Orthogonal polynomials, Laguerre Fock space and quasi-classical asymptotics

TL;DR

This paper explores a novel Berezin-Toeplitz quantization scheme based on Laguerre polynomials, describing a Laguerre Fock space and its semi-classical asymptotics, extending to Legendre polynomials.

Contribution

It introduces a Laguerre analogue of the Fock space and analyzes its semi-classical asymptotics, connecting to Barut-Girardello coherent states and extending to Legendre polynomials.

Findings

Laguerre Fock space coincides with the space for Barut-Girardello coherent states

Semi-classical asymptotics of Toeplitz operators are described

Extension to Legendre polynomials is discussed

Abstract

Continuing our earlier investigation of the Hermite case [J. Math. Phys. 55 (2014), 042102], we study an unorthodox variant of the Berezin-Toeplitz quantization scheme associated with Laguerre polynomials. In particular, we describe a "Laguerre analogue" of the classical Fock (Segal-Bargmann) space and the relevant semi-classical asymptotics of its Toeplitz operators; the former actually turns out to coincide with the Hilbert space appearing in the construction of the well-known Barut-Girardello coherent states. Further extension to the case of Legendre polynomials is likewise discussed.