A special type of Jacobi matrices, discrete Schrödinger operators, is found to play an important role in quantum physics. In this paper, we show that given the spectrum of a discrete Schrödinger operator and the spectrum of the operator obtained by deleting the first row and the first column of it can determine the discrete Schrödinger operator uniquely, even though one eigenvalue of the latter is missing. Moreover, we find the forms of the discrete Schrödinger operators when their smallest and largest eigenvalues attain the extrema under certain constraints by use of the notion of generalized directional derivative and the method of Lagrange multiplier.

发现一种特殊的Jacobi矩阵，即离散Schrödinger运算符，在量子物理学中起着重要作用。在本文中，我们表明，给定离散薛定ding算子的频谱以及通过删除第一行和第一列而获得的算子的频谱可以唯一地确定离散薛定ding算子，即使后者的一个特征值缺失。此外，我们利用广义方向导数的概念和拉格朗日乘数法，找到了离散薛定ding算子的形式，当它们的最小和最大特征值在一定约束下达到极值时。