# Heat kernel expansions on the integers

- Authors
- Type
- Published Article
- Publication Date
- Jun 09, 2002
- Submission Date
- Jun 09, 2002
- Identifiers
- DOI: 10.1023/A:1016258207606
- arXiv ID: math/0206089
- Source
- arXiv
- License
- Unknown
- External links

## 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.