A simple framework for accurate solution of a general class of one-dimensional Fokker-Planck and/or Schr\"odinger equations is presented. The main idea is representing the propagator in the form
P(x,t∣x0)=P0(x,t∣x0)exp[W(x,t∣x0)] and expanding the exponent
W in a power series in a given function of
t, where
P0 is the exact solution of a reference harmonic-oscillator problem. The expansion coefficients are analytically evaluated from recursive relations. This approach is shown to be a dramatic improvement over the standard Taylor series expansion for the propagator in that just a few terms of the present expansion are sufficient to attain a very accurate description in the whole time domain.
A simple algebraic approach based on the well known angular momentum SU (2) algebra is presented to describe 1D systems for arbitrary potentials. The approach is based on the dimension increase of the 1D harmonic oscillator space through the addition of a scalar boson, keeping constant the total number of bosons. In this new space the realization of the coordinate and momentum correspond to components of the angular momentum algebra, which in turn define the coordinate and momentum representation bases. This remarkable result allows the Hamiltonian matrix representationto be expressed in terms of a diagonal matrix characterizing the potential we are considering. The solutions are obtained as an expansion of harmonic oscillator functions by purely algebraic means. As an example of our approach the Morse and asymmetric double Morse potentials are considered.