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

@article{83f0c45a7e674f0894cc9bb7a0e18485,

title = "How to test for diagonalizability: the discretized PT-invariant square-well potential",

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.",

keywords = "PT-symmetry, diagonalizability, discretized square-well potential, MODEL",

author = "S Weigert",

year = "2005",

month = sep,

doi = "10.1007/s10582-005-0126-7",

language = "English",

volume = "55",

pages = "1183--1186",

journal = "Czechoslovak journal of physics",

issn = "0011-4626",

publisher = "Springer",

number = "9",

}

TY - JOUR

T1 - How to test for diagonalizability: the discretized PT-invariant square-well potential

AU - Weigert, S

PY - 2005/9

Y1 - 2005/9

N2 - 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.

AB - 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.

KW - PT-symmetry

KW - diagonalizability

KW - discretized square-well potential

KW - MODEL

U2 - 10.1007/s10582-005-0126-7

DO - 10.1007/s10582-005-0126-7

M3 - Article

SN - 0011-4626

VL - 55

SP - 1183

EP - 1186

JO - Czechoslovak journal of physics

JF - Czechoslovak journal of physics

IS - 9

ER -