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arXiv:quant-ph/0111151 · International Symposium on Quantum Theory and Symmetries (QTS)

COHERENT STATES FROM COMBINATORIAL SEQUENCES

K. A. Penson, A. I. Solomon

Published Jun 1, 2002• 30 citations
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Abstract

We construct coherent states using sequences of combinatorial numbers such as various binomial and trinomial numbers, and Bell and Catalan numbers. We show that these states satisfy the condition of the resolution of unity in a natural way. In each case the positive weight functions are given as solutions of associated Stieltjes or Hausdorff moment problems, where the moments are the combinatorial numbers.

DOI
10.1142/9789812777850_0066

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Abstract. The canonical coherent states can be written as infinite series in powers of a single complex number z and a positive integer ρ(m). The requirement that these states realize a resolution of the identity typically ends up in a moment problem, where the moments form the positive sequence of real numbers {ρ(m)} ∞ m=0. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number z and the moments ρ(m) of the canonical coherent states by n × n matrices. A matrix moment problem is proposed and it is solved by introducing a matrix weight. Associated oscillator algebras are discussed. Finally, as a physical application, a generalized version of the Jaynes-Cummings model is discussed. 1.