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

TL;DR

This paper develops a multiscale approximation of the free-particle propagator in quantum mechanics using multiwavelet bases, employing contour deformation to efficiently discretize oscillatory integrals, advancing simulation methods.

Contribution

It introduces a novel multiresolution approach for the free-particle propagator using multiwavelet bases and contour deformation for efficient integral discretization.

Findings

Effective discretization of oscillatory integrals

Enhanced multiscale representation of the propagator

Improved computational efficiency in quantum simulations

Abstract

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.