Solution of the Time-dependent Schrödinger Equation Using Interpolating Wavelets

An adaptive numerical method for solution of the time-dependent Schrödinger equation is presented. By using an interpolating wavelet transform the number of used points can be reduced, constructing an adaptive sparse point representation. Two di erent discretisation approaches are studied in 1D ; one point-based and one based on equidistant blocks. Due to the nature of the wavelet transform a strict point method is not viable in practice. A block-based method though, utilizes both the wave-package characteristics of the Schrödinger-solution and the symmetry of the interpolating wavelet transform. The methods decrease memory usage by a huge factor, but both yield unacceptable computational overhead in the one-dimensional case. Arithmetic complexity and memory usage for the generalized problem in n-space suggests that a usage of an adaptive method based on the interpolating wavelet transform can speedup the solution of the Schrödinger equation considerably. 1