JournalsprimsVol. 39 , No. 2DOI 10.2977/prims/1145476105

Eigenvalue Asymptotics for the Schrödinger Operator with Steplike Magnetic Field and Slowly Decreasing Electric Potential

  • Shin-ichi Shirai

    Ritsumeikan University, Shiga, Japan
Eigenvalue Asymptotics for the Schrödinger Operator with Steplike Magnetic Field and Slowly Decreasing Electric Potential cover

Abstract

In this paper we consider the two-dimensional Schrödinger operator of the form:

HV = −∂2+ (1b(_x_1))2 + V(_x_1, _x_2),
∂_x_12i∂_x_2

where the magnetic field B(_x_1) = rot(0, b(_x_1)) is monotone increasing and steplike, namely the limits lim _x_1 →±∞ B(_x_1) = _B_± exist with 0 < _B_− < B+ < ∞, and V is the slowly power-decaying electric potential. The spectrum σ(_H_0) of the unperturbed operator H_0 (= HV with V = 0) has the band structure and HV has the discrete spectrum in the gaps of the essential spectrum σ_ess (HV) = σ(_H_0). The aim of this paper is to study the asymptotic distribution of the eigenvalues near the edges of the spectral gaps. Using the min-max argument, we prove that the classical Weyl-type asymptotic formula is satisfied under suitable assumptions on B and V.