Thursday, February 16, 2012

1202.3410 (Vincent X. Genest et al.)

Generalized squeezed-coherent states of the finite one-dimensional
oscillator and matrix multi-orthogonality

Vincent X. Genest, Luc Vinet, Alexei Zhedanov
A set of generalized squeezed-coherent states for the finite u(2) oscillator
is obtained. These states are given as linear combinations of the mode
eigenstates with amplitudes determined by matrix elements of exponentials in
the su(2) generators. These matrix elements are given in the (N+1)-dimensional
basis of the finite oscillator eigenstates and are seen to involve 3x3 matrix
multi-orthogonal polynomials Q_n(k) in a discrete variable k which have the
Krawtchouk and vector-orthogonal polynomials as their building blocks. The
algebraic setting allows for the characterization of these polynomials and the
computation of mean values in the squeezed-coherent states. In the limit where
N goes to infinity and the discrete oscillator approaches the standard harmonic
oscillator, the polynomials tend to 2x2 matrix orthogonal polynomials and the
squeezed-coherent states tend to those of the standard oscillator.
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