Kovalevskaya’s Dynamics and Schrödinger Equations of Heun Class

Abstract

In previous publications of the author [6 ], [7 ], [8 ] the Painleve equations have been derived from linear second-order equations related to the Heun class. This has been done twofold. One approach is based on a phenomenological transformation from a quantum hamiltonian, related to Heun’s class equation, to a classical hamiltonian related to Painleve equations. The other approach is based on isomonodromie conditions formulated for auxiliary linear equations where an additional apparent singularity is added. The latter is an expansion of the idea first proposed by R. Fuchs [1] 1. However, these studies are based on specific forms of linear equations and therefore are not invariant. Here, firstly, the links of the proposed approach to classical dynamics and studies of S. Kovalevskaya are traced; hence, the isomonodromic conditions play more the role of a conservation law. Secondly, the derivations are presented in invariant terms.