Limiting absorption principle for the magnetic Dirichlet Laplacian in a half-plane.
We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dispersion curves, that are real analytic functions. Each of them is simple and monotonically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator both outside and at the thresholds. Finally, we establish analytic and decay properties for functions lying in the absorption spaces. We point out that the analysis carried out in this paper is rather general and can be adapted to a wide class of fibered magnetic Laplacians with thresholds in their spectrum that are finite limits of their band functions..
AMS 2000 Mathematics Subject Classification: 35J10, 81Q10,
Keywords: Two-dimensional Schrödinger operators, constant magnetic field, limit absorption, thresholds.
In the present article we consider the Hamiltonian with magnetic potential , defined in the half-plane . We impose Dirichlet boundary conditions at and introduce the self-adjoint realization
initially defined in and then closed in . This operator models the planar motion of a quantum charged particle (the various physical constants are taken equal to ) constrained to and submitted to an orthogonal magnetic field of strength , it has already been studied in several articles (e.g., [8, 15, 6, 16, 20]).
The Schrödinger operator is translationally invariant in the -direction and admits a fiber decomposition with fiber operators which have purely discrete spectrum. The corresponding dispersion curves (also named band functions in this text) are real analytic functions in , monotonically decreasing from positive infinity to the -th Landau level for . As a consequence, the spectrum of is absolutely continuous, equals the interval . Hence the resolvent operator depends analytically on in , and is well defined for every .
Since has a continuous spectrum, the spectral projector of , associated with the interval , , expresses as
by the spectral theorem. Suitable functions of the operator may therefore be expressed in terms of the limits of the resolvent operators for . As a matter of fact the Schrödinger propagator associated with reads
This motivates for a quantitative version of the convergence , for , known as the limiting absorption principle (abbreviated to LAP in the sequel). Notice moreover that a LAP is a useful tool for the analysis of the scattering properties of , and more specifically for the proof of the existence and the completeness of the wave operators (see e.g. [25, Chap. XI]). The main purpose of this article is to establish a LAP for . That is, for each , we aim to prove that has a limit as , in a suitable sense we shall make precise further.
There is actually a wide mathematical literature on LAP available for various operators of mathematical physics (see e.g. [29, 10, 19, 1, 28, 27, 3, 9]). More specifically, the case of analytically fibered self-adjoint operators was addressed in e.g. [7, 13, 26]. Such an operator is unitarily equivalent to the multiplier by a family of real analytic dispersion curves, so its spectrum is the closure of the range of its band functions. Generically, energies associated with a “flat” of any of the band functions , are thresholds in the spectrum of . More precisely, a threshold of the operator is any real number satisfying for some and all neighborhoods of in . We call the set of thresholds.
The occurrence of a LAP outside the thresholds of analytically fibered operators is a rather standard result. It is tied to the existence of a Mourre inequality at the prescribed energies (see [13, 12]), arising from the non-zero velocity of the dispersion curves for the corresponding frequencies. More precisely, given an arbitrary compact subset , we shall extend to a Hölder continuous function on in the norm-topology of for any . Here and henceforth the Hilbert space
is endowed with the scalar product .
Evidently, local extrema of the dispersion curves are thresholds in the spectrum of fibered operators. Any such energy being a critical point of some band function, it is referred as an attained threshold. Actually, numerous operators of mathematical physics modeling the propagation of acoustic, elastic or electromagnetic waves in stratified media [7, 4, 5, 26] and various magnetic Hamiltonians [11, 15, 30] have all their thresholds among local minima of their band functions. A LAP at an attained threshold may be obtained upon imposing suitable vanishing condition (depending on the level of degeneracy of the critical point) on the Fourier transform of the functions in , at the corresponding frequency. See [4, 26] for the analysis of this problem in the general case.
Nevertheless, none of the above mentioned papers seems relevant for our operator . This comes from the unusual behavior of the band functions of at infinity: In the framework examined in this paper, there exists a countable set of thresholds in the spectrum of , but in contrast with the situations examined in [4, 26], none of these thresholds are attained. This peculiar behavior raises several technical problems in the derivation of a LAP for at , . Nevertheless, for any arbitrary compact subset (which may contain one or several thresholds for ) we shall establish in Theorem 2.5 a LAP for in for the topology of the norm in , where is a suitable subspace of , for an appropriate , which is dense in . The space is made of -functions, with smooth Fourier coefficients vanishing suitably at the thresholds of lying in . Otherwise stated there is an actual LAP at , , even though is a non attained threshold of . Moreover, it turns out that the method developed in the derivation of a LAP for is quite general and may be generalized to a wide class of fibered operators (such as the ones examined in [15, 30, 6, 17, 23]) with non attained thresholds in their spectrum.
Finally, functions in exhibit interesting geometrical properties. Namely, assuming that , it turns out that the asymptotic behavior of the -th band function of at positive infinity (computed in [16, Theorem 1.4]) translates into super-exponential decay in the -variable (orthogonal to the edge) of their -th harmonic, see Theorem 3.5. Such a behavior is typical of magnetic Laplacians, as explained in Remark 3.7.
1.1. Spectral decomposition associated with the model
Let us now collect some useful information on the fiber decomposition of the operator .
The Schrödinger operator is translationally invariant in the longitudinal direction and therefore allows a direct integral decomposition
where denotes the partial Fourier transform with respect to and the fiber operator acts in with a Dirichlet boundary condition at . Since the effective potential is unbounded as goes to infinity, each , , has a compact resolvent, hence a purely discrete spectrum. We note the non-decreasing sequence of the eigenvalues of , each of them being simple. Furthermore, for introduce a family of eigenfunctions of the operator , which satisfy
and form an orthonormal basis in .
As is a Kato analytic family, the functions , , are analytic (see e.g. [24, Theorem XII.12]). Moreover they are monotonically decreasing in according to [8, Lemma 2.1 (ii)] and the Max-Min principle yields (see [8, Lemma 2.1 (iii) and (v)])
Therefore, the general theory of fibered operators (see e.g. [24, Section XIII.16]) implies that the spectrum of is purely absolutely continuous, with
For all we define the Fourier coefficient of by
and introduce its harmonic as
In view of (1), we have the standard Fourier decomposition in and the following Parseval identity
involving that the linear mapping is continuous from into for each . Let us now recall the following useful properties of the restriction of to for (see e.g. [7, Proposition 3.2]).
Fix . Then the operators are uniformly bounded with respect to from into :
Moreover for any , each operator , , is bounded from into , the set of locally Hölder continuous functions in , of exponent . Namely, there exists a function such that,
2. Limiting absorption principle
For all and , standard functional calculus yields
Since for each , the function is analytic on so is well defined. In light of (6), it suffices that each , with , be suitably extended to some locally Hölder continuous function in , to derive a LAP for the operator .
2.1. Singular Cauchy integrals
Let be fixed. Bearing in mind that is an analytic diffeomorphism from onto , we note the function inverse to and put for any function . Then, upon performing the change of variable in the integral appearing in (6), we get for every that
Therefore, the Cauchy integral is singular for . Our main tool for extending singular Cauchy integrals of this type to locally Hölder continuous functions in is the Plemelj-Privalov Theorem (see e.g. [22, Chap. 2, §22]), stated below.
Let and let , where is an open bounded subinterval of . Then the mapping , defined in , satisfies for every :
Moreover the function
is analytic in and locally Hölder continuous of order in in the sense that there exists such that
In addition, if , then extends to a locally Hölder continuous function of order in .
2.2. Limiting absorption principle outside the thresholds
In this subsection we establish a LAP for outside its thresholds . This is a rather standard result that we state here for the convenience of the reader. For the sake of completeness we also recall its proof, which requires several ingredients that are useful in the derivation of the main result of subsection 2.3.
Let be a compact subset of . Then for all and any , the resolvent extends to in a Hölder continuous function of order , still denoted by , for the topology of the norm in . Namely there exists a constant , such that the estimate
holds for all and all .
Thus it suffices to examine each , for , separately. Using that is a compact subset of we pick an open bounded subinterval , with , such that . With reference to (7) we have the following decomposition for each ,
Since , and are both Lipschitz continuous in , and . Therefore we deduce from Lemma 1.1 that and verifies
for some constant which is independent of and . From this and Lemma 2.1 then follows that extends to a locally Hölder continuous function of order in , satisfying
where the positive constant depends neither on nor on .
Next, as the Euclidean distance between and is positive, from the very definition of , we get that , since is compact. Therefore is Lipschitz continuous in and satisfies
according to (3).
holds uniformly in and . The result follows from this and the fact that and the space of continuous linear forms on are isometric, with the duality pairing
2.3. Limiting absorption principle at the thresholds
We now examine the case of a compact subset containing one or several thresholds of , i.e. such that . Since then the bounded set contains at most a finite number of thresholds. For the sake of clarity, we first investigate the case where contains exactly one threshold:
The target here is the same as in subsection 2.2, that is to establish a LAP for in . Actually, for all and any , it is clear from the proof of Proposition 2.2 that can be regarded as a -Hölder continuous function in with values in .
Thus we are left with the task of suitably extending in . But, as may actually blow up as tends to , obviously the method used in the proof of Proposition 2.2 does not apply to when lies in the vicinity of . This is due to the vanishing of the denominator of in (7) as approaches , or, equivalently, to the flattening of when goes to . We shall compensate this peculiar asymptotic behavior of the dispersion curve by imposing appropriate conditions on the functions and so the numerator decays sufficiently fast at . This require that the following useful functional spaces be preliminarily introduced.
Suitable functional spaces.
For any open subset and the non vanishing function
on , we denote by 111Since is not defined at then it is understood in the peculiar case where that if and only if extends continuously to a function lying in .(resp. ), the -weighted space of Hölder continuous functions of order (resp., square integrable functions) in . Endowed with the norm (resp., ), (resp., ) is a Banach space since this is the case for (resp., ).
Further, the above definitions translate through the linear isometry from into , to
which is evidently a Banach space for the norm . As a consequence the set
equipped with its natural norm is a Banach space as well.
On we define the linear form . Notice from the embedding , that is well defined since extends to a continuous function in . Furthermore, we have for any so the linear form is continuous on . Let us now introduce the subspace
where, as usual, stands for . Since is continuous then is closed in . Therefore it is a Banach space for the norm
Moreover, being dense in , we deduce from the imbedding that is dense in for the usual norm-topology.
Summing up, we have obtained the:
The set is a Banach space and is dense in .
Absorption at .
Having defined for fixed, we now derive a LAP at for the restriction of the operator to associated with suitable values of and .
Let be a compact subset of obeying (13) and let . Then, for every and , both limits exist in the uniform operator topology on . Moreover the resolvent extends to a Hölder continuous on with order ; Namely there exists such that we have
for some constant that depends only on , and .
We turn now to examining . Taking into account that is bounded we pick so large that and refer once more to the proof of Proposition 2.2. We get that extends to a Hölder continuous function of exponent in , with
where is a constant depending only on , and and .
Finally, since and are taken in then the function , defined in (7), is -Hölder continuous in , and we have
Here we used the basic identity and the continuity of embedding . This entails the desired result. ∎
For any compact subset , the set is finite. Then upon substituting for in the proof of Proposition 2.4, it is apparent that we obtain the:
Let be compact, and let and be the same as in Proposition 2.4. Then the resolvent