Solutions to the Painlevé V equation through supersymmetric quantum mechanics

Abstract

In this paper we shall use the algebraic method known as supersymmetric quantum mechanics (SUSY QM) to obtain solutions to the Painlevé V (PV) equation, a second-order non-linear ordinary differential equation. For this purpose, we will apply first the SUSY QM treatment to the radial oscillator. In addition, we will revisit the polynomial Heisenberg algebras (PHAs) and we will study the general systems ruled by them: for first-order PHAs we obtain the radial oscillator, while for third-order PHAs the potential will be determined by solutions to the PV equation. This connection allows us to introduce a simple technique for generating solutions of the PV equation expressed in terms of confluent hypergeometric functions. Finally, we will classify them into several solution hierarchies.

39 pages, 18 figures, 4 tables, 70 references

References in corpus (9)

• Rational extensions of the quantum harmonic oscillator and exceptional Hermite polynomials
• Exceptional orthogonal polynomials and the Darboux transformation
• Superintegrability with third order integrals of motion, cubic algebras and supersymmetric quantum mechanics I:Rational function potentials
• Non-hermitian Hamiltonians and Painlevé IV equation with real parameters
• Complex SUSY Transformations and the Painlevé IV Equation
• Rational extension and Jacobi-type {\boldmath{$X_m$}} solutions of a quantum nonlinear oscillator
• Ladder operators for solvable potentials connected with exceptional orthogonal polynomials
• Polynomial Heisenberg algebras and Painlevé equations
• Beyond conventional factorization: Non-Hermitian Hamiltonians with radial oscillator spectrum

Cited by in corpus (11)

• Trends in supersymmetric quantum mechanics
• Fourth order superintegrable systems separating in Polar Coordinates. I. Exotic Potentials
• Fourth-order superintegrable systems separating in Polar Coordinates. II. Standard Potentials
• SUSY partners of the truncated oscillator, Painlevé transcendents and Bäcklund transformations
• Coherent states for the supersymmetric partners of the truncated oscillator
• Polynomial Heisenberg algebras, multiphoton coherent states and geometric phases
• Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
• Fourth Painlevé and Ermakov equations: quantum invariants and new exactly-solvable time-dependent Hamiltonians
• Perturbation-based Non-perturbative Method
• On the general family of third-order shape-invariant Hamiltonians related to generalized Hermite polynomials
• Quantization of Liénard's nonlinear harmonic oscillator and its solutions in the framework of supersymmetric quantum mechanics