Polynomial Heisenberg algebras and Painlev\'e equations

TL;DR

This paper explores polynomial Heisenberg algebras linked to Painlevé equations, deriving solutions and coherent states for quantum systems using SUSY QM, and establishing reduction theorems for algebraic structures.

Contribution

It introduces new reduction theorems for polynomial Heisenberg algebras and constructs solutions to Painlevé IV and V equations via SUSY quantum mechanics.

Findings

Derived solutions to Painlevé IV and V equations using hypergeometric functions.

Established reduction theorems simplifying higher-order PHA to second or third order.

Constructed Painlevé IV coherent states and generalized displacement operators.

Abstract

We study first the supersymmetric quantum mechanics (SUSY QM), specially the cases of the harmonic and radial oscillators. Then, we obtain a new Wronskian formula for the confluent SUSY transformation and apply the SUSY QM to the inverted oscillator. After that, we present the polynomial Heisenberg algebras (PHA). We study the general systems described by PHA: for zeroth- and first-order we obtain the harmonic and radial oscillators, respectively; for second- and third-order PHA, the potential is determined in terms of solutions to Painlev\'e IV and V equations ($P_{IV}$ and $P_{V}$), respectively. Later on, we review the six Painlev\'e equations and we study the cases of $P_{IV}$ and $P_V$. We prove a reduction theorem for $2k$th-order PHA to be reduced to second-order algebras. We also prove an analogous theorem for the $(2k+1)$th-order PHA to be reduced to third-order ones. Through these theorems we find solutions to $P_{IV}$ and $P_V$ given in terms of confluent hypergeometric functions. For some special cases, those can be classified in several solution hierarchies. In this way, we find real solutions with real parameters and complex solutions with real and complex parameters for both equations. Finally, we study the coherent states (CS) for the SUSY partners of the harmonic oscillator that are connected with $P_{IV}$, which we will call Painlev\'e IV coherent states. Since these systems have third-order ladder operators $l_k^\pm$, we seek first the CS as eigenstates of the annihilation operator $l_k^-$. We also define operators analogous to the displacement operator and we get CS departing from the extremal states in each subspace in which the Hilbert space is decomposed. We conclude our treatment applying a linearization process to the ladder operators in order to define a new displacement operator to obtain CS involving the entire Hilbert space.