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arXiv:math-ph/0612071 · Days on Diffraction

Coherent states of Krawtchouk oscillator and beyond

Вадим Васильевич Борзов, Eugeniy Viktorovich Damaskinsky

Published Jan 1, 2006• 0 citations
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Abstract

In the frame of our approach we constructed the generalized oscillator connected with Krawtchouk polynomials (named Krawtchouk oscillator) and coherent states for this oscillator too. Ours results are compared with analogues ones obtained for another variant of Krawtchouk oscillator in the paper [N.M.Atakishiev and S.K.Suslov, Difference analogs of the harmonic oscillator, Teor. Mat. Fiz., 85, 64-73 (1990)] from other point of view. Our definition of coherent states is close to one given in [B.Roy and P.Roy, Phase properties of a new nonlinear coherent state, quant-ph/0002043] This investigation is partially supported by RFBR grant No 06-01-00451

DOI
10.1109/dd.2006.348175

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In the frame of our approach we constructed the generalized oscillator connected with Krawtchouk polynomials (named Krawtchouk oscillator) and coherent states for this oscillator too. Ours results are compared with analogues ones obtained for another variant of Krawtchouk oscillator in the paper [N.M.Atakishiev and S.K.Suslov, Difference analogs of the harmonic oscillator, Teor. Mat. Fiz., 85, 64-73 (1990)] from other point of view. Our definition of coherent states is close to one given in [B.Roy and P.Roy, Phase properties of a new nonlinear coherent state, quant-ph/0002043] This investigation is partially supported by RFBR grant No 06-01-00451
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The expression of Hermite polynomials via Laguerre ones allows to connect the standard representations of the Heisenberg algebra with representations of generalized oscillator algebra, related to the Laguerre polynomials. We give the connection between the measures participating in the orthogonality relations for these polynomials and the connection between energy levels of related oscillators. Such considerations can be extended to a general case in which Hermite polynomials are replaced by polynomials orthogonal with respect to a symmetric measure. This work is partially supported by RFBR grant No 09-01-00504-a.
We define the coherent states for the oscillator-like systems, connected with the Chebyshev polynomials Tn(x)T_n(x)Tn​(x) and Un(x)U_n(x)Un​(x) of the 1-st and 2-nd kind.