Nonlocal boundary value problems for the Schrödinger equation. (English) Zbl 1155.65368

Summary: In the present paper, the nonlocal boundary value problem \[ \begin{cases} \text i\frac{\text du}{\text dt} + Au = f(t),\quad 0<t<T, \\ u(0) =\sum^p_{m=1} \alpha_m u(\lambda_m)+\varphi \\ 0< \lambda_1 < \lambda_2 < \cdots < \lambda_p \leq T\end{cases} \] 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.


65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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