Heat semigroup representation of Laplacian

Abstract

This work introduces novel numerical algorithms for computational quantum mechanics, grounded in a representation of the Laplace operator -- frequently used to model kinetic energy in quantum systems -- via the heat semigroup. The key advantage of this approach lies in its computational efficiency, particularly within the framework of multiresolution analysis, where the heat equation can be solved rapidly. This results in an effective and scalable method for evaluating kinetic energy in quantum systems. The proposed multiwavelet-based Laplacian approximation is tested through two fundamental quantum chemistry applications: self-consistent field calculations and the time evolution of Schr\"odinger-type equations.