Bispectral Operators of Rank 1 and Dual Isomonodromic Deformations

Abstract

A comparison is made between bispectral operator pairs and dual pairs of isomonodromic deformation equations. Through examples, it is shown how operators belonging to rank one bispectral algebras may be viewed equivalently as defining 1-parameter families of rational first order differential operators with matricial coefficients on the Riemann sphere, whose monodromy is trivial. By interchanging the r\^oles of the two variables entering in the bispectral pair, a second 1-parameter family of operators with trivial monodromy is obtained, which may be viewed as the dual isomonodromic deformation system.