Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics

Details Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics Spectra and generalized eigenfunctions of the one- and two-mode squeezing operators in quantum optics

1997-11-14

arxiv.org/abs/quant-ph/9711018v1

Physics

Quantum Physics

This is a reproduction, with minor corrections and an added Appendix, of a paper published in J. Bertrand et al.(Eds.), Modern

Scientific paper

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.

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