Dispersion estimates for the discrete Hermite operator

Abstract

In this article, we obatin the l∞ estimate of the kernel an,m(t) for m= 0 , 1 , m= n and t∈ [1 , ∞] for the propagator e-itHd 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 ‖e-itHd‖l1→l∞ and ‖e-itHd‖lσ2→l-σ2 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 Hd. © 2021, The Indian National Science Academy.