2021/09/15 by Jussi Behrndt, Behrndt, Jussi, Markus Holzmann +5
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2109.07233
openalex publication_date 2021/09/15 · openalex created_date 2023/04/23 · openalex updated_date 2026/07/28
We consider the self-adjoint Landau Hamiltonian H0 in L2(ℝ2) whose spectrum consists of infinitely degenerate eigenvalues Λq, q ∈ ℤ+, and the perturbed operator Hυ = H0 + υδΓ, where Γ⊂ ℝ2 is a regular Jordan C1,1-curve, and υ ∈ Lp(Γ;ℝ), p>1, has a constant sign. We investigate \rm Ker(Hυ -Λq), q ∈ ℤ+, and show that generically 0 ≤ \rm dim Ker(Hυ -Λq) - \rm dim Ker(Tq(υ δΓ)) lt; ∞, where Tq(υ δΓ) = pq (υ δΓ)pq, is an operator of Berezin-Toeplitz type, acting in pq L2(ℝ2), and pq is the orthogonal projection on \rm Ker (H0 -Λq). If υ ≠ 0 and q = 0, we prove that \rm Ker (T0(υ δΓ)) = \0\. If q ≥ 1, and Γ= Cr is a circle of radius r, we show that \rm dim Ker (Tq(δCr)) ≤ q, and the set of r ∈ (0,∞) for which \rm dim Ker(Tq(δCr)) ≥ 1, is infinite and discrete.