 3 we will describe precisely some properties of a Birman-Schwinger operator for which a careful analysis of the band functions and of their derivatives is necessary. As a consequence, in Corollary 2.4, we can show that the SSF is bounded on compact sets not containing the thresholds . Then, in Section 4, assuming that is smooth and admits a power-like decay at infinity, we will prove our main result Theorem 2.3. It gives the asymptotic behavior of as the energy approaches the singularity present at the spectral threshold . This result is proved using pseudodifferential methods.

2 Main results

2.1 Reduction to a counting function for a compact operator

Our first main result concerns an effective Hamiltonian that permits us to estimate the behavior of the SSF for a wide class of non-negative potentials . The effective Hamiltonian is given by the real part of a limit of the trace class operator in (2.3) below. To describe this operator we need to introduce some notations, which will be used systematically in the sequel.

Fix and . Denote by the one-dimensional orthogonal projection onto . Then

(2.1)

where , , is an eigenfunction of that satisfies

(2.2)

Moreover, could be chosen to be real-valued, and since the family is analytic, it can be chosen analytic as a function of . The projection is analytic with respect to as well.

Fix . For define

(2.3)

By Proposition 3.6 below, if satisfies (1.5), the limit exists in the trace class-norm for energies , .

For a compact self-adjoint operator , define the eigenvalue counting function

(2.4)
Theorem 2.1.

Assume that satisfies (1.5), and write . For all and all we have

(2.5)

as .

The proof of Theorem 2.1 is contained in Section 3.4.

Arguing as in the proof Theorem 2.1 we have the following:

Corollary 2.2.

Suppose that satisfies (1.5). Then, on any compact set ,

i.e. the SSF is bounded away from the thresholds.

2.2 Spectral asymptotics

One consequence of Corollary 2.2 is that the only possible points of unbounded growth of are the Landau Levels . On the other side, Theorem 2.1 can be used to describe the explicit asymptotic behavior of the SSF at these points. In our second main theorem we obtain this behavior for potentials that decay moderately at infinity. To measure this decaying rate it is typically considered the following volume function:

(2.6)

where is measurable, , and denotes the Lebesgue measure in . Further, we will need the Hörmander class: for

(2.7)

Then, if , we have that , for . But this bound is insufficient for our purposes, since it will be necessary to have also some control “from below” and on the regularity of the volume function. In consequence, we will suppose that satisfies the following conditions:

(2.8)

Conditions (2.8) are commonly assumed in the study of the distribution of eigenvalues of some pseudodifferential operators (see for instance [10, 32, 21, 23, 37]). A typical situation of satisfying (2.8), is when where is smooth and .

In the following, for two functions and defined on some interval , we will write

(2.9)

for all in and for positive constants .

Theorem 2.3.

Let and write . If satisfies (2.8), then the following asymptotic formulas for the SSF

(2.10)

hold true for any fixed . This implies in particular that

The proof of Theorem 2.3 can be found in Section 4.3.

Remarks

  1. The results in Theorem 2.3 resemble those for the eigenvalue counting function of . More explicitly, if satisfies (1.5) we can define the function that counts the number of eigenvalues in the gaps of as In this case, from Pushnitski formula (3.1) and the Birman-Schwinger principle one easily obtain that

    (2.11)

    and therefore the study of the eigenvalue counting function of the perturbed Landau Hamiltonian , is the same as the study of the Spectral Shift Function for the pair ). In fact, under conditions similar to those in Theorem 2.3, it is proved in [32] that the asymptotic behavior of is also given by a semiclassical formula of the form (2.10). Since can be decomposed as in (1.2) but with constant band functions, from (2.11) one can say that, to some extent, our work extends the 2D results on the SSF of [32], to a case where the band functions of the unperturbed operator are not constant.

  2. Related to the previous remark, we can mention the study of the SSF under compactly supported perturbations of (including obstacle perturbations), as a natural and important open problem (in particular from the physical point of view, see [11, 19, 1, 14]). These cases present a different difficulty since the pseudodifferential analysis we used here, does not work (there is no convenient class of symbols for compactly supported potentials), and some non-semiclassical asymptotics are expected (see [35, 7]). In fact, using the effective Hamiltonian of our Theorem 2.1 together with ideas of [6, 7], one can show that for of compact support

  3. The singularities of the SSF are naturally related to clusters of resonances (see for instance [4]). It would be interesting to analyze this phenomenon, but the first difficulty to overcome is to define the resonances for our fiber operator. Due to the exponential decay properties of the band functions, some non-standard tools of complex analysis would be necessary.

Let us complete the results of Theorem 2.3 by other consequences of our analysis.

Corollary 2.4.
  1. Assume that satisfies (1.5). Then

  2. On the other side, if satisfies (2.8)

    (2.12)

The proof of part 1 of this Corollary is a direct consequence of Theorem 2.1, while part 2 can be proved using the same tools of pseudodifferential analysis used for the proof of Theorem 2.3.

From Theorem 2.3 and Corollary 2.4 we can compare the behavior of the SSF as a Landau level is approached from different sides:

(2.13)

This result is different to the results obtained for magnetic Hamiltonians for which the behavior of the SSF was studied at the thresholds (see [12], [34], [39]). For those models the corresponding limit (2.13) is a constant different from zero, at least for , which gives a generalization of the Levinson’s formula.

At last, let us mention that following the proofs of our work, it is easy to obtain some analog results for the half-plane magnetic Schrödinger operator with a Neumann boundary condition.

2.3 Results for Neumann boundary conditions

Let us consider , the Neumann realization of (1.1), namely the self-adjoint operator generated in by the closure of the quadratic form

defined originally on , see [14].

