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

Understand

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 $\{\ell_{\mathcal{D}},\ell_{\mathcal{D}}+1,\ell_{\mathcal{D}}+2,\ldots\}$, 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.