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arXiv:math-ph/0001001 · Journal of Physics A Mathematical and General

Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers

J Katriel, M. Kibler

Published May 7, 1992• 95 citations
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Abstract

The normal ordering formulae for powers of the boson number operator n are extended to deformed bosons. It is found that for the 'M-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=1, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the 'P-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=q -n are found to depend on the operator n. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.

DOI
10.1088/0305-4470/25/9/036

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The normal ordering formulae for powers of the boson number operator n are extended to deformed bosons. It is found that for the 'M-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=1, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the 'P-type' deformed bosons, which satisfy aa Dagger -qa Dagger a=q -n are found to depend on the operator n. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.
The problem of normal ordering of the following boson operator functions (a+a + ) m , (a r +a + ) m , (a+N) m , (a 2 +N) m , where N=a + a, is solved explicitly. Simple algebraic methods are used. As an intermediate result the ordering formulae for the symmetrised products of boson operators are presented. There is agreement with earlier results of other authors and generalisation of these results. A new representation of the generalised Stirling numbers used for the ordering of the binomial (a 2 +N) m is given and several principal properties of these numbers are derived.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobinski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
We resolve the ladder operators normal ordering problem for strings in the form for ( q , p )-deformed supersymmetric and shape-invariant potential systems, where n is a positive integer. We provide exact and explicit expressions for their normal form , where in all are at the right-hand side, and find that the solution involves expansion-coefficients sequence which corresponds to the generalization, for any ( q , p )-deformed shape-invariant potential system, of the classical Stirling and Bell numbers. We show that these numbers are given for expressions depending on parameters related to the forms of supersymmetric partner potentials and the system quantum deformation. We apply the general formalism to Pöschl–Teller, Morse and Scarf potential systems.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobihski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.
We resolve the normal ordering problem for symmetric (Dˆ+Dˆ−)n and asymmetric (Dˆ+rDˆ−)n strings of the nonlinear deformed ladder operators Dˆ± for supersymmetric and shape-invariant potential systems, where r and n are positive integers. We provide exact and explicit expressions for their normal forms N{(Dˆ+Dˆ−)n} and N{(Dˆ+rDˆ−)n}, where in N{...} all Dˆ− are at the right side. We find that the solutions involve sequence of expansion coefficients which, for r, n > 1, corresponds to the f-deformed generalization of the classical Stirling and Bell numbers of the second kind. We apply the general formalism for the translational shape-invariant potential systems as well as for the particular case of the harmonic oscillator potential system. We show that these numbers are obtained for families of polynomial expressions generated with the deformations parameters and the parameters related to the forms of the supersymmetric partner potentials.