On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit II. Analytic regularity
Provides rigorous error estimates for numerical methods in semi-classical quantum mechanics, addressing a known regularity issue.
The paper proves that Lie-Trotter splitting for the nonlinear Schrödinger equation in the semi-classical limit preserves WKB structure and allows computation of quadratic observables with a time step independent of the Planck constant, using analytic regularity to avoid loss of regularity.
We consider the time discretization based on Lie-Trotter splitting, for the nonlinear Schrodinger equation, in the semi-classical limit, with initial data under the form of WKB states. We show that both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. The functional framework is based on time-dependent analytic spaces, in order to overcome a previously encountered loss of regularity phenomenon.