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CP methods for the Schrödinger equation. (English) Zbl 0971.65067
This paper focuses on a class of piecewise perturbation methods for approximating the potential energy function when solving the one dimensional Schrödinger equation. These methods, known as constant perturbation (CP) methods, approximate the potential energy function by piecewise constants on a set of subintervals and then perform polynomial-type perturbation corrections. An error analysis is given and a simple numerical illustration.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
Software:
SLDRIVER
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References:
[1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1972), Dover New York · Zbl 0515.33001
[2] Bailey, P.B.; Gordon, M.K.; Shampine, L.F., Automatic solution of the sturm – liouville problem, ACM trans. math. software, 4, 193-207, (1978) · Zbl 0384.65045
[3] Canosa, J.; Gomes de Oliveira, R., A new method for the solution of the Schrödinger equation, J. comput. phys., 5, 188-207, (1970) · Zbl 0195.17302
[4] Gordon, R.G., A new way for constructing wave functions for bound states and scattering, J. chem. phys., 51, 14-25, (1969)
[5] L.Gr. Ixaru, An algebraic solution of the Schrödinger equation, Internal Report IC/69/6, International Centre for Theoretical Physics, Trieste, 1969.
[6] Ixaru, L.Gr., The error analysis of the algebraic method to solve the Schrödinger equation, J. comput. phys., 9, 159-163, (1972) · Zbl 0235.65059
[7] Ixaru, L.Gr., Numerical methods for differential equations and applications, (1984), Reidel Dordrecht · Zbl 0301.34010
[8] Ixaru, L.Gr.; De Meyer, H.; Vanden Berghe, G., CP methods for the Schrödinger equation revisited, J. comput. appl. math., 88, 289, (1998) · Zbl 0909.65045
[9] Ixaru, L.Gr.; De Meyer, H.; Vanden Berghe, G., SLCPM12, a program for the solution of regular sturm – liouville problems, Comput. phys. comm., 118, 259, (1999) · Zbl 1008.34016
[10] Ixaru, L.Gr.; Rizea, M.; Vertse, T., Piecewise perturbation methods for calculating eigensolutions of a complex optical potential, Comput. phys. comm., 85, 217-230, (1995) · Zbl 0873.65078
[11] Paine, J.W.; de Hoog, F.R., Uniform estimation of the eigenvalues of sturm – liouville problem, J. austral. math. soc. ser. B, 21, 365-383, (1980) · Zbl 0417.34046
[12] Pruess, S., Estimating the eigenvalues of sturm – liouville problems by approximating the differential equation, SIAM J. numer. anal., 10, 55-68, (1973) · Zbl 0259.65078
[13] Pruess, S.; Fulton, C.T., Mathematical software for sturm – liouville problems, ACM trans. math. software, 19, 360-376, (1993) · Zbl 0890.65087
[14] Pryce, J.D., Numerical solution of sturm – liouville problems, (1993), Oxford University Press Oxford · Zbl 0795.65053
[15] J.D. Pryce, SLDRIVER installation guide and user guide, Technical Report SEAS/CISE/JDP03/96, Royal Military College of Sciences, Shrivenham, UK, 1996.
[16] Pryce, J.D.; Marletta, M., Automatic solution of sturm – liouville problems using the pruess method, J. comput. appl. math., 39, 57-78, (1992) · Zbl 0747.65070
[17] L.Gr. Ixaru, The algebraic approach to the scattering problem, Internal Report IC/69/7, International Centre for Theoretical Physics, Trieste, 1969.
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