Bounding the Bogoliubov coefficients
Abstract
While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [M. Visser, Phys. Rev. A 59 (1999) 427-438, (arXiv:quant-ph/9901030)], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2x2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.
- Authors:
-
- School of Mathematics, Statistics, and Computer Science, Victoria University of Wellington, P.O. Box 600, Wellington (New Zealand)
- Publication Date:
- OSTI Identifier:
- 21163678
- Resource Type:
- Journal Article
- Journal Name:
- Annals of Physics (New York)
- Additional Journal Information:
- Journal Volume: 323; Journal Issue: 11; Other Information: DOI: 10.1016/j.aop.2008.02.002; PII: S0003-4916(08)00028-6; Copyright (c) 2008 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0003-4916
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EXACT SOLUTIONS; MATRICES; PARTICLE PRODUCTION; QUANTUM MECHANICS; WAVE EQUATIONS
Citation Formats
Boonserm, Petarpa, and Visser, Matt. Bounding the Bogoliubov coefficients. United States: N. p., 2008.
Web. doi:10.1016/j.aop.2008.02.002.
Boonserm, Petarpa, & Visser, Matt. Bounding the Bogoliubov coefficients. United States. https://doi.org/10.1016/j.aop.2008.02.002
Boonserm, Petarpa, and Visser, Matt. 2008.
"Bounding the Bogoliubov coefficients". United States. https://doi.org/10.1016/j.aop.2008.02.002.
@article{osti_21163678,
title = {Bounding the Bogoliubov coefficients},
author = {Boonserm, Petarpa and Visser, Matt},
abstractNote = {While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [M. Visser, Phys. Rev. A 59 (1999) 427-438, (arXiv:quant-ph/9901030)], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2x2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.},
doi = {10.1016/j.aop.2008.02.002},
url = {https://www.osti.gov/biblio/21163678},
journal = {Annals of Physics (New York)},
issn = {0003-4916},
number = 11,
volume = 323,
place = {United States},
year = {Sat Nov 15 00:00:00 EST 2008},
month = {Sat Nov 15 00:00:00 EST 2008}
}