Heat Kernel Expansions on the Integers

Abstract. In the case of the heat equation ut = uxx + V u 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.