Multiresolution of the one dimensional free-particle propagator. Part 1: Construction

The free-particle propagator, a key operator in various algorithms for simulating the time evolution of the Schr\"odinger equation, is studied. A multiscale approximation of this propagator is constructed, representing the semigroup associated with the free-particle Schr\"odinger operator in a multiwavelet basis. This representation involves integrals of highly oscillatory functions. These integrals are efficiently discretized using a contour deformation technique, which addresses the challenges posed by earlier discretization methods.