Eigenvalue bounds for two-dimensional magnetic Schroedinger operators

TL;DR

This paper establishes bounds on the number of negative eigenvalues for 2D magnetic Schrödinger operators, highlighting the influence of magnetic fields and potential properties, with implications for Hardy inequalities.

Contribution

It provides new upper bounds on negative eigenvalues for 2D magnetic Schrödinger operators, emphasizing the role of magnetic fields, which was not previously understood.

Findings

Negative eigenvalues are bounded by electric potential strength.

Bounds depend on magnetic field properties.

Connections to Hardy inequalities are discussed.

Abstract

We prove that the number of negative eigenvalues of two-dimensional magnetic Schroedinger operators is bounded from above by the strength of the corresponding electric potential. Such estimates fail in the absence of a magnetic field. We also show how the corresponding upper bounds depend on the properties of the magnetic field and discuss their connection with Hardy-type inequalities.