The virial theorem and the method of multipliers in spectral theory

TL;DR

This paper connects the virial theorem with multiplier methods in spectral theory, applying these techniques to analyze spectral properties of various quantum Hamiltonians, especially in non-self-adjoint cases.

Contribution

It introduces a novel link between the virial theorem and multiplier methods, advancing spectral analysis of complex quantum operators.

Findings

Established conditions for absence of eigenvalues in electromagnetic Hamiltonians

Extended spectral analysis techniques to non-self-adjoint operators

Provided new insights into spectral properties with matrix-valued potentials

Abstract

We provide a link between the virial theorem in functional analysis and the method of multipliers in theory of partial differential equations. After giving a physical insight into the techniques, we show how to use them to deduce the absence of eigenvalues and other spectral properties of electromagnetic quantum Hamiltonians. We focus on our recent developments in non-self-adjoint settings, namely on Schroedinger operators with matrix-valued potentials, relativistic operators of Pauli and Dirac types, and complex Robin boundary conditions.