On some strong ratio limit theorems for heat kernels

We study strong ratio limit properties of the quotients of the heat kernels of subcritical and critical operators which are defined on a noncompact Riemannian manifold.Comment: 16 pages. This version coincides with the published one, except for Remark 4 added after the paper has appeare

Absence of eigenvalues of two-dimensional magnetic Schroedinger operators

By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schroedinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered

Absence of eigenvalues of two-dimensional magnetic Schr ̈odinger operators

By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schr ̈odinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered

Spectral stability of Schrödinger operators with subordinated complex potentials

We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for Schroedinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities

On the improvement of the Hardy inequality due to singular magnetic fields

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic Hardy inequality for a magnetic field that vanishes at infinity and diverges along a plane

Sharp bounds for eigenvalues of biharmonic operators with complex potentials in low dimensions

We derive sharp quantitative bounds for eigenvalues of biharmonic operators perturbed by complex-valued potentials in dimensions one, two and three