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Journal of Physics A Mathematical and General

Coherent states for isospectral oscillator Hamiltonians

David J. Fernández C., Véronique Hussin, Luis Miguel Nieto

Published May 21, 1994• 95 citations
Publisher

Abstract

Coherent states for a family of isospectral oscillator Hamiltonians are derived from a suitable choice of annihilation and creation operators. The Fock-Bargmann representation is also considered.
DOI
10.1088/0305-4470/27/10/029

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Recently we have obtained, on the basis of a group approach to quantization, a Bargmann-Fock-like realization of the Relativistic Harmonic Oscillator as well as a generalized Bargmann transform relating fock wave functions and a set of relativistic Hermite polynomials. Nevertheless, the relativistic creation and annihilation operators satisfy typical relativistic commutation relations of the Lie product (vector-z, vector-z(sup dagger)) approximately equals Energy (an SL(2,R) algebra). Here we find higher-order polarization operators on the SL(2,R) group, providing canonical creation and annihilation operators satisfying the Lie product (vector-a, vector-a(sup dagger)) = identity vector 1, the eigenstates of which are 'true' coherent states.
We present, for the isospectral family of oscillator Hamiltonians, a systematic procedure for constructing raising and lowering operators satisfying any prescribed `distorted' Heisenberg algebra (including the qqq-generalization). This is done by means of an operator transformation implemented by a shift operator. The latter is obtained by solving an appropriate partial isometry condition in the Hilbert space. Formal representations of the non-local operators concerned are given in terms of pseudo-differential operators. Using the new annihilation operators, new classes of coherent states are constructed for isospectral oscillator Hamiltonians. The corresponding Fock-Bargmann representations are also considered, with specific reference to the order of the entire function family in each case.