Localized and Non-Localized Solutions of q-Deformed Oscillators

Eigenfunctions of the Fock and non-Fock irreducible representations are explicitly derived for the q-deformed oscillators of the Macfarlane and Dubna types. While the Fock representations are composed of localized functions, the non-Fock representations consist of nonlocal oscillating functions which are constructed for the first time in this paper. The concept of the q-Hermite functions is generalized to include both the Fock and non-Fock types in a unified manner.