Generation of Painlevé V transcendents

Abstract

An algorithm for generating solutions to the Painlevé V equation (the Painlevé V transcendents) is presented. The first step is to look for general one-dimensional Schrödinger Hamiltonians ruled by third degree polynomial Heisenberg algebras, which have fourth order differential ladder operators. It is realized then that there is a key function that must satisfy the Painlevé V equation. Conversely, by identifying systems ruled by a third degree polynomial Heisenberg algebra, in particular their four extremal states, this key function can be built straightforwardly. The simplest Painlevé V transcendents will be generated through this algorithm.