An asymptotic expansion of the trace of the heat kernel of a singular two-particle contact interaction in one dimension

TL;DR

This paper derives a detailed small-time asymptotic expansion of the heat kernel trace for a one-dimensional Schrödinger operator with a singular two-particle contact interaction, using resolvent analysis and Laplace transforms.

Contribution

It introduces a novel method to obtain the heat kernel trace asymptotics without standard parametrix constructions, focusing on resolvent expansion and inverse Laplace techniques.

Findings

Complete small-time asymptotic expansion in fractional powers of t

Avoids standard parametrix methods for heat kernel analysis

Provides explicit connection between resolvent expansion and heat kernel asymptotics

Abstract

The regularized trace of the heat kernel of a one-dimensional Schr\"odinger operator with a singular two-particle contact interaction being of Lieb-Liniger type is considered. We derive a complete small-time asymptotic expansion in (fractional) powers of the time, $t$. Most importantly, we do not invoke standard parametrix constructions for the heat kernel. Instead, we first derive the large-energy expansion of the regularized trace of the resolvent for the considered operator. Then, we exploit that the resolvent may be obtained by a Laplace transformation of the heat semi-group, and an application of a suitable inverse Watson lemma eventually yields the small-$t$ asymptotic expansion of the heat-kernel trace.