On-diagonal singularities of the Green functions for Schroedinger operators

TL;DR

This paper analyzes the on-diagonal singularities of Green functions for Schrödinger operators in two and three dimensions, revealing their dependence on potentials and perturbations, with implications for spectral analysis.

Contribution

It characterizes the behavior of Green function singularities near the diagonal in low dimensions and identifies conditions under which these singularities are preserved or affected by potentials.

Findings

In 2D, singularities are independent of potentials.

In 3D, certain conditions preserve the singularity under perturbations.

Examples show how potentials influence the singularity behavior.

Abstract

We investigate the behavior of the Green functions of Schroedinger operators near the diagonal. The only non-trivial cases, where the on-diagonal singularities are non-zero and do not depend on the spectral parameter, are two and three dimensions. In the case of two dimensions, we show that the singularity is independent of both the scalar and the gauge potentials. In dimension three, we obtain conditions for preserving the singularity under perturbations by non-regular potentials. Some examples illustrating dependence of the singularity on general scalar and gauge potentials are presented.