Computing semiclassical quantum dynamics using Hagedorn wavepackets

We consider the approximation of multi-particle quantum dynamics in the semi-classical regime by Hagedorn wavepackets, which are products of complex Gaussians with polynomials that form an orthonormal L^2 basis and preserve their type under propagation in Schrödinger equations with quadratic potentials. We build a fully explicit, time-reversible time-stepping algorithm to approximate the solution of the Hagedorn wavepacket dynamics. The algorithm is based on a splitting between the kinetic and potential part of the Hamiltonian operator, as well as on a splitting of the potential into its local quadratic approximation and the remainder. The algorithm is robust in the semi-classical limit. It reduces to the Strang splitting of the Schrödinger equation in the limit of the full basis set, and it advances positions and momenta by the Störmer--Verlet method for the classical equations of motion. The algorithm allows for the treatment of multi-particle problems by thinning out the basis according to a hyperbolic cross approximation.

The talk is based on joint work with Erwan Faou and Vasile Gradinaru.