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Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

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Abstract

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x 2,…,x p,exp (±wx),xexp (±wx),…,x mexp (±w x)} and the second is based on the exact integration of the functions of the form: {1,x,x 2,…,x p,exp (±wx),exp (±2wx),…,exp (±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrödinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrödinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrödinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrödinger equation. Finally, numerical results for the resonance problem of the radial Schrödinger equation are presented.

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  1. 10 Konitsis Street, Amfithea—Paleon Faliron, 175 64, Athens, Greece

    T. E. Simos

  2. Laboratory of Computational Sciences, Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, 221 00, Tripolis, Greece

    T. E. Simos

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Highly Cited Researcher in Mathematics (ISI), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences Corresponding Member of European Academy of Arts, Sciences and Humanities.

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Simos, T.E. Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation. Acta Appl Math 110, 1331–1352 (2010). https://doi.org/10.1007/s10440-009-9513-6

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