Dispersion estimates for the discrete Hermite operator

Abstract

In this article, we obatin the $l^{\infty }$ estimate of the kernel $a_{n,m}(t)$ for $m=0,1$, $m=n$ and $t\in [1,\infty ]$ for the propagator $e^{-itH_d}$ of one dimensional difference operator associated with the Hermite functions. We conjecture that this estimate holds true for any positive integer m and in that case, we obtain better decay for $\Vert e^{-itH_d}\Vert _{l^1\rightarrow l^{\infty }}$ and $\Vert e^{-itH_d}\Vert _{l_{\sigma }^2 \rightarrow l_{-\sigma }^2}$ for large |t| compare to the Euclidean case, see Egorova (J Spectr Theory 5:663–696, 2015). These estimates are useful in the analysis of one-dimensional discrete Schrödinger equation associated with operator $H_d$.