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A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. (English) Zbl 1198.81107
Summary: We present a new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation. The scheme uses a symmetrical multi-point difference formula to represent the partial differentials of the two-dimensional variables, which can improve the accuracy of the numerical solutions to the order of \(\Delta x^{2N_q+2}\) when a (\(2N_q+1\))-point formula is used for any positive integer \(N_q\) with \(\Delta x=\Delta y\), while \(N_q=1\) equivalent to the traditional scheme. On the other hand, the new scheme keeps the same form of the traditional matrix equation so that the standard algebraic eigenvalue algorithm with a real, symmetric, large sparse matrix is still applicable. Therefore, for the same dimension, only a little more CPU time than the traditional one should be used for diagonalizing the matrix. The numerical examples of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heiles potential demonstrate that by using the new method, the error in the numerical solutions can be reduced steadily and extensively through the increase of \(N_q\), which is more efficient than the traditional methods through the decrease of the step size.

MSC:
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
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References:
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