[Paper Review] An infinite family of superintegrable systems with the fifth Painleve transcendent from higher order ladder operators and supersymmetry

Ian Marquette|arXiv (Cornell University)|Aug 18, 2010

Quantum Mechanics and Non-Hermitian Physics1 references1 citations

TL;DR

This paper introduces a new infinite family of quantum superintegrable systems with fourth-order ladder operators derived via second-order supersymmetric quantum mechanics, leading to systems governed by the fifth Painlevé transcendent. The key contribution is the construction of a polynomial Heisenberg algebra and explicit separation of variables in Cartesian coordinates, demonstrating integrability beyond second-order systems.

ABSTRACT

We will discuss how we can obtain new quantum superintegrable Hamiltonians allowing the separation of variables in Cartesian coordinates with higher order integrals of motion from ladder operators. We will discuss also how higher order supersymmetric quantum mechanics can be used to obtain systems with higher order ladder operators and their polynomial Heisenberg algebra. We will present a new family of superintegrable systems involving the fifth Painleve transcendent which possess fourth order ladder operators constructed from second order supersymmetric quantum mechanics. We present the polynomial algebra of this family of superintegrable systems.

Motivation & Objective

• To construct new quantum superintegrable Hamiltonians with higher-order integrals of motion using ladder operators.
• To extend higher-order supersymmetric quantum mechanics to generate systems with fourth-order ladder operators.
• To explore the role of the fifth Painlevé transcendent in integrable systems with separable Cartesian coordinates.
• To establish the polynomial Heisenberg algebra structure for the new family of superintegrable systems.
• To demonstrate the existence of an infinite family of such systems with explicit algebraic and spectral properties.

Proposed method

• Utilize second-order supersymmetric quantum mechanics to generate higher-order ladder operators from seed potentials.
• Construct Hamiltonians that allow separation of variables in Cartesian coordinates using these ladder operators.
• Derive the polynomial algebra associated with the ladder operators, showing closure under commutation.
• Employ the fifth Painlevé transcendent as a key component in the potential structure of the resulting superintegrable systems.
• Analyze the algebraic structure of the integrals of motion to confirm superintegrability and higher-order symmetry.
• Verify the existence of an infinite family of such systems through systematic construction from the supersymmetric framework.

Experimental results

Research questions

• RQ1 Can higher-order ladder operators be systematically generated from second-order supersymmetric quantum mechanics to yield new superintegrable systems?
• RQ2 How does the fifth Painlevé transcendent emerge in the potential structure of these systems?
• RQ3 What algebraic structure—specifically, what polynomial Heisenberg algebra—arises from the fourth-order ladder operators in this context?
• RQ4 Can the resulting Hamiltonians be separated in Cartesian coordinates despite the presence of higher-order integrals of motion?
• RQ5 What is the role of the infinite family parameter in generating distinct yet structurally similar superintegrable systems?

Key findings

• An infinite family of superintegrable systems is constructed with fourth-order ladder operators derived from second-order supersymmetric quantum mechanics.
• The systems exhibit separation of variables in Cartesian coordinates, confirming their integrability beyond second-order systems.
• The fifth Painlevé transcendent appears explicitly in the potential functions of the Hamiltonians, linking them to special functions in mathematical physics.
• The polynomial algebra of the system is shown to close under commutation, forming a non-linear algebraic structure generalizing the Heisenberg algebra.
• The ladder operators generate a consistent algebraic framework that supports the existence of discrete, non-degenerate energy spectra.
• The construction provides a systematic method to generate new superintegrable models with higher-order symmetries and special function potentials.