Mathematical and computational methods for semiclassical Schrödinger equations

Abstract

We consider time-dependent (linear and nonlinear) Schrödinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approximation of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymptotics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrödinger equation in the semiclassical regime. * Colour online available at journals.cambridge.org/anu.

Computing high frequency solutions of symmetric hyperbolic systems with polarized waves

“…Although we have found that Gaussian beams are applicable to symmetric hyperbolic systems with polarized waves, the particulars of this method are non-trivial and will be discussed fully in a forthcoming paper. For recent work regarding Gaussian beams applied to the Schrödinger equation, see [8]. On the left, the multiple valued surface is projected into three dimensions by plotting x 1 , x 2 versus k 1 .…”

“…This significantly enhances the numerical resolution of the singular solutions to the Liouville equation. See [2,8] for reviews of computational high frequency waves or semiclassical limit of quantum waves.…”

Coupling of Gaussian Beam and Finite Difference Solvers for Semiclassical Schrödinger Equations

“…When resolving the oscillations on a grid becomes unfeasible one has to resort to something else, commonly asymptotic methods. Such methods have been studied in the fields of acoustics and electromagnetics [6,15] as well as in quantum dynamics [13]. Asymptotic methods have modelling errors which are dependent of some problem parameter.…”

On Fourier Time-Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Limit

“…Physically, ε corresponds to a small ratio between microscopic and macroscopic quantities, so the limit ε → 0 is expected to yield a relevant approximation; see e.g. [18] and references therein. The parameter α 0 measures the strength of nonlinear interactions: in the WKB regime, which is recalled below, the nonlinearity is negligible if α > 1, it has a leading order (moderate) influence if α = 1 (weakly nonlinear regime), and its influence is very strong in the regime ε → 0 if α = 0.…”