Quasi-Classical Versus Non-Classical Spectral Asymptotics for Magnetic Schrödinger Operators With Decreasing Electric Potentials

Abstract

We consider the Schrödinger operator, with constant magnetic field and electric potential V which typically decays at infinity exponentially fast or has a compact support. We investigate the asymptotic behaviour of the discrete spectrum of H(V ) near the boundary points of its essential spectrum. If the decay of V is Gaussian or faster, this behaviour is non-classical in the sense that it is not described by the quasiclassical formulas known for the case where V admits a power-like decay.

Finite Rank Bergman–Toeplitz and Bargmann–Toeplitz Operators in Many Dimensions

“…The distribution of perturbed eigenvalues in clusters is essentially governed by the spectrum of Toeplitz-Bargmann operators with weight function V expressed in an explicit way in the terms of the perturbation. Many results in this direction have been obtained in [7], [6], [9], [8] and other publications. In particular, simple operator-theoretical arguments,(see e.g.…”

“…In particular, simple operator-theoretical arguments,(see e.g. [7,Proposition 4.1]) show that the Landau level is, in fact, the accumulation point of a cluster if and only if T B V has infinite rank. So, if T B V has finite rank, the Landau level remains, even after the perturbation, being an isolated eigenvalue of infinite multiplicity.…”

Spectral Asymptotics of Pauli Operators and Orthogonal Polynomials in Complex Domains

“…We refer the reader to the discussion in [17]. The case of a constant magnetic field and compactly supported potentials v was considered in [17] and [12].…”

“…We refer the reader to the discussion in [17]. The case of a constant magnetic field and compactly supported potentials v was considered in [17] and [12]. The case of a two-dimensional operator with variable magnetic field and potentials v with power or exponential decay and also with compactly supported potentials was treated in [16].…”

“…The case of a two-dimensional operator with variable magnetic field and potentials v with power or exponential decay and also with compactly supported potentials was treated in [16]. The results of [17,12,16] for the case of compactly supported potentials read as…”