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.
Highlights
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
We shall give an overview of the current state of the art of numerical methods for the Schrodinger equation in the semiclassical regime
Possible nonlinear effects can be taken into account by considering nonlinear potentials V = f (|uε|2)
Summary
We recall the basic existence theory for linear Schrodinger equations of the form iε∂tuε. Expectation values of physical observables are computed via quadratic functionals of uε. To this end, denote by aW (x, εDx) the operator corresponding to a classical (phase space) observable a ∈ Cb∞(Rd × Rd), obtained via Weyl quantization, aW (x, εDx)f (x). The convenience of the Weyl calculus lies in the fact that an (essentially) self-adjoint Weyl operator aW (x, εDx) has a real-valued symbol a(x, ξ): see Hormander (1985). The quantum mechanical wave function uε can be considered only an auxiliary quantity, whereas (real-valued) quadratic quantities of uε yield probability densities for the respective physical observables. Quadratic operations defining densities of physical observables do not, in general, commute with weak limits, and it remains a challenging task to identify the (weak) limits of certain physical observables, or densities, respectively
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