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Physical Review A

Eigenfunctions and eigenvalues of squeeze operators

Chi‐Fai Lo

Published Dec 1, 1990• 14 citations
Publisher

Abstract

In this paper we have constructed the continuum eigenfunctions and eigenvalue spectra for the squeeze operators of both single mode and two modes. Our analyses also show explicitly that proper eigenstates of both single-mode and two-mode squeeze operators do not exist, which implies that the squeeze operators do not have a discrete spectrum.

DOI
10.1103/physreva.42.6752

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In this paper we have constructed the continuum eigenfunctions and eigenvalue spectra for the squeeze operators of both single mode and two modes. Our analyses also show explicitly that proper eigenstates of both single-mode and two-mode squeeze operators do not exist, which implies that the squeeze operators do not have a discrete spectrum.
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.