  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.
