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arXiv:math-ph/0508032 · Symmetry Integrability and Geometry Methods and Applications

Spectra of Observables in the q-Oscillator and q-Analogue of the Fourier Transform

Anatoliy U. Klimyk

Published Oct 21, 2005• 5 citations
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Abstract

Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + -qa + a = 1) are studied when q > 1.These operators are symmetric but not self-adjoint.They have a one-parameter family of selfadjoint extensions.These extensions are derived explicitly.Their spectra and eigenfunctions are given.Spectra of different extensions do not intersect.The results show that the creation and annihilation operators a + and a of the q-oscillator for q > 1 cannot determine a physical system without further more precise definition.In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.

DOI
10.3842/sigma.2005.008

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Spectra of the position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation aa + -qa + a = 1) are studied when q > 1.These operators are symmetric but not self-adjoint.They have a one-parameter family of selfadjoint extensions.These extensions are derived explicitly.Their spectra and eigenfunctions are given.Spectra of different extensions do not intersect.The results show that the creation and annihilation operators a + and a of the q-oscillator for q > 1 cannot determine a physical system without further more precise definition.In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators.
We elaborate on the Macfarlane–Biedenharn q-oscillator when q>1. In this case the position operator Q = a†+a and the momentum operator P = i(a†-a) are symmetric, but not self-adjoint. For this reason, one cannot specify spectra of Q and P. Since these operators have one-parameter families of self-adjoint extensions with different spectra, the common definition of such q-oscillator is not complete. We derive an action of Q and P (as well as of the related Hamiltonian) upon functions given on the corresponding coordinate spaces, on which Q and P are self-adjoint operators. To each self-adjoint extension of Q there corresponds an appropriate coordinate space (a spectrum of this self-adjoint extension). Thus, for every fixed q>1 one obtains a one-parameter family of non-equivalent q-oscillators in their coordinate spaces.
The aim of this paper is to study the position and momentum operators in q-deformed oscillator algebras. The natural form of the position operator is Xp=qpN(a++a)qpN, where p is a real number. This operator is an operator representable by a Jacobi matrix. Using the theory of Jacobi matrices, the theory of classical moment problem and the theory of basic hypergeometric functions, it is shown that, depending on values of q and p, Xp can be unbounded symmetric operator [which has the deficiency indices (1,1) and, hence, is not self-adjoint, but has self-adjoint extensions], bounded self-adjoint operator with continuous simple spectrum or self-adjoint operator of trace class (therefore, with discrete spectrum with zero as the point of accumulation of eigenvalues). The connection of the q-deformed Heisenberg relation PX−qXP=1 for the position and momentum operators with a q-deformation of the quantum harmonic oscillator is also considered.
The author gives formulae for point spectra of Hermitian operators built from creation and annihilation type operators satisfying general algebraic relation which he calls (g, f, F) equivalence. By choosing the parameters in a specific way the description of the quantum q-oscillator is given in a model built from the q-derivative and dilatation operator. He shows the explicit form of eigenfunctions of the model from L 2 (R). Then he constructs the transformation between his model and the Q-oscillator with arbitrary Q real.
The finite q -oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q -algebra su q (2). The spectrum of position in this discrete system, in a fixed representation j , consists of 2 j + 1 'sensor'-points , and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wavefunctions involve dual q -Kravchuk polynomials, which are solutions to a finite-difference Schrödinger equation. Time evolution (times a phase) defines the fractional Fourier– q -Kravchuk transform. In the classical limit as q → 1 we recover the finite oscillator Lie algebra, the N = 2 j → ∞ limit returns the Macfarlane–Biedenharn q -oscillator and both limits contract the generators to the standard quantum-mechanical harmonic oscillator.
We consider the one-dimensional Dirac equation for the harmonic oscillator and the associated second-order separated operators. Such operators, defined by complex dilation, give the resonances of the problem. Their unique extensions as closed operators with a purely point spectrum are PT -symmetric with positive eigenvalues converging to the Schrödinger ones as c → ∞ . Precise numerical computations show that these eigenvalues coincide with the positions of the resonances up to the order of the width. The corresponding eigenfunctions are a definite choice of metastable states of the problem. Similar results are found for the Klein–Gordon oscillator: here also we have two closed, isospectral and complex conjugate extensions of the formal operator with PT -symmetry, but an infinite number of self-adjoint extensions and physical dynamics. The infinitely many pairs of eigenvectors of the two closed PT -symmetric operators give metastable states for any choice of the dynamics. The eigenvalues of the operator defined by complex dilation are resonances, although not according to the standard definition, for any dynamics.