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A finite-difference method for the one-dimensional time-dependent Schrödinger equation on unbounded domain. (English) Zbl 1092.65071

The authors introduce a finite-difference approximation scheme to numerically solve the one-dimensional time-dependent Schrödinger equation on an unbounded domain. Some artificial boundary conditions are introduced in order to reduce the original problem to an initial-boundary value problem on a finite-computational domain. By applying the method of reduction of order the authors implement their scheme to the reduced problem. It is proved that the proposed scheme is uniquely solvable. Moreover, it is also proved that the proposed method is unconditionally stable and convergent. A typical example is presented and the results obtained are compared with other numerical schemes.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical 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
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