Construction of the Spectral Measure for Deformed Oscillator Position Operator in the Case of Undetermined Hamburger Moment Problem

Abstract

The spectral measure of the position (momentum) operator X for q-deformed oscillator is calculated in the case of the indetermine Hamburger moment problem. The exposition is given for concrete choice of generators for q-oscillator algebra, although developed technique apply for every other cases with indetermine moment problem. The Stieltjes transformation m(z) of spectral measure is expressed in terms of the entries of Jacobi matrix X only. The direct connection between values of parameters labeling the spectral measures and related selfadjoint extensions of X is established.

Generalized coherent states for oscillators connected with Meixner and Meixner—Pollaczek polynomials

“…In works [25]- [35] we suggested a new method for constructions of oscillator-like systems (or, to put it briefly "oscillator"), which are connected with a family of orthogonal polynomials just as the usual boson oscillator with Hermite polynomials. This approach contains the construction of the generalized coherent states for such oscillators.…”

“…However for the deformed polynomial systems more difficult undetermined moment problem sturt up (see [55]- [56]). In this case we used the results of research of this problem obtained together with P.P.Kulish in [25].…”

“…Let us remark that Jacobi matrix defined by the relation (13) has a nonzero diagonal and the appropriate moment problem is determined [25].…”

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

“…There is no problem with a spectrum of the Hamiltonian H = 1 2 (aa + + a + a): this spectrum is discrete and the corresponding eigenvectors are easily determined. But in some cases (similar to the case of representations of noncompact quantum algebras (see, for example, [6]) there are difficulties with spectra of the position and momentum operators (see [7,8,9]). It was shown that if the position operator Q = a + + a (or the momentum operator P = i(a + − a)) is not bounded, then this symmetric operator is not essentially self-adjoint.…”

Generalized coherent states for q-oscillator connected with q-Hermite polynomials

“…One year ago we talked here about GCS connected with classical polynomials. Our approach to such construction is developed in the following papers [1] - [8].…”