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Journal of Mathematical Physics

New Approach to Low-Energy Potential Scattering

Roger F. Dashen

Published Mar 1, 1963• 11 citations
Publisher

Abstract

The problem of low-energy potential scattering is reformulated in a manner suggested by the ``invariant imbedding'' techniques of transport theory. The differential equations thus obtained have several conceptual and computational advantages over the Schrödinger equation. Some new bounds and approximations are derived and a rigorous investigation of the Born approximation is carried out.

DOI
10.1063/1.1703965

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The problem of low-energy potential scattering is reformulated in a manner suggested by the ``invariant imbedding'' techniques of transport theory. The differential equations thus obtained have several conceptual and computational advantages over the Schrödinger equation. Some new bounds and approximations are derived and a rigorous investigation of the Born approximation is carried out.
In order to put Zwaan's scheme for deriving the Kramers connection formulas on a rigorous basis, differential equations are set up governing the variation in the coefficients used to fit a linear combination of the B. W. K. type approximation functions to an exact solution of Schr\"odinger's equation in one dimension. Approximate solutions of the differential equations are worked out which lend themselves to the setting up of the connection formulas and give definite upper bounds to the errors involved in their use. The method is also used to set up the Sommerfeld phase integral quantum condition independently of the connection formulas. An upper limit to the error in the energy is worked out. A similar treatment of the problem of the transmission of matter waves through rounded potential barriers is formulated.
A new approach to the problem of description of the stationary motion of a quantum particle in the field of a one-dimensional arbitrary potential is developed. It is shown that the wave function of infinite motion, with the accuracy to two arbitrary constants, can be expressed by means of an arbitrary single solution for some set of linear differential equations of the first order. It is shown that one general property of the Schrödinger equation solutions lies in the basis of many known methods of the problem consideration, such as the method of integral equations, transfer-matrix method, imbedding method, and method of combination of scattering parameters. Within the framework of the proposed approach, the connection between the above-mentioned methods becomes more transparent.