A heat trace anomaly on polygons

Journal article, 2015

Let Ω0 be a polygon in $\mathbb{R}$2, or more generally a compact surface with piecewise smooth boundary and corners. Suppose that Ωε is a family of surfaces with ${\mathcal C}$∞ boundary which converges to Ω0 smoothly away from the corners, and in a precise way at the vertices to be described in the paper. Fedosov [6], Kac [8] and McKean–Singer [13] recognised that certain heat trace coefficients, in particular the coefficient of t0, are not continuous as ε ↘ 0. We describe this anomaly using renormalized heat invariants of an auxiliary smooth domain Z which models the corner formation. The result applies to both Dirichlet and Neumann boundary conditions. We also include a discussion of what one might expect in higher dimensions.