  The Darboux method is commonly used in the coordinate variable to produce new
exactly solvable (stationary) potentials in quantum mechanics. In this work we
follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to
include the time-variable as a parameter of the transformation. The new
potentials are nonstationary and define Hamiltonians which are not integrals of
motion for the system under study. We take the stationary oscillator of
constant frequency to produce nonstationary oscillators, and also provide an
invariant that serves to define uniquely the state of the system. In this sense
our approach completes the program of the BSS method since the eigenfunctions
of the invariant are an orthonormal basis for the space of solutions of the
related Schrodinger equation. The orthonormality holds when the involved
functions are evaluated at the same time. The dynamical algebra of the
nonstationary oscillators is generated by properly chosen ladder operators and
coincides with the Heisenberg algebra. We also construct the related coherent
states and show that they form an overcomplete set that minimizes the
quadratures defined by the ladder operators. These states are not invariant
under time-evolution since their time-dependence relies on the basis of states
and not on the complex eigenvalue that labels them. Some concrete examples are
provided.
