Estimation of the number of negative eigenvalues of magnetic Schr\"odinger operators in a strip

TL;DR

This paper provides an upper bound on the count of negative eigenvalues for magnetic Schrödinger operators with Aharonov-Bohm fields in a strip, highlighting the field's critical role in spectral properties.

Contribution

The paper introduces a specific upper estimate for negative eigenvalues in a magnetic Schrödinger operator with Aharonov-Bohm magnetic field, demonstrating its failure without the field.

Findings

Upper estimate established for negative eigenvalues with Aharonov-Bohm field

Estimate does not hold without the magnetic field

Highlights the importance of magnetic field in spectral estimates

Abstract

An upper estimate for the number of negative eigenvalues below the essential spectrum for the magnetic Schr\"odinger operator with Aharonov-Bohm magnetic field in a strip is obtained. Its further shown that the estimate does not hold in absence of Aharonov-Bohm magnetic field.