A fibered decomposition of the form (1.2) holds true in this case as well. Thanks to [9] we know that each band function of the Neumann Hamiltonian, , is a decreasing function until a unique non degenerated minimum, and then increase satisfying and . Then, the minimum of each band is a threshold of the spectrum of . Due to the non-degeneracy condition, it is well known how to study the behavior of the SSF at this points (see [5], [7]).

On the other side, for the extremal points at infinity, it can be shown that the behavior of the band functions and of the associated eigenfunctions are the same as those of the Dirichlet operator (see Propositions 3.5 and 3.3 below). Thus, just like in Theorem 2.1, we can justify that the main contribution of the SSF near a fixed Landau level depends on the behavior of the corresponding band functions , the associated eigenfunction, and the interplay of this objects at infinity (this last one still given by (3.4) below). The main difference with the Dirichlet case comes from the fact that, at infinity the band functions are now below the corresponding Landau level. Then, up to a change of sign of (or equivalently of and of ), the above results remain true. More precisely, for , we have:

Theorem 2.5.

The statements of Theorems 2.1 and 2.3 hold true for the Neumann boundary conditions. Moreover, the results of Corollary 2.4 have to be replaced by:

  1. If satisfies (1.5), then as .

  2. If satisfies (2.8), then , as

By the previous arguments it seems reasonable that these results could also be extended without mayor changes to other 2D magnetic models like, (1.1) in the half-plane with a Robin boundary condition and some types of Iwatsuka Hamiltonians [22].

3 Estimates on the SSF

3.1 Pushnitski’s representation of the SSF

The proof of Theorem 2.1 is based on the following representation of the SSF given by A. Pushnitski in [29]. For and , define

As is shown in Lemma 3.8 below, the norm limits

exist for every , provided that satisfies (1.5). Moreover, is a Hilbert-Schmidt operator, and .

Theorem 3.1.

[29, Theorem 1.2] Assume that satisfies (1.5). Then for almost all we have

(3.1)

3.2 Spectral properties of

We will use (3.1) to obtain information about the SSF at the threshold . This requieres to understand the behavior of the spectrum of near , which, in view of (1.2) and (1.4), are intimately connected with the behavior of the band function and the eigenprojection at infinity.

We begin then, with the definition and properties of the limit operator of . Let be the self-adjoint realization in of

The operator has discrete spectrum and its eigenvalues are given by . Denote by , , , the orthogonal projection onto . Then, similarly to (2.1)

where the eigenfunction satisfies

The function could be chosen real-valued. Furthermore, the functions admit the following explicit description. Namely, if we put

(3.2)

where

are the Hermite polynomials (see e.g. [2, Chapter I, Eqs. (8.5), (8.7)]). Then, the real-valued function satisfies

and hence

(3.3)

Define the non-negative operator

where is the zero operator in .

Proposition 3.2.

[7, Propositions 3.4-3.5-3.6] For any , as :

As a consequence of Proposition 3.2 we have:

(3.4)

Denote by , , the Schatten – von Neumann class of compact operators, equipped with the norm

where are the square roots of the eigenvalues of .

For future references, it will be useful to have the following estimate of the difference of the eigenfunctions.

Proposition 3.3.

For any it is satisfied

Proof.

Define , where is the scalar product in . Since , we have that , as . By the continuity of we may assume from the beginning that , as .

Now,

(3.5)

From the definition of , straightforward calculations show that

(3.6)

Where we have used (3.8) below. The statements 2. and 3. of Proposition 3.2 imply that

(3.7)

Then, putting together (3.5), (3.6), (3.7) and the fact that is bounded for , we finish the proof. ∎

Proposition 3.4.

For any , there exists such that

for a constant independent of .

Proof.

Since is analytic, it is sufficient to prove that its derivative is uniformly bounded with respect to large enough. Set to be any number that satisfies . By (1.4) and the strict monotonicity of the bands functions, we can take a contour around such that no other eigenvalues of lie inside the region enclosed by whenever . Then, for , we have

and

To conclude the proof we need to show that and are uniformly bounded with respect to and to . The first condition follows since our choice of implies that . For the second condition, we use the following relations. First, applying the spectral theorem, for any , we have

and second, using self-adjointness properties, we have

and consequently

Proposition 3.5.

[21, Corollary 15.A.6] For any , and

(3.8)

In addition

(3.9)

To finish this part we define as the inverse function of . Evidently, the function is strictly decreasing with and . Moreover, the preceding proposition yields:

(3.10)

3.3 Analysis of on the real axis

The following step in our proof of Theorem 2.1 consists in the analysis of the operator appearing in (3.1). We will decompose this operator according to the band structure of . In Proposition 3.6 and its proof we describe each one of the components of the decomposition.

Proposition 3.6.

Suppose that satisfies (1.5). Then, for all , and all , the operator is in . Furthermore, for any , the limit

exists in the -norm, and is continuous with respect to in the standard operator-norm. Moreover, for it is satisfied

(3.11)
Proof.

Define the operator-valued function , by

(3.12)

For any fixed with Im , let us prove that

(3.13)

First note that the function is locally Lipschitz and then measurable. Moreover,

(3.14)

where and is independent of big enough. Using that is a finite rank operator, these properties follow from

(3.15)

The Lipschitz condition follows from (3.15) and the fact that the projection depends analytically on . Inequality (3.14) follows from Proposition 3.4.

Next, set as the operator valued function given by

(3.16)

The function being defined in (3.3).

Obviously, for any the operators are of rank one, hence their Hilbert-Schmidt and trace norms coincide. Based on this property we will use the estimate

(3.17)

For the first term on the r.h.s. of (3.17), we can see that (1.5) implies