Isospectral polygons, planar graphs and heat content |
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Authors: | Patrick McDonald Robert Meyers |
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Institution: | Department of Mathematics, New College of Florida, Sarasota, Florida 34243 ; The Courant Institute of Mathematical Sciences, New York, New York 10276-0907 |
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Abstract: | Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. Using the natural heat operators on the weighted graphs, we associate to each graph a heat content. We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs. As a corollary, we prove that the sequence of exit time moments for the natural Markov chains associated to each graph, averaged over starting points in the interior of the graph, provides a collection of invariants that distinguish isospectral pairs in general. |
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Keywords: | Isospectral pairs Dirichlet spectrum heat content |
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