How to test for diagonalizability: the discretized PT-invariant square-well potential |
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Authors: | Stefan Weigert |
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Institution: | (1) Department of Mathematics, University of York, Heslington, York, YO105DD, UK |
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Abstract: | Given a non-Hermitian matrix M, the structure of its minimal polynomial encodes whether M is diagonalizable or not. This note explains 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 described by a matrix of dimension three which
turns out not to be diagonalizable for a critical strength of the interaction. 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.
Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22,
2005. |
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Keywords: | PT-symmetry diagonalizability discretized square-well potential |
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