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}}).