Continuity properties of integral kernels associated with Schroedinger operators on manifolds

TL;DR

This paper investigates the continuity properties of integral kernels such as the heat kernel and Green function associated with Schrödinger operators on manifolds of bounded geometry, including those with magnetic fields and singular potentials.

Contribution

It establishes joint continuity of the heat kernel and continuity of the Green function outside the diagonal for these operators, using the Lippmann-Schwinger equation.

Findings

Heat kernel is jointly continuous on manifolds of bounded geometry.

Green function is continuous outside the diagonal.

Results apply to Schrödinger operators with singular potentials and magnetic fields.

Abstract

For Schroedinger operators (including those with magnetic fields) with singular (locally integrable) scalar potentials on manifolds of bounded geometry, we study continuity properties of some related integral kernels: the heat kernel, the Green function, and also kernels of some other functions of the operator. In particular, we show the joint continuity of the heat kernel and the continuity of the Green function outside the diagonal. The proof makes intensive use of the Lippmann-Schwinger equation.