Generalized squeezed-coherent states of the finite one-dimensional oscillator and matrix multi-orthogonality

TL;DR

This paper introduces a new class of generalized squeezed-coherent states for the finite u(2) oscillator, linking them to matrix multi-orthogonal polynomials and exploring their limits towards standard harmonic oscillator states.

Contribution

It develops a novel algebraic framework for finite oscillator states using matrix multi-orthogonal polynomials, extending the understanding of squeezed-coherent states.

Findings

States expressed via matrix elements involving multi-orthogonal polynomials

Polynomials involve Krawtchouk and vector-orthogonal polynomials

States tend to standard harmonic oscillator states as N approaches infinity

Abstract

A set of generalized squeezed-coherent states for the finite u(2) oscillator is obtained. These states are given as linear combinations of the mode eigenstates with amplitudes determined by matrix elements of exponentials in the su(2) generators. These matrix elements are given in the (N+1)-dimensional basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the Krawtchouk and vector-orthogonal polynomials as their building blocks. The algebraic setting allows for the characterization of these polynomials and the computation of mean values in the squeezed-coherent states. In the limit where N goes to infinity and the discrete oscillator approaches the standard harmonic oscillator, the polynomials tend to 2x2 matrix orthogonal polynomials and the squeezed-coherent states tend to those of the standard oscillator.