Wiener-Type Theorems for the Laplace Transform. With Applications to Ground State Problems

TL;DR

This paper investigates the behavior of probability measures via Laplace transforms, connecting it to quantum ground state problems and providing criteria for their existence or non-existence.

Contribution

It introduces Wiener-type theorems for the Laplace transform and applies these results to quantum ground state analysis, linking probability theory with quantum physics.

Findings

Derived criteria for ground state existence and non-existence.

Connected Laplace transform behavior to rank-one perturbation and renewal theory.

Provided new insights into probability measures near support boundaries.

Abstract

We study the behavior of a probability measure near the bottom of its support in terms of time averaged quotients of its Laplace transform. We discuss how our results are connected to both rank-one perturbation theory as well as renewal theory. We further apply our results in order to derive criteria for the existence and non-existence of ground states for a finite dimensional quantum system coupled to a bosonic field.