Darboux-Lamé equation and isomonodromic deformation

Abstract

The Darboux-Lamé equation is defined as the double Darboux transformation of the Lamé equation, and is studied from the viewpoint of the isomonodromic deformation theory. It is shown that the second-order ordinary differential equation of Fuchsian type on ${\mathbf{P}}_1$ corresponding to the second Darboux-Lamé equation is obtained as isomonodromic deformation of some specific Gauss' hypergeometric differential equation.