Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics
Abstract
The spectra and generalized eigenfunctions of the hyperbolic and parabolic generators of the standard representation of SU(1,1) in the one-mode boson Hilbert space are derived. The eigenfunctions are given in three different forms, corresponding to the coordinate, photon number, and Fock-Bargmann representations of the state vectors. The possible spectra of general second degree Hamiltonians are determined. Some corresponding results in the two-mode case are also given. - In the Appendix we prove completeness and orthonormality relations for the polynomials giving the number representation expansion coefficients of the generalized eigenfunctions of the hyperbolic generator (= squeezing generator). These polynomials are special cases of Pollaczek polynomials.
Cite
@article{arxiv.quant-ph/9711018,
title = {Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics},
author = {Bengt Nagel},
journal= {arXiv preprint arXiv:quant-ph/9711018},
year = {2007}
}
Comments
This is a reproduction, with minor corrections and an added Appendix, of a paper published in J. Bertrand et al.(Eds.), Modern Group Theoretical Methods in Physics, Kluwer Academic Publishers 1995. 13 p