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

### Shin-ichi Shirai

Ritsumeikan University, Shiga, Japan

## Abstract

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

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

∂_x_12 | i | ∂_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*.

## Cite this article

Shin-ichi Shirai, Eigenvalue Asymptotics for the Schrödinger Operator with Steplike Magnetic Field and Slowly Decreasing Electric Potential. Publ. Res. Inst. Math. Sci. 39 (2003), no. 2, pp. 297–330

DOI 10.2977/PRIMS/1145476105