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.