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Title: Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation

Abstract

We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. -- Highlights: •Symmetric quadratic operators are useful models for many physical applications. •Any such operator exhibits a pseudo-Hermitian matrix representation. •Its eigenvalues are the natural frequencies of the Hamiltonian operator. •The eigenvalues may be real or complex and describe a phase transition.

Authors:
Publication Date:
OSTI Identifier:
22560328
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 369; Journal Issue: Complete; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHARGED PARTICLES; COORDINATES; EIGENVALUES; GAIN; HAMILTONIANS; HERMITIAN MATRIX; PHASE TRANSFORMATIONS; RESONATORS; SYMMETRY

Citation Formats

Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar. Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation. United States: N. p., 2016. Web. doi:10.1016/J.AOP.2016.03.002.
Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar. Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation. United States. doi:10.1016/J.AOP.2016.03.002.
Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar. 2016. "Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation". United States. doi:10.1016/J.AOP.2016.03.002.
@article{osti_22560328,
title = {Symmetric quadratic Hamiltonians with pseudo-Hermitian matrix representation},
author = {Fernández, Francisco M., E-mail: fernande@quimica.unlp.edu.ar},
abstractNote = {We prove that any symmetric Hamiltonian that is a quadratic function of the coordinates and momenta has a pseudo-Hermitian adjoint or regular matrix representation. The eigenvalues of the latter matrix are the natural frequencies of the Hamiltonian operator. When all the eigenvalues of the matrix are real, then the spectrum of the symmetric Hamiltonian is real and the operator is Hermitian. As illustrative examples we choose the quadratic Hamiltonians that model a pair of coupled resonators with balanced gain and loss, the electromagnetic self-force on an oscillating charged particle and an active LRC circuit. -- Highlights: •Symmetric quadratic operators are useful models for many physical applications. •Any such operator exhibits a pseudo-Hermitian matrix representation. •Its eigenvalues are the natural frequencies of the Hamiltonian operator. •The eigenvalues may be real or complex and describe a phase transition.},
doi = {10.1016/J.AOP.2016.03.002},
journal = {Annals of Physics},
number = Complete,
volume = 369,
place = {United States},
year = 2016,
month = 6
}
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