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Progress of Theoretical Physics

Localized and Non-Localized Solutions of q-Deformed Oscillators

Kozo Koizumi, Ikuo S. Sogami

Published Apr 1, 2007• 0 citations
Publisher

Abstract

Eigenfunctions of the Fock and non-Fock irreducible representations are explicitly derived for the q-deformed oscillators of the Macfarlane and Dubna types. While the Fock representations are composed of localized functions, the non-Fock representations consist of nonlocal oscillating functions which are constructed for the first time in this paper. The concept of the q-Hermite functions is generalized to include both the Fock and non-Fock types in a unified manner.

DOI
10.1143/ptp.117.589

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Eigenfunctions of the Fock and non-Fock irreducible representations are explicitly derived for the q-deformed oscillators of the Macfarlane and Dubna types. While the Fock representations are composed of localized functions, the non-Fock representations consist of nonlocal oscillating functions which are constructed for the first time in this paper. The concept of the q-Hermite functions is generalized to include both the Fock and non-Fock types in a unified manner.
A wide class of q-deformed harmonic oscillators including those of Macfarlane type and of Dubna type is shown to be describable in a unified way. The Hamiltonian of the oscillator is assumed to be given by a q-deformed anti-commutator of the q-deformed ladder operators. By solving qqq-difference equations, explicit coordinate representations of ladder operators and wave functions are derived, and unified parametric representations are found for qqq-Hermite functions and related formulas for the oscillators of Macfarlane and Dubna types. In addition to the well-known solutions with globally periodic structure, there exist an infinite number of solutions with globally aperiodic structure.
Hamilton functions of classical deformed oscillators (c-deformed oscillators) are derived from Hamiltonians of q-deformed oscillators of the Macfarlane and Dubna types. A new scale parameter, lq, with the dimension of length, is introduced to relate a dimensionless parameter characterizing the deformation with the natural length of the harmonic oscillator. Contraction from q-deformed oscillators to c-deformed oscillators is accomplished by keeping lq finite while taking the limit bar→0. The c-deformed Hamilton functions for both types of oscillators are found to be invariant under discrete translations: the step of the translation for the Dubna oscillator is half of that for the Macfarlane oscillator. The c-deformed oscillator of the Macfarlane type has propagating solutions in addition to localized ones. Reinvestigation of the q-deformed oscillator carried out in the light of these findings for the c-deformed systems proves that the q-deformed systems are invariant under the same translation symmetries as the c-deformed systems and have propagating waves of the Bloch type.
Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described.
The aim of this paper is to review our results on description of the multi-parameter deformed oscillators and their oscillator algebras. We define generalized (q;α,β,γ;ν)-deformed oscillator algebra and study its irreducible representations. The Arik-Coon oscillator with the main relation aa^+ - q a^+a = 1, where q > 1 is embedded in this framework. We find connection of this oscillator with the Askey (1/q)-Hermite polynomials. We construct family of the generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of appropriate classical Stielties moment problem we prove the (over)completeness relation of these states.
We consider Fock representations of the Q-deformed commutation relations ∂s∂t†=Q(s,t)∂t†∂s+δ(s,t) for s,t∈T. Here T:=Rd (or more generally T is a locally compact Polish space), the function Q:T2→C satisfies |Q(s,t)|≤1 and Q(s,t)=Q(t,s)¯, and ∫T2h(s)g(t)δ(s,t) σ(ds)σ(dt):=∫Th(t)g(t) σ(dt), σ being a fixed reference measure on T. In the case, where |Q(s,t)|≡1, the Q-deformed commutation relations describe a generalized statistics studied by Liguori and Mintchev. These generalized statistics contain anyon statistics as a special case (with T=R2 and a special choice of the function Q). The related Q-deformed Fock space F(H) over H:=L2(T→C,σ) is constructed. An explicit form of the orthogonal projection of H⊗n onto the n-particle space Fn(H) is derived. A scalar product in Fn(H) is given by an operator Pn≥0 in H⊗n which is strictly positive on Fn(H). We realize the smeared operators ∂t† and ∂t as creation and annihilation operators in F(H), respectively. Additional Q-commutation relations are obtained between the creation operators and between the annihilation operators. They are of the form ∂s†∂t†=Q(t,s)∂t†∂s†, ∂s∂t=Q(t,s)∂t∂s, valid for those s,t∈T for which |Q(s, t)| = 1.