Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators. (English) Zbl 1267.81174
Summary: The type III Hermite \(X_m\) exceptional orthogonal polynomial family is generalized to a double-indexed one \(X_{m_{1},m_{2}}\) (with \(m_1\) even and \(m_2\) odd such that \(m_2 > m_1\)) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order \(m_2 - m_1 + 1\), which may alternatively be interpreted in terms of a special type of (\(m_2 - m_1 + 2\))th-order shape invariance property.
MSC:
81Q60 | Supersymmetry and quantum mechanics |
81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
81R15 | Operator algebra methods applied to problems in quantum theory |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |