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Nonlocal boundary value problems for the Schrödinger equation. (English) Zbl 1155.65368
Summary: In the present paper, the nonlocal boundary value problem $$\cases \text i\frac{\text du}{\text dt} + Au = f(t),\quad 0<t<T, u(0) =\sum^p_{m=1} \alpha_m u(\lambda_m)+\varphi 0< \lambda_1 < \lambda_2 < \cdots < \lambda_p \le T\endcases$$ for the Schrödinger equation in a Hilbert space $H$ with the self-adjoint operator $A$ is considered. Stability estimates for the solution of this problem are established. Two nonlocal boundary value problems are investigated. The first and second order of accuracy difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. In practice, stability inequalities for the solutions of difference schemes for the Schrödinger equation are obtained. A numerical method is proposed for solving a one-dimensional Schrödinger equation with nonlocal boundary condition. A procedure involving the modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by giving numerical examples.

##### MSC:
 65M06 Finite difference methods (IVP of PDE)
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##### References:
 [1] M.E. Mayfield, Non-reflective boundary conditions for Schrödinger’s equation, Ph.D. Thesis, University of Rhode Island, 1989 [2] Gordeziani, D. G.; Avalishvili, G. A.: Time-nonlocal problems for Schrödinger type equations: I. Problems in abstract spaces, Differential equations 41, No. 5, 703-711 (2005) · Zbl 1081.35004 · doi:10.1007/s10625-005-0205-3 [3] Gordeziani, D. G.; Avalishvili, G. A.: Time-nonlocal problems for Schrödinger type equations: II. Results for specific problems, Differential equations 41, No. 6, 852-859 (2005) · Zbl 1082.35055 · doi:10.1007/s10625-005-0224-0 [4] Han, H.; Jin, J.; Wu, X.: A finite difference method for the one-dimensional Schrödinger equation on unbounded domain, Computers and mathematics with applications 50, 1345-1362 (2005) · Zbl 1092.65071 · doi:10.1016/j.camwa.2005.05.006 [5] Antoine, X.; Besse, C.; Mouysset, V.: Numerical schemes for the simulation of the two-dimensional Schrödinger equation using non-reflecting boundary conditions, Mathematics of computation 73, No. 248, 1779-1799 (2004) · Zbl 1053.65072 · doi:10.1090/S0025-5718-04-01631-X [6] Ashyralyev, A.: An estimation of the convergence for the solution of the modified Crank--Nicholson difference schemes for parabolic equations with nonsmooth initial data, Izv. akad. Nauk turkmen. SSR ser. Fiz.-tekhn. Khim. geol. Nauk 1, 3-8 (1989) · Zbl 0674.65031 [7] Ashyralyev, A.; Piskarev, S.; Wei, S.: On well-posedness of the difference schemes for abstract parabolic equations in $Lp([0,1],E)$ spaces, Numerical functional analysis and optimization 23, No. 7--8, 669-693 (2002) · Zbl 1022.65095 · doi:10.1081/NFA-120016264 [8] Rannacher, R.: Discretization of the heat equation with singular initial data, Zeitschrift für angewandte Mathematik und mechanik 62, No. 5, 346-348 (1982)