Absence of positive eigenvalues of magnetic Schrödinger operators

Abstract

We study sufficient conditions for the absence of positive eigenvalues of magnetic Schrödinger operators in ${\mathbb{R}}^d,d\ge 2$. In our main result we prove the absence of eigenvalues above certain threshold energy which depends explicitly on the magnetic and electric field. A comparison with the examples of Miller--Simon shows that our result is sharp as far as the decay of the magnetic field is concerned. As applications, we describe several consequences of the main result for two-dimensional Pauli and Dirac operators, and two and three dimensional Aharonov--Bohm operators.

arXiv commentaccepted for publication in Calculus of Variations & PDE