How to test for diagonalizability: the discretized PT-invariant square-well potential

S Weigert

doi:10.1007/s10582-005-0126-7

How to test for diagonalizability: the discretized PT-invariant square-well potential

S Weigert

Mathematics

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1183-1186
Number of pages	4
Journal	Czechoslovak journal of physics
Volume	55
Issue number	9
DOIs	https://doi.org/10.1007/s10582-005-0126-7
Publication status	Published - Sept 2005

Keywords

PT-symmetry
diagonalizability
discretized square-well potential
MODEL

Access to Document

10.1007/s10582-005-0126-7

http://arxiv.org/abs/quant-ph/0507202

Cite this

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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.",

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KW - PT-symmetry

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KW - discretized square-well potential

KW - MODEL

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