Variationally evolving Gaussians revisited (Christian Lubich, Tuebingen)

20.12.2019 11:15

The talk reviews Gaussian wave packets that evolve according to the Dirac--Frenkel time-dependent variational principle for the semiclassically scaled Schr\"odinger equation. Old and new results on the 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, currently in preparation, which is about computational methods for quantum dynamics in the semiclassical regime.

Lieu

salle 624, Séminaire d'analyse numérique

Organisé par

Section de mathématiques

Intervenant-e-s

Christian Lubich , Univ. Tuebingen

entrée libre

Classement

Catégorie: Séminaire

Mots clés: analyse numérique