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Heat kernel expansions on the integers

F. Alberto Grunbaum·Plamen Iliev
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Abstract

In the case of the heat equation $u_t=u_{xx}+Vu$ 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 $L_0$. We show if $L$ denotes the result of applying a finite number of Darboux transformations to $L_0$ then the fundamental solution of $u_t=Lu$ is given by a finite sum of terms involving the Bessel function $I$ of imaginary argument.

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F. Alberto Grunbaum, Plamen Iliev. 2002-06-09. Heat kernel expansions on the integers. https://doi.org/10.1023/a%3A1016258207606

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