Dispersion Estimates for the Discrete Laguerre Operator

We derive an explicit expression for the kernel of the evolution group \({\exp(-\mathrm{i} t H_0)}\) of the discrete Laguerre operator H₀ (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 \({(1+t^2)^{-1/2}}\).