On position operator spectral measure for deformed oscillator in the case of indetermine 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.