A heat trace anomaly on polygons

TL;DR

This paper investigates the discontinuity in heat trace coefficients during the smooth approximation of polygonal domains, revealing an anomaly linked to corner formation and modeling it with renormalized invariants.

Contribution

It introduces a precise description of the heat trace anomaly on polygons using renormalized invariants, applicable to both Dirichlet and Neumann boundary conditions.

Findings

Heat trace coefficients are discontinuous during domain smoothing.

The anomaly is modeled via renormalized invariants of an auxiliary domain.

Results extend to both Dirichlet and Neumann boundary conditions.

Abstract

Let $\Omega_0$ be a polygon in $\RR^2$, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that $\Omega_\e$ is a family of surfaces with $\calC^\infty$ boundary which converges to $\Omega_0$ smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer \cite{MS} recognized that certain heat trace coefficients, in particular the coefficient of $t^0$, are not continuous as $\e \searrow 0$. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain $Z$ which models the corner formation. The result applies both for Dirichlet and Neumann conditions. We also include a discussion of what one might expect in higher dimensions.