  We introduce a one-parameter generalized oscillator algebra A(k) (that covers
the case of the harmonic oscillator algebra) and discuss its finite- and
infinite-dimensional representations according to the sign of the parameter k.
We define an (Hamiltonian) operator associated with A(k) and examine the
degeneracies of its spectrum. For the finite (when k < 0) and the infinite
(when k > 0 or = 0) representations of A(k), we construct the associated phase
operators and build temporally stable phase states as eigenstates of the phase
operators. To overcome the difficulties related to the phase operator in the
infinite-dimensional case and to avoid the degeneracy problem for the
finite-dimensional case, we introduce a truncation procedure which generalizes
the one used by Pegg and Barnett for the harmonic oscillator. This yields a
truncated generalized oscillator algebra A(k,s), where s denotes the truncation
order. We construct two types of temporally stable states for A(k,s) (as
eigenstates of a phase operator and as eigenstates of a polynomial in the
generators of A(k,s)). Two applications are considered in this article. The
first concerns physical realizations of A(k) and A(k,s) in the context of
one-dimensional quantum systems with finite (Morse system) or infinite
(Poeschl-Teller system) discrete spectra. The second deals with mutually
unbiased bases used in quantum information.
