如何检验可对角化：离散PT不变方阱势
How to Test for Diagonalizability: The Discretized PT-Invariant Square-Well Potential
摘要：
给定一个非厄米矩阵M，它的最小多项式的结构编码了M是否可对角化。本文将解释如何确定矩阵的最小多项式，而不需要通过其特征多项式。该方法应用于在分段恒定PT对称势影响下在方势阱中运动的量子力学粒子。在对位形空间进行离散化后，系统由三维矩阵描述。它原来是不可对角化的临界强度的相互作用，也表明了两个真正的过渡到一对复杂的能量本征值。该系统开发了一个三重退化的本征值，和两个三个本征函数消失在这个特殊的点，给出了代数和几何之间的差异的本征值的重数等于2。
英文摘要：
Given a non-hermitean matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note will explain how to determine the minimal polynomial of a matrix without going through its characteristic polynomial. The approach is applied to a quantum mechanical particle moving in a square well under the influence of a piece-wise constant PT-symmetric potential. Upon discretizing the configuration space, the system is decribed by a matrix of dimension three. It turns out not to be diagonalizable for a critical strength of the interaction, also indicated by the transition of two real into a pair of complex energy eigenvalues. The systems develops a three-fold degenerate eigenvalue, and two of the three eigenfunctions disappear at this exceptional point, giving a difference between the algebraic and geometric multiplicity of the eigenvalue equal to two.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
