Efficient Computation of the Semiclassical Limit of the Schrödinger Equation
B1 R3426
This talk presents the Gaussian Wave Packet Transform, an efficient numerical framework for solving the semiclassical Schrödinger equation that reduces spatial and temporal stiffness, enables accurate simulation of highly oscillatory quantum dynamics in multiple dimensions, and supports efficient computation using Gaussian and Hagedorn wave packet representations.
Overview
The semiclassical Schrödinger equation is important in chemistry and physics, but its numerical solution becomes difficult when the non dimensional Planck constant ε is small. For localized initial data of width O(√ε), solutions oscillate on spatial and temporal scales of O(ε), making fully resolved computations especially costly in several dimensions. We consider the initial-value problem for a quantum particle subject to oscillate in a potential well, where the initial condition is a modulated Gaussian wave packet.
We present the Gaussian Wave Packet Transform, a time-dependent change of variables closely related to Gaussian wave packets. The transformed wave function satisfies a Schrödinger equation with a time-dependent potential that is a harmonic oscillator perturbed by a term of order O(√ε). This substantially reduces spatial and temporal stiffness; over a fixed time interval, fewer modes are needed as ε decreases. For Gaussian initial data, very small number of modes can suffice, while observables can be computed directly from the transformed solution. The transform was first developed in one dimension and then extended to several dimensions. Numerical results demonstrate the accuracy and efficiency of these approaches. A related spectral method, developed with B. Miao and Z. Zhou, expands the transformed solution in Hagedorn wave packets, which are Hermite-type functions.
Publication Authors
Giovanni Russo and Peter Smereka.
Presenters
Giovanni Russo, Full Professor of Numerical Analysis, Department of Mathematics and Computer Science, University of Catania (UNICT)
Brief Biography
Giovanni Russo is a Full Professor of Numerical Analysis at the University of Catania, Italy. His research focuses on numerical methods for partial differential equations, with applications in computational fluid dynamics, hyperbolic systems, kinetic equations, and multiscale problems.
Before joining the University of Catania as a full professor in 2000, he held academic positions at the University of L'Aquila and the Courant Institute of Mathematical Sciences at New York University. He received his Ph.D. in Physics from the University of Catania, where he studied the propagation and stability of shock waves in gas dynamics.
Russo has made significant contributions to the development of high-order numerical methods, asymptotic-preserving schemes, and computational approaches for highly oscillatory solutions. He has authored more than 150 scientific publications and led several national and international research projects. He also serves as an associate editor for several scientific journals, including SIAM publications.
