L2 method for a soluble model
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The technique of describing scattering solutions of the Schrödinger equation by means of L2 functions is illustrated for a variable coupling constant in a simple separable potential model which admits an exact solution. The orthogonal polynomials generated by the L2 solution are shown to be positive definite only for a limited range of the coupling constant. Further, it is shown that the underlying Gaussian quadrature rule for the positive-definite region can be extended to the general case albeit with the consequence that the weights are no longer all positive. A simple expression for the weights in terms of the associated abscissas is derived for the model. To complete the analysis it is demonstrated that a renormalization procedure of the L2 functions to recover the scattering solutions is justified for arbitrary values of the coupling constant. Finally, some numerical examples are presented to test the Heller equivalent weight prescription and it is concluded that it works for all ranges of coupling strengths in the model.
|Publication Type:||Journal Article|
|Murdoch Affiliation:||School of Mathematical and Physical Sciences|
|Publisher:||The American Physical Society|
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