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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Grünbaum, F.A., Iliev, P. Heat Kernel Expansions on the Integers. Mathematical Physics, Analysis and Geometry 5, 183–200 (2002). https://doi.org/10.1023/A:1016258207606
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DOI: https://doi.org/10.1023/A:1016258207606