An Application of the rayleigh-ritz method to the integral-equation representation of the one-dimensional schrödinger equation

Kaya, Ruşen
In this thesis, the theory of the relations between differential and integral equations is analyzed and is illustrated by the reformulation of the one-dimensional Schrödinger equation in terms of an integral equation employing the Green’s function. The Rayleigh- Ritz method is applied to the integral-equation formulation of the one-dimensional Schrödinger equation in order to approximate the eigenvalues of the corresponding singular problem within the desired accuracy. The outcomes are compared with those resulting from the methods applied to the original formulation of the problem. Consecutive symmetries are observed throughout the symmetric structure of the problem, the symmetric Green’s function, the symmetric potentials used in the method and the symmetric matrices obtained eventually.


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A Rayleigh–Ritz Method is suggested for solving linear Fredholm integral equations of the second kind numerically in a desired accuracy. To test the performance of the present approach, the classical one-dimensional Schrödinger equation -y″(x)+v(x)y(x)=λy(x),x∈(-∞,∞) has been converted into an integral equation. For a regular problem, the unbounded interval is truncated to x∈ [ - ℓ, ℓ] , where ℓ is regarded as a boundary parameter. Then, the resulting integral equation has been solved and the results are co...
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In this thesis, the magnetic-field integral equation (MFIE) for three-dimensional perfectly conducting objects is studied with a particular focus on the solutions of the formulation with the method of moments employing low-order discretization elements. Possible discretization functions and their applications in the testing of MFIE while considering different numbers of testing points are analyzed for accurate and efficient solutions. Successful results are obtained by using rotational Buffa-Christiansen te...
Citation Formats
R. Kaya, “An Application of the rayleigh-ritz method to the integral-equation representation of the one-dimensional schrödinger equation,” Thesis (M.S.) -- Graduate School of Applied Mathematics. Mathematics., Middle East Technical University, 2019.