Conference ID: 295-528-576
PIN Code: 775196
The semiclassically scaled multi-particle Schroedinger equation poses the combined computational challenges of high oscillations and high dimension. This talk reviews Gaussian wave packets that evolve according to the Dirac–Frenkel time-dependent variational principle for the semiclassical Schroedinger equation. This provides a simple yet fundamental approximation that is basic for many more elaborate approximations.
Old and new results on the Gaussian approximation to the wave function are given, in particular an $L^2$ error bound that goes back to Hagedorn (1980) in a non-variational setting, and a new error bound for averages of observables, which shows the double approximation order in the semiclassical scaling parameter in comparison with the norm estimate.
The variational equations of motion in Hagedorn’s parametrization of the Gaussian are presented. They show a perfect quantum–classical correspondence and allow us to read off directly that the Ehrenfest time is determined by the Lyapunov exponent of the classical equations of motion.
A variational splitting integrator is formulated and its remarkable conservation and approximation properties are discussed. A new result shows that the integrator approximates averages of observables with the full order in the time stepsize, with an error constant that is uniform in the semiclassical parameter.
The material presented here for variational Gaussians is part of an Acta Numerica review article with Caroline Lasser, to appear in 2020, which is about computational methods for quantum dynamics in the semiclassical regime.
Christian Lubich is Professor of Numerical Mathematics at the University of Tuebingen (Germany). He is an expert in the numerical analysis of time-dependent problems, whether ordinary or partial differential equations or related evolutionary integral equations. He is a coauthor, with Ernst Hairer and Gerhard Wanner, on an influential monograph on Geometric Numerical Integration. His recent research interests include dynamical low-rank approximation of high-dimensional matrix and tensor differential equations with applications to quantum dynamics and plasma physics, highly oscillatory problems such as quantum dynamics in the semiclassical regime and charged-particle dynamics in a strong magnetic field, stable numerical interior-exterior coupling of wave equations, and the numerical analysis of geometric evolution equations such as mean curvature flow. Professor Lubich is a member of the editorial boards of leading journals in numerical analysis and a member of the Scientific Committee of the Oberwolfach Mathematical Research Institute. He won a SIAM Dahlquist Prize, and gave a plenary lecture at the International Congress of Mathematicians in Rio de Janeiro in 2018 on “Dynamics, numerical analysis, and some geometry”.