Heat kernel expansions on the integers

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.

18 pages

References in corpus (1)

• D-modules and Darboux transformations

Cited by in corpus (6)

• Harmonic Analysis associated with a discrete Laplacian
• Fractional discrete Laplacian versus discretized fractional Laplacian
• Heat kernel expansions on the integers and the Toda lattice hierarchy
• The CMV bispectral problem
• Lattice sums of $I$-Bessel functions, theta functions, linear codes and heat equations
• Higher-order heat equation and the Gelfand-Dickey hierarchy