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

Haine, Luc

Annales de l'Institut Fourier, Tome 55 (2005) no. 6, pp. 1765-1788.

A l’aide de la théorie de Sato, on calcule la matrice spectrale de Stieltjes associée à une matrice de Jacobi doublement infinie, donnant lieu à une solution $g$-soliton du réseau de Toda. On utilise ce résultat pour donner un développement explicite de la solution fondamentale de versions discrètes de l’équation de la chaleur, en termes d’une série des $q$-déformations de Jackson des fonctions de Bessel. Pour les solitons dits de Askey-Wilson, ce développement se réduit à une somme finie.

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.

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