Exactly Solvable Discrete Quantum Mechanical Systems and Multi-indexed Orthogonal Polynomials of the Continuous Hahn and Meixner-Pollaczek Types

Abstract

We present new exactly solvable systems of the discrete quantum mechanics with pure imaginary shifts, whose physical range of the coordinate is the whole real line. These systems are shape invariant and their eigenfunctions are described by the multi-indexed continuous Hahn and Meixner-Pollaczek orthogonal polynomials. The set of degrees of these multi-indexed polynomials are $\{{\ell }_{\mathcal{D}},{\ell }_{\mathcal{D}}+1,{\ell }_{\mathcal{D}}+2,\dots \}$, where ${\ell }_{\mathcal{D}}$ is an even positive integer ($\mathcal{D}$ : a multi-index set), but they form a complete set of orthogonal basis in the weighted Hilbert space.

27 pages. Comments added, typo corrected. To appear in PTEP

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Cited by in corpus (6)

• Type II Multi-indexed Little $q$-Jacobi and Little $q$-Laguerre Polynomials
• Wronskian/Casoratian Identities and their Application to Quantum Mechanical Systems
• New Finite Type Multi-Indexed Orthogonal Polynomials Obtained From State-Adding Darboux Transformations
• Some Difference Relations for Orthogonal Polynomials of a Continuous Variable in the Askey Scheme
• Discrete Orthogonality Relations for the Multi-Indexed Orthogonal Polynomials in Discrete Quantum Mechanics with Pure Imaginary Shifts
• Recurrence Relations of the Multi-Indexed Orthogonal Polynomials VI : Meixner-Pollaczek and continuous Hahn types