Spectral Properties of a Magnetic Quantum Hamiltonian on a Strip (original) (raw)
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Mourre estimates for a 2D magnetic quantum Hamiltonian on strip-like domains
Contemporary Mathematics, 2009
We consider a 2D Schrödinger operator H0 with constant magnetic field defined on a strip of finite width. The spectrum of H0 is absolutely continuous and contains a discrete set of thresholds. We perturb H0 by an electric potential V , and establish a Mourre estimate for H = H0 + V when V is periodic in the infinite direction of the strip, or decays in a suitable sense at infinity. In the periodic case, for each compact subinterval I contained in between two consecutive thresholds, we show as a corollary that the spectrum of H remains absolutely continuous in I, provided the period and the size of the perturbation are sufficiently small. In the second case we obtain that the singular continuous spectrum of H is empty, and any compact subset of the complement of the thresholds set contains at most a finite number of eigenvalues of H, each of them having finite multiplicity. Moreover these Mourre estimates together with some of their spectral consequences generalize to the case of 2D magnetic Schrödinger operators defined on R 2 for suitable confining potentials modeling Dirichlet boundary conditions.
Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian
Journal of Functional Analysis, 2018
We consider the Schrödinger operator with constant magnetic field defined on the halfplane with a Dirichlet boundary condition, H 0 , and a decaying electric perturbation V. We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of H 0 , by studying the Spectral Shift Function (SSF) associated to the pair (H 0 + V, H 0). 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 V 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.
On the discrete spectrum of Schroedinger operators with Ahlfors regular potentials in a strip
2019
In this paper, quantitative upper estimates for the number of eigenvalues lying below the essential spectrum of Schroedinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L^1 norms and Orlicz norms of the potential.
On the discrete spectrum of Schrödinger operators with Ahlfors regular potentials in a strip
Journal of Mathematical Analysis and Applications, 2019
In this paper, quantitative upper estimates for the number of eigenvalues lying below the essential spectrum of Schrödinger operators with potentials generated by Ahlfors regular measures in a strip subject to two different types of boundary conditions (Robin and Dirichlet respectively) are presented. The estimates are presented in terms of weighted L 1 norms and Orlicz norms of the potential.
2010
We consider a magnetic Schr\"odinger operator HhH^hHh, depending on the semiclassical parameter h>0h>0h>0, on a two-dimensional Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value b_0b_0b_0 of the magnetic field bbb is strictly positive, and there exists a unique minimum point of bbb, which is non-degenerate. The main result of the paper is a complete asymptotic expansion for the low-lying eigenvalues of the operator HhH^hHh in the semiclassical limit. We also apply these results to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.
On the Singularities of the Magnetic Spectral Shift Function at the Landau Levels
Annales Henri Poincaré, 2004
We consider the three-dimensional Schrödinger operators H 0 and H ± where H 0 = (i∇+A) 2 −b, A is a magnetic potential generating a constant magnetic field of strength b > 0, and H ± = H 0 ± V where V ≥ 0 decays fast enough at infinity. Then, A. Pushnitski's representation of the spectral shift function (SSF) for the pair of operators H ± , H 0 is well defined for energies E = 2qb, q ∈ Z +. We study the behaviour of the associated representative of the equivalence class determined by the SSF, in a neighbourhood of the Landau levels 2qb, q ∈ Z +. Reducing our analysis to the study of the eigenvalue asymptotics for a family of compact operators of Toeplitz type, we establish a relation between the type of the singularities of the SSF at the Landau levels and the decay rate of V at infinity. Résumé. On considère les opérateurs de Schrödinger tridimensionnels H 0 et H ± où H 0 = (i∇ + A) 2 − b, A est un potentiel magnétique engendrant un champ magnétique constant d'intensité b > 0, et H ± = H 0 ±V où V ≥ 0 décroît assez viteà l'infini. Alors, la représentation obtenue par A. Pushnitski de la fonction du décalage spectral pour les opérateurs H ± , H 0 est bien définie pour desénergies E = 2qb, q ∈ Z +. Onétudie le comportement du représentant associé de la classe d'équivalence déterminée par la fonction du décalage spectral, au voisinage des niveaux de Landau 2bq, q ∈ Z +. En réduisant l'analyseà l'investigation de l'asymptotique des valeurs propres d'une famille d'opérateurs de Toeplitz compacts, onétablit une relation entre le type des singularités de la fonction du décalage spectral aux niveaux de Landau et la vitesse de la décroissance de Và l'infini.
Weak Convergence of Spectral Shift Functions for One-Dimensional Schrödinger Operators
2011
We study the manner in which spectral shift functions associated with self-adjoint one-dimensional Schrödinger operators on the finite interval (0,R) converge in the infinite volume limit R→∞ to the half-line spectral shift function. Relying on a Fredholm determinant approach combined with certain measure theoretic facts, we show that prior vague convergence results in the literature in the special case of Dirichlet boundary conditions extend to the notion of weak convergence and arbitrary separated self-adjoint boundary conditions at x=0 and x=R.
Communications in Partial Differential Equations, 2012
We continue our study of a magnetic Schrödinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.
Spectrum of the Iwatsuka Hamiltonian at thresholds
Journal of Mathematical Analysis and Applications, 2018
We consider the bi-dimensional Schrödinger operator with unidirectionally constant magnetic field, H 0 , sometimes known as the "Iwatsuka Hamiltonian". This operator is analytically fibered, with band functions converging to finite limits at infinity. We first obtain the asymptotic behavior of the band functions and its derivatives. Using this results we give estimates on the current and on the localization of states whose energy value is close to a given threshold in the spectrum of H 0. In addition, for non-negative electric perturbations V we study the spectral density of H 0˘V , by considering the Spectral Shift Function associated to the operator pair pH 0V , H 0 q. We compute the asymptotic behavior of the Spectral Shift Function at the thresholds, which are the only points where it can grows to infinity.
Resonances and SSF Singularities for Magnetic Schrödinger Operators
The aim of this note is to review recent articles on the spectral properties of magnetic Schrödinger operators. We consider H0, a 3D Schrödinger operator with constant magnetic field, an H0, a perturbation of H0 by an electric potential which depends only on the variable along the magnetic field. Let H (resp H) be a short range perturbation of H0 (resp. o H0). In the case of (H, H0), we study the local singularities of the Krein spectral shift function (SSF) and the distribution of the resonances of H near the Landau levels which play the role of spectral thresholds. In the case of H H0), we study similar problems near the eigenvalues o H0 of infinite multiplicity. RESUMEN El objetivo de esta nota es reseñar artículos recientes sobre las propiedades espectrales de operadores de Schrödinger magnéticos. Consideramos H0, el operador tridimensional de Schrödinger con campo magnético constante, H0, perturbación de H0 por un potencial eléctrico que depende sólo de la variable a lo largo d...