2011/01/05 by Pablo Miranda, Miranda, Pablo, Georgi Raikov +1
Mathematics · Physics and Astronomy · #35J10 #35P20 #47F05 #81Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #math.SP #msc:35J10 #msc:35P20 #msc:47F05 #msc:81Q10
paper · pdf · doi:10.48550/arxiv.1101.1079
Lemma 2.1 added, the proof of Theorem 3.1 streamlined, typos corrected. 21 pages
openalex publication_date 2011/01/05 · arxiv created 2011/05/27 · arxiv updated 2011/05/31 · openalex created_date 2022/10/07 · openalex updated_date 2026/07/28
We consider the unperturbed operator H0: = (-i ∇ - \bf A)2 + W, self-adjoint in L2(\mathbb R2). Here \bf A is a magnetic potential which generates a constant magnetic field b>0, and the edge potential W = W is a \mathcal T-periodic non constant bounded function depending only on the first coordinate x ∈ \mathbb R of (x,y) ∈ \mathbb R2. Then the spectrum σ(H0) of H0 has a band structure, the band functions are b \mathcal T-periodic, and generically there are infinitely many open gaps in σ(H0). We establish explicit sufficient conditions which guarantee that a given band of σ(H0) has a positive length, and all the extremal points of the corresponding band function are non degenerate. Under these assumptions we consider the perturbed operators H± = H0 ± V where the electric potential V ∈ L∞(\mathbb R2) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H_± in the spectral gaps of H0. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schroedinger operator with infinite-matrix-valued potential. Further, we restrict our attention on perturbations V of compact support. We find that there are infinitely many discrete eigenvalues in any open gap in the spectrum of σ(H0), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.