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A heat trace anomaly on polygons

Published online by Cambridge University Press:  19 June 2015

RAFE MAZZEO
Affiliation:
Department of Mathematics, Stanford University, Building 380, Stanford, CA 94305, U.S.A. e-mail: mazzeo@math.standord.edu
JULIE ROWLETT
Affiliation:
Mathematical Sciences, Chalmers University of Technology and the University of Gothenburg, SE-412 96 Göteborg, Sweden e-mail: julie.rowlett@chalmers.se

Abstract

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.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2015 

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References

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