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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.

##### MSC:
 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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