Affiliations: Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Santiago de Chile, Chile. E-mail: pmirandar@ug.uchile.cl | Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago de Chile, Chile. E-mail: graikov@mat.puc.cl
Note: [] Corresponding author: Georgi Raikov, Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile, Chile. E-mail: graikov@mat.puc.cl.
Abstract: We consider the unperturbed operator H0:=(−i∇−A)2+W, self-adjoint in L2(R2). Here A is a magnetic potential which generates a constant magnetic field b>0, and the edge potential W=W¯ is a 𝒯-periodic non-constant bounded function depending only on the first coordinate x∈R of (x,y)∈R2. Then the spectrum σ(H0) of H0 has a band structure, the band functions are b𝒯-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∞(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 Schrödinger 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 σ(H0), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.
Keywords: magnetic Schrödinger operators, spectral gaps, eigenvalue distribution
