Dispersion estimates for the discrete Hermite operator

In this article, we obatin the $$l^{\infty }$$ l ∞ estimate of the kernel $$a_{n,m}(t)$$ a n , m ( t ) for $$m=0,1$$ m = 0 , 1 , $$m=n$$ m = n and $$t\in [1,\infty ]$$ t ∈ [ 1 , ∞ ] for the propagator $$e^{-itH_d}$$ e - i t H 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 $$\Vert e^{-itH_d}\Vert _{l^1\rightarrow l^{\infty }}$$ ‖ e - i t H d ‖ l 1 → l ∞ and $$\Vert e^{-itH_d}\Vert _{l_{\sigma }^2 \rightarrow l_{-\sigma }^2}$$ ‖ e - i t H d ‖ l σ 2 → l - σ 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 $$H_d$$ H d .