SUSY partners of the truncated oscillator, Painlev\'e transcendents and B\"acklund transformations

TL;DR

This paper applies supersymmetric techniques to the truncated oscillator to generate Hamiltonians linked to Painlevé IV and V equations, providing solutions and Bäcklund transformations for these nonlinear differential equations.

Contribution

It introduces a novel application of supersymmetric methods to connect truncated oscillator Hamiltonians with Painlevé equations and their transformations.

Findings

Generated Hamiltonians with polynomial Heisenberg algebras

Produced particular solutions to Painlevé IV and V equations

Derived Bäcklund transformations relating these equations

Abstract

In this work the supersymmetric technique is applied to the truncated oscillator to generate Hamiltonians ruled by second and third-order polynomial Heisenberg algebras, which are connected to the Painlev\'e IV and Painlev\'e V equations respectively. The aforementioned connection is exploited to produce particular solutions to both non-linear differential equations and the B\"acklund transformations relating them.