Numerical solution of Schroedinger equation by wavelet-Galerkin and spectral methods

To generate data for the efficiency and speedup possible for the wavelet-Galerkin algorithm we solved several simple cases where the potential, $U$, is a product of harmonic functions. We expand these potentials in a wavelet-Galerkin series.

We then solve the Schroedinger equation

$$\imath \psi_t = \Delta \psi + U\psi$$

by the wavelet-Galerkin method, using three term connection coefficients to evaluate the product term, $U\psi$.

We use the implicit time differencing to a tolerance of $1.e-3$

$$\imath (\psi_{n+1}-\psi_n)/dt = .5\Delta(\psi_{n+1}+\psi_n) + .5(\psi_{n+1}+\psi_n)U,$$

which conserves the energy of the wave function, $\int\psi\overline{\psi}$. The implicit time differencing is solved for $\psi_{n+1}$ by iteration

$$\imath (f_{k+1}-\psi_n)/dt = .5\Delta(f_{k+1}+\psi_n) + .5(f_{k}+\psi_n)U,$$

until $\sup\vert f_{k+1}-f_{k}\vert < \epsilon$ where $f_0=\psi_n$.