  We construct an explicit solution of the Cauchy initial value problem for the
one-dimensional Schroedinger equation with a time-dependent Hamiltonian
operator for the forced harmonic oscillator. The corresponding Green function
(propagator) is derived with the help of the generalized Fourier transform and
a relation with representations of the Heisenberg-Weyl group N(3) in a certain
special case first, and then is extended to the general case. A three parameter
extension of the classical Fourier integral is discussed as a by-product.
Motion of a particle with a spin in uniform perpendicular magnetic and electric
fields is considered as an application; a transition amplitude between Landau
levels is evaluated in terms of Charlier polynomials. In addition, we also
solve an initial value problem to a similar diffusion-type equation.
