Heat Kernel Expansions on the Integers

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 L₀. We show if L denotes the result of applying a finite number of Darboux transformations to L₀ then the fundamental solution of u_t=Lu is given by a finite sum of terms involving the Bessel function I of imaginary argument.