Heat Kernel Expansions on the Integers |
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Authors: | Grünbaum F. Alberto Iliev Plamen |
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Affiliation: | (1) Department of Mathematics, University of California, Berkeley, CA, 94720-3840, U.S.A.;(2) Department of Mathematics, University of California, Berkeley, CA, 94720-3840, U.S.A. |
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Abstract: | In the case of the heat equation ut=uxx+Vu on the real line, there are some remarkable potentials V for which the asymptotic expansion of the fundamental solution becomes a finite sum and gives an exact formula.We show that a similar phenomenon holds when one replaces the real line by the integers. In this case the second derivative is replaced by the second difference operator L0. We show if L denotes the result of applying a finite number of Darboux transformations to L0 then the fundamental solution of ut=Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument. |
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Keywords: | heat kernel expansions Toda lattice hierarchy Darboux transformations |
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