The spectral matrices of toda solitons and the fundamental solution of some discrete heat equations

摘要

The Stieltjes spectral matrix measure of the doubly infinite Jacobi matrix associated with a Toda $g$-soliton is computed, using Sato theory. The result is used to give an explicit expansion of the fundamental solution of some discrete heat equations, in a series of Jackson's $q$-Bessel functions. For Askey-Wilson type solitons, this expansion reduces to a finite sum.