On position operator spectral measure for deformed oscillator in the case of indetermine Hamburger moment problem

TL;DR

This paper calculates the spectral measure of the position operator for a $q$-deformed oscillator in cases where the Hamburger moment problem is indeterminate, linking spectral measures to selfadjoint extensions.

Contribution

It develops a technique to explicitly compute the spectral measure of the position operator in the indeterminate case for $q$-oscillators, applicable to various algebra generators.

Findings

Spectral measure expressed via Jacobi matrix entries

Connection established between spectral measure parameters and selfadjoint extensions

Technique applicable to all cases with indeterminate 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.