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Numerical solution of the two-dimensional time independent Schrödinger equation with Numerov-type methods. (English) Zbl 1077.65110
Summary: The solution of the two-dimensional time-independent Schrödinger equation is considered by partial discretization. The discretized problem is treated as an ordinary differential equation problem and Numerov type methods are used to solve it. Specifically the classical Numerov method, the exponentially and trigonometrically fitting modified Numerov methods of G. Vanden Berghe, H. De Meyer and J. Vanthournout [Int. J. Comput. Math. 32, No. 3/4, 233–242 (1990; Zbl 0752.65059)], of G. Vanden Berghe and H. De Meyer [ibid. 37, No. 1/2, 63–77 (1990; Zbl 0726.65100)], and the minimum phase-lag method of M. M. Chawla and P. S. Rao [J. Comput. Appl. Math. 11, 277–281 (1984; Zbl 0565.65041); ibid. 15, 329–337 (1986; Zbl 0598.65054)] are applied to this problem. All methods are applied for the computation of the eigenvalues of the two-dimensional harmonic oscillator and the two-dimensional Henon-Heils potential. The results are compared with the results produced by full discterization.

MSC:
65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
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