标题： 解析可解的哈密顿量用于具有最近邻相互作用的量子系统 显示英文标题

标题： Analytically solvable Hamiltonians for quantum systems with a nearest neighbour interaction

摘要： 我们研究由线性链上 n 个耦合谐振子组成的量子系统，相互作用形式为$-q_r q_{r+1}$（$q_r$表示第$r$个振子的位置）。 原则上，这类系统总是可以用数值方法求解，并涉及相互作用矩阵的特征值。 本文探讨了此类系统何时可以解析求解，即当相互作用矩阵的特征值和特征向量具有解析闭合表达式时。 这种情况发生在相互作用矩阵与离散正交多项式系统的雅可比矩阵一致时。 我们的系统研究导致了三个新的解析可解哈密顿量：具有 Krawtchouk 相互作用、Hahn 相互作用或 q-Krawtchouk 相互作用。 对于每种情况，我们给出了哈密顿量的谱（以解析形式）并讨论了谱的一些典型性质。 显示英文摘要

摘要： We consider quantum systems consisting of a linear chain of n harmonic oscillators coupled by a nearest neighbour interaction of the form $-q_r q_{r+1}$ ($q_r$ refers to the position of the $r$th oscillator). In principle, such systems are always numerically solvable and involve the eigenvalues of the interaction matrix. In this paper, we investigate when such a system is analytically solvable, i.e. when the eigenvalues and eigenvectors of the interaction matrix have analytically closed expressions. This is the case when the interaction matrix coincides with the Jacobi matrix of a system of discrete orthogonal polynomials. Our study of possible systems leads to three new analytically solvable Hamiltonians: with a Krawtchouk interaction, a Hahn interaction or a q-Krawtchouk interaction. For each of these cases, we give the spectrum of the Hamiltonian (in analytic form) and discuss some typical properties of the spectra.