Continuity properties of the semi-group and its integral kernel in non-relativistic QED

TL;DR

This paper investigates the continuity properties of the semi-group and its integral kernel in non-relativistic quantum electrodynamics, providing new results on their dependence on parameters and implications for spectral subspaces.

Contribution

It establishes the continuity of the semi-group and its kernel in non-relativistic QED models without smallness assumptions, extending previous results to more general potentials and models.

Findings

Proves continuity of the semi-group between weighted L^p-spaces.

Shows pointwise continuity of the semi-group kernel.

Derives exponential decay and continuity results for spectral subspace elements.

Abstract

Employing recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics by B. G\"uneysu, J.S. M{\o}ller, and the present author, we study the continuity of the corresponding semi-group between weighted vector-valued L^p-spaces, continuity properties of elements in the range of the semi-group, and the pointwise continuity of an operator-valued semi-group kernel. We further discuss the continuous dependence of the semi-group and its integral kernel on model parameters. All these results are obtained for Kato decomposable electrostatic potentials and the actual assumptions on the model are general enough to cover the Nelson model as well. As a corollary we obtain some new pointwise exponential decay and continuity results on elements of low-energetic spectral subspaces of atoms or molecules that also take spin into account. In a simpler situation where spin is neglected we explain how to verify the joint continuity of positive ground state eigenvectors with respect to spatial coordinates and model parameters. There are no smallness assumptions imposed on any model parameter.