Abstract. We consider the approximation of multi-particle quantum dynamics in the semiclassical 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, and of high-dimensional problems by Hartree-type approximations in a moving coordinate frame.
The time-dependent Schrödinger equation is discretized in space by a sparse grid pseudo-spectral method. The Strang splitting for the resulting evolutionary problem features first or second order convergence in time, depending on the smoothness of the potential and of the initial data. In contrast to the full grid case, where the frequency domain is the working place, the proof of the sufficient conditions for the convergence is done in the space realm.
We propose a new algorithm for solving the semiclassical timedependent Schrödinger equation. The algorithm is based on semiclassical wavepackets. The focus of the analysis is only on the time discretization: convergence is proved to be quadratic in the time step and linear in the semiclassical parameter ε.
Tunneling through a one-dimensional Eckart barrier is investigated using a recently developed propagation scheme based on semi-classical wave-packets. This version of the time-dependent discrete variable representation method yields linear equations for the parameters, is fully adaptive, and does not require a frozen Ansatz in order to approximate the exact solution of the Schrödinger equation accurately. We rely on an analytical result to derive a new algorithm to spawn a second family of semi-classical wave-packets after the tunneling has occurred. Numerical results for a benchmark problem demonstrate the accuracy of the new method.
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