On position and momentum operators in the q-oscillator algebra

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 X(p) = q(pN) (a(+) + a)q(pN), 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, X(p) 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. (C) 1996 American Institute of Physics.