Local realizations of $q$-Oscillators in Quantum Mechanics
  Representations of the quantum q-oscillator algebra are studied with particular attention to local Hamiltonian representations of the Schroedinger type. In contrast to the standard harmonic oscillators such systems exhibit a continuous spectrum. The general scheme of realization of the q-oscillator algebra on the space of wave functions for a one-dimensional Schroedinger Hamiltonian shows the existence of non-Fock irreducible representations associated to the continuous part of the spectrum and directly related to the deformation. An algorithm for the mapping of energy levels is described. 
A. A. Andrianov (1), F. Cannata (2), J.-P. Dedonder (3), M. V. Ioffe   (1) ((1) Sankt-Petersburg State University, Russia, (...