Heat kernel expansions on the integers

Details Heat kernel expansions on the integers Heat kernel expansions on the integers

2002-06-09

arxiv.org/abs/math/0206089v1

Math. Phys. Anal. Geom. 5 (2002), no.2, 183-200

Mathematics

Classical Analysis and ODEs

18 pages

Scientific paper

10.1023/A:1016258207606

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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