Spectral Invariants of the Magnetic Dirichlet-to-Neumann Map on Riemannian Manifolds

TL;DR

This paper derives explicit formulas for spectral invariants associated with the magnetic Steklov eigenvalue problem on Riemannian manifolds, linking eigenvalues to geometric and physical properties of the boundary.

Contribution

It introduces a method to compute all coefficients of the heat trace asymptotic expansion for the magnetic Steklov problem, including explicit formulas for the first four coefficients.

Findings

Explicit expressions for the first four heat trace coefficients.

Coefficients encode boundary volume and curvature information.

Spectral invariants relate eigenvalues to physical quantities.

Abstract

This paper is devoted to investigate the heat trace asymptotic expansion corresponding to the magnetic Steklov eigenvalue problem on Riemannian manifolds with boundary. We establish an effective procedure, by which we can calculate all the coefficients $a_0$, $a_1$, $\dots$, $a_{n-1}$ of the heat trace asymptotic expansion. In particular, we explicitly give the expressions for the first four coefficients. These coefficients are spectral invariants which provide precise information concerning the volume and curvatures of the boundary of the manifold and some physical quantities by the magnetic Steklov eigenvalues.