Threshold Singularities of the Spectral Shift Function for Geometric Perturbations of Magnetic Hamiltonians

Bruneau, Vincent; Raikov, Georgi

doi:10.1007/s00023-020-00904-6

Cited by 3 publications

(2 citation statements)

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“…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%

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: Denote Bymentioning

confidence: 99%

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

et al. 2023

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“…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 (υδ Γ ) corresponds to the quadratic form (2.12)…”

Section: Denote Bymentioning

confidence: 99%

The fate of Landau levels under $δ$-interactions

Behrndt¹,

Holzmann²,

Lotoreichik³

et al. 2021

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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 Λ q , q ∈ Z + , and the perturbed operator H υ = H 0 + υδ Γ , where Γ ⊂ R 2 is a regular Jordan C 1,1 -curve and υ ∈ L p (Γ; R), p > 1, has a constant sign. We investigate ker(H υ − Λ q ), q ∈ Z + , and show that genericallywhere T q (υδ Γ ) = p q (υδ Γ )p q , is an operator of Berezin-Toeplitz type, acting in p q L 2 (R 2 ), and p q is the orthogonal projection on ker (H 0 − Λ q ). If υ = 0 and q = 0, we prove that ker (T 0 (υδ Γ )) = {0}. If q ≥ 1 and Γ = C r is a circle of radius r, we show that dim ker(T q (δ Cr )) ≤ q, and the set of r ∈ (0, ∞) for which dim ker(T q (δ Cr )) ≥ 1 is infinite and discrete.

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