## Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions.(English)Zbl 1053.65072

The paper is devoted to the numerical computation of the solution to the two-dimensional Schrödinger equation in a bounded domain $$\Omega$$ with artificial conditions set on the arbitrarily shaped boundary of $$\Omega$$ . Using the results of their earlier paper [cf. X. Antoine and C. Besse, J. Math. Pures Appl. 80, No. 7, 701–738 (2001; Zbl 1129.35324)], which allows one to construct a hierarchy of artificial boundary conditions, they study the truncated initial boundary value problem with one of the previous artificial boundary conditions and prove the uniqueness of the solution as well as the decay of the total energy associated with the system.
The wellposedness of the semi-discrete problem discretized by the Crank-Nicolson scheme is investigated. The result indicates that the decay of the energy is preserved at the semi-discrete level, implying hence the stability of the whole scheme. A few numerical experiments are done.

### MSC:

 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35Q40 PDEs in connection with quantum mechanics 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

Zbl 1129.35324
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