A heat trace anomaly on polygons

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.Comment: Revision includes treatment of the Neumann problem and a discussion of the higher dimensional case; some new reference

A Szegő limit theorem for translation‐invariant operators on polygons

“…In addition, the coefficient c 0 for the polygon P cannot be obtained from via approximation of P by domains with smooth boundary. This anomaly resembles the analogous result for the constant order term in the heat trace asymptotics for the Dirichlet Laplacian on a two‐dimensional domain with corners, see .…”

On the corner contributions to the heat coefficients of geodesic polygons

“…and [13]); moreover, c 1 (γ) and the coefficient at K(p) 2 in (1.3) coincide, of course, with Uçar's corresponding formulas for constant curvature. The main novelty here is the coefficient at ∆ g K(p) in (1.3) which, of course, did not appear in the constant curvature case.…”

“…(i) Formula (5.4) for c 0 (γ) seems well-known, even for general γ (not only those of the form γ = π/k) and without any symmetry assumptions; see, e.g., the discussion in [13]. Of course, in the case of euclidean polygons this is obvious from the classical formula (1.1) found by D. Ray and proved by van den Berg and Srisatkunarajah [1].…”

One can hear the corners of a drum

“…The reason is that if there are corners, then these each contribute an extra, purely local term, to the coefficient a 0 . This has been called a "corner defect" or an "anomaly" [8] due to the fact that the coefficient a 0 is not continuous under Lipschitz convergence of domains.…”