A note on the modified Crank-Nicolson difference schemes for Schrödinger equation. (English) Zbl 1144.65313

Ashyralyev, Allaberen; Sirma, Ali

Aliyev Azeroğlu, Tahir (ed.) et al., Complex analysis and potential theory. Proceedings of the conference satellite to ICM 2006, Gebze, Turkey, September 8–14, 2006. Hackensack, NJ: World Scientific (ISBN 978-981-270-598-3/hbk). 256-271 (2007).

Summary: The nonlocal boundary value problem
\[ \begin{cases} i\,\frac{du}{dt}+Au=f(t), & 0<t<T,\\ u(0)=\alpha u(\lambda)+\mu, &|a|<1,\;0<\lambda\leq T\end{cases} \]
for Schrödinger equation in a Hilbert space \(H\) with the self-adjoint positive definite operator \(A\) is considered. The second order of accuracy \(r\)-modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. In practice, the stability inequalities for the solutions of difference schemes for Schrödinger equation are obtained.
A numerical method is proposed for solving a one-dimensional Schrödinger equation with nonlocal boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.
For the entire collection see [Zbl 1124.30001].

MSC:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q40	PDEs in connection with quantum mechanics
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
34G10	Linear differential equations in abstract spaces
34L40	Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
65L12	Finite difference and finite volume methods for ordinary differential equations
65L20	Stability and convergence of numerical methods for ordinary differential equations