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

### MSC:

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

### Keywords:

Schrödinger equation; difference schemes; stability; self adjoint operator; spectral representation
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\textit{A. Ashyralyev} and \textit{A. Sirma}, Comput. Math. Appl. 55, No. 3, 392--407 (2008; Zbl 1155.65368)

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### References:

[2] | Gordeziani, D. G.; Avalishvili, G. A., Time-nonlocal problems for Schrödinger type equations: I. Problems in abstract spaces, Differential Equations, 41, 5, 703-711 (2005) · Zbl 1081.35004 |

[3] | Gordeziani, D. G.; Avalishvili, G. A., Time-nonlocal problems for Schrödinger type equations: II. Results for specific problems, Differential Equations, 41, 6, 852-859 (2005) · Zbl 1082.35055 |

[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 |

[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, 248, 1779-1799 (2004) · Zbl 1053.65072 |

[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), (in Russian) · Zbl 0674.65031 |

[7] | Ashyralyev, A.; Piskarev, S.; Wei, S., On well-posedness of the difference schemes for abstract parabolic equations in \(L_p([0, 1], E)\) spaces, Numerical Functional Analysis and Optimization, 23, 7-8, 669-693 (2002) · Zbl 1022.65095 |

[8] | Rannacher, R., Discretization of the heat equation with singular initial data, Zeitschrift für Angewandte Mathematik und Mechanik, 62, 5, 346-348 (1982) · Zbl 0503.65060 |

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