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

## TL;DR

This paper introduces new exactly solvable discrete quantum systems with pure imaginary shifts, whose eigenfunctions are described by multi-indexed continuous Hahn and Meixner-Pollaczek orthogonal polynomials, forming a complete basis.

## Contribution

It presents the construction of shape invariant, exactly solvable discrete quantum systems with multi-indexed orthogonal polynomial eigenfunctions, expanding the class of solvable models.

## Key findings

- Eigenfunctions are multi-indexed continuous Hahn and Meixner-Pollaczek polynomials.
- Systems are shape invariant and solvable with pure imaginary shifts.
- Eigenfunctions form a complete orthogonal basis in the weighted Hilbert space.

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