Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain

Pushnitski, Alexander; Rozenblum, Grigori

doi:10.4171/dm/235

Cited by 24 publications

(2 citation statements)

References 18 publications

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“…In particular, it was shown that if either 0 or 0 on , 6 0, and certain additional regularity assumptions hold, then in a neighborhood of any ƒ q there are infinitely many discrete eigenvalues of H and their accumulation rate to the Landau levels is described in terms of the logarithmic capacity of the interaction support; cf. [10,20,35,37] for similar results on the clustering of eigenvalues of Landau Hamiltonians on unbounded domains with Dirichlet, Neumann, and Robin boundary conditions.…”

Section: Introductionmentioning

confidence: 76%

“…Singular Toeplitz operators as in (2.11) play an important role in modern operator theory and are also of independent interest. They were already considered in [2], and in connection with magnetic Laplacians with different types of boundary conditions these types of operators appear in [20,21,37]; we also refer the reader to [3,10,11,13,16,32,36,41,42] for some other recent related works in this context. Note that the operator T q .ı / corresponds to the quadratic form t q .ı /OEu WD Z .x/ju.x/j 2 ds; u 2 p q L 2 .R 2 /:…”

Section: Denote Bymentioning

confidence: 99%

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The fate of Landau levels under $\delta$-interactions

et al. 2023

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We consider the self-adjoint Landau Hamiltonian H 0 in L 2 .R 2 / whose spectrum consists of infinitely degenerate eigenvalues ƒ q , q 2 Z C , and the perturbed Landau Hamiltonian H D H 0 C ı , where R 2 is a regular Jordan C 1;1 -curve and 2 L p .I R/, p > 1, has a constant sign. We investigate ker.H ƒ q /, q 2 Z C , and show that generically 0 dim ker.H ƒ q / dim ker.T q .ı // < 1;where T q .ı / D p q .ı /p q , is an operator of Berezin-Toeplitz type, acting in p q L 2 .R 2 /, and p q is the orthogonal projection onto ker.H 0 ƒ q /. If ¤ 0 and q D 0, then we prove that ker.T 0 .ı // D ¹0º. If q 1 and D C r is a circle of radius r, then we show that dim ker.T q .ı Cr // q, and the set of r 2 .0; 1/ for which dim ker.T q .ı Cr // 1 is infinite and discrete.

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Section: Introductionmentioning

confidence: 76%

Section: Denote Bymentioning

confidence: 99%

The fate of Landau levels under $\delta$-interactions

et al. 2023

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Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

Miranda

2015

Ann. Henri Poincaré

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We consider the discrete spectrum of the two-dimensional Hamiltonian H = H0 + V , where H0 is a Schrödinger operator with a non-constant magnetic field B that depends only on one of the spatial variables, and V is an electric potential that decays at infinity. We study the accumulation rate of the eigenvalues of H in the gaps of its essential spectrum. First, under certain general conditions on B and V , we introduce effective Hamiltonians that govern the main asymptotic term of the eigenvalue counting function. Further, we use the effective Hamiltonians to find the asymptotic behavior of the eigenvalues in the case where the potential V is a power-like decaying function and in the case where it is a compactly supported function, showing a semiclassical behavior of the eigenvalues in the first case and a non-semiclassical behavior in the second one. We also provide a criterion for the finiteness of the number of eigenvalues in the gaps of the essential spectrum of H.

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Counting Function of Magnetic Resonances for Exterior Problems

Bruneau

Sambou

2016

Ann. Henri Poincaré

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We study the asymptotic distribution of the resonances near the Landau levels Λq = (2q + 1)b, q ∈ N, of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of R 3 of the 3D Schrödinger Schrödinger operator with constant magnetic field of scalar intensity b > 0. We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels.

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Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain

Abstract: We consider the Schrödinger operator with a constant magnetic field in the exterior of a compact domain on the plane. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We discuss the rate of accumulation of eigenvalues in a fixed cluster.

Cited by 24 publications

References 18 publications

The fate of Landau levels under $\delta$-interactions

The fate of Landau levels under $\delta$-interactions

Eigenvalue Asymptotics for a Schrödinger Operator with Non-Constant Magnetic Field Along One Direction

Counting Function of Magnetic Resonances for Exterior Problems

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