Trace formulas for the magnetic Laplacian and Dirichlet to Neumann operator -- Explicit expansions --

TL;DR

This paper explores how magnetic effects influence heat trace formulas related to the magnetic Laplacian and Dirichlet-to-Neumann operator, providing explicit examples and new formulas including non-local and logarithmic terms.

Contribution

It introduces explicit expansions and formulas for magnetic heat trace asymptotics, highlighting non-local and logarithmic terms with detailed calculations.

Findings

Derived explicit heat trace formulas for magnetic Laplacian

Calculated non-local and logarithmic terms in Steklov asymptotics

Provided illustrative examples demonstrating magnetic effects

Abstract

Inspired by a recent paper of G. Liu and X. Tan (2023), we would like to measure how the magnetic effect appears in the heat trace formula associated with the magnetic Laplacian and the magnetic Dirichlet-to-Neumann operator. We propose to the reader an overview of magnetic heat trace formulas through explicit examples. On the way we obtain new formulas and in particular we calculate explicitely some non local terms and logarithmic terms appearing in the Steklov heat trace asymptotics.