Some remarks on the complex heat kernel in the scalar potential case

TL;DR

This paper discusses the complex heat kernel in scalar potentials, focusing on deformation formulas, Borel summability, and unique solutions in unusual cases, emphasizing the role of complex analysis in heat equation solutions.

Contribution

It provides remarks on the complex heat kernel, explores deformation formulas in complex settings, and establishes uniqueness in atypical cases, extending previous work on Borel summability.

Findings

Deformation formula yields solutions in complex heat kernel cases

Complex setting is crucial for understanding heat kernel behavior

Uniqueness results for solutions in unusual scenarios

Abstract

In previous works, we used a so-called deformation formula in order to study, in particular, the Borel summability of the heat kernel of some operators. A goal of this paper is to collect miscellaneous remarks related to these works. Here the complex setting plays an important role. Moreover, the deformation formula provides a solution of the heat equation in "unusual" cases. We also give a uniqueness statement concerning these cases.