Discrete bispectral Darboux transformations from Jacobi operators

F. Alberto Grünbaum and Milen Yakimov

Vol. 204 (2002), No. 2, 395–431

DOI: 10.2140/pjm.2002.204.395

Abstract

We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n)T⁻¹ where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We conjecture that along with operators previously constructed by Grünbaum, Haine, Horozov and Iliev they exhaust all bispectral regular (i.e., a(n)≠0,c(n)≠0,∀n ∈ ℤ) operators of the form above.

Milestones

Received: 18 October 2000

Revised: 3 January 2001

Published: 1 June 2002