Dispersion Estimates for the Discrete Laguerre Operator

TL;DR

This paper derives an explicit kernel expression for the evolution group of the discrete Laguerre operator using Jacobi polynomials and establishes its dispersive decay rate from 1 to .

Contribution

It provides a new explicit formula for the evolution kernel of the discrete Laguerre operator and determines its dispersive decay rate.

Findings

Kernel expressed in terms of Jacobi polynomials

Dispersive estimate 1 to  decay rate

Explicit  decay norm of the evolution group

Abstract

We derive an explicit expression for the kernel of the evolution group $\exp(-\mathrm{i} t H_0)$ of the discrete Laguerre operator $H_0$ (i.e. the Jacobi operator associated with the Laguerre polynomials) in terms of Jacobi polynomials. Based on this expression we show that the norm of the evolution group acting from $\ell^1$ to $\ell^\infty$ is given by $(1+t^2)^{-1/2}$.