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 \(H_d\).

在本文中，我们对\(m=0,1\) , \(m=n )的核\(a_{n,m}(t)\)的\(l^{\infty }\)估计\)和\(t\in [1,\infty ]\)用于与 Hermite 函数关联的一维差分算子的传播子\(e^{-itH_d}\)。我们推测这个估计对于任何正整数m 都成立，在这种情况下，我们获得了更好的衰减\(\Vert e^{-itH_d}\Vert _{l^1\rightarrow l^{\infty }}\)和\(\Vert e^{-itH_d}\Vert _{l_{\sigma }^2 \rightarrow l_{-\sigma }^2}\)对于大 | 吨| 与欧几里得情况相比，请参阅 Egorova（J Spectr Theory 5:663–696, 2015）。这些估计在分析与算子\(H_d\)相关的一维离散薛定谔方程时很有用。