Semiclassical reduction for magnetic Schroedinger operator with periodic zero-range potentials and applications

TL;DR

This paper analyzes the spectral properties of a 2D magnetic Schrödinger operator with periodic zero-range potentials, revealing Cantor spectrum structures in the weak magnetic field regime through semiclassical reduction.

Contribution

It introduces a semiclassical reduction method for the spectral analysis of magnetic Schrödinger operators with periodic zero-range potentials, highlighting Cantor spectrum features.

Findings

Existence of Cantor spectrum parts for specific magnetic flux values

Reduction of the spectral problem to semiclassical analysis of Harper-like operators

Spectral structure influenced by weak magnetic fields

Abstract

The two-dimensional Schroedinger operator with a uniform magnetic field and a periodic zero-range potential is considered. For weak magnetic fields we reduce the spectral problem to the semiclassical analysis of one-dimensional Harper-like operators. This shows the existence of parts of Cantor structure in the spectrum for special values of the magnetic flux.