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On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime. (English) Zbl 1006.65112
Authors’ summary: We study time-splitting spectral approximations for the linear Schrödinger equation in the semiclassical regime, where the Planck constant is small. In this regime, the equation propagates oscillations with a wavelength of \(O(\varepsilon)\), and finite difference approximations require the spatial mesh size \(h=o(\varepsilon)\) and the time step \(k=o(\varepsilon)\) in order to obtain physically correct observables. Much sharper mesh-size constraints are necessary for a uniform \(L^2\)-approximation of the wave function.
The spectral time-splitting approximation under study will be proved to be unconditionally stable, time reversible, and gauge invariant. It conserves the position density and gives uniform \(L^2\)-approximation of the wave function for \(k=o (\varepsilon)\) and \(h=O (\varepsilon)\).
Extensive numerical examples in both one and two space dimensions and analytical considerations based on the Wigner transform even show that weaker constraints (e.g., \(k\) independent of, and \(h=O (\varepsilon)\)) are admissible for obtaining “correct” observables. Finally, we address the application to nonlinear Schrödinger equations and conduct some numerical experiments to predict the corresponding admissible meshing strategies.

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
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