A new kind of discretization scheme for solving a two-dimensional time-independent Schrödinger equation.

*(English)*Zbl 1198.81107Summary: 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 |

##### Keywords:

two-dimensional Schrödinger equation; eigenvalue problem; large sparse matrix; discretization
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\textit{Z. Wang} and \textit{H. Shao}, Comput. Phys. Commun. 180, No. 6, 842--849 (2009; Zbl 1198.81107)

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##### References:

[1] | Parlett, B.N., The symmetric eigenvalue problem, (1980), Prentice-Hall Englewood Cliffs, NJ · Zbl 0431.65016 |

[2] | Liu, X.S.; Su, L.W.; Liu, X.Y.; Ding, P.Z., Int. J. quantum chem., 83, 303, (2001) |

[3] | Monovasilis, Th.; Simos, T.E., Chem. phys., 313, 293, (2005) |

[4] | van Dijk, W.; Toyama, F.M., Phys. rev. E, 75, 036707, (2007) |

[5] | Davis, M.J.; Heller, E.J., J. chem. phys., 71, 5356, (1982) |

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