Equivalence of two variational methods in scattering theory

Abstract

Starting from the integral version of the SchrSdinger equation, a nonlinear equation of first degree has been derived for the so-called (~ phase-function ~)(1). This function is directly related to the phase-shift. The equation has been usefully applied in many cases of interest (~). Among other methods of solving it, a variational approach has been suggested, which is derived by (~ quasi-linearizing ~) the phase-equation (1). Sehwinger's variational method is also derived from the integral version of the SchrSdinger equation, and one would expect some connection between the two. I t is shown, in the present note, that they are equivalent with a particular identification of the trial function and the one could be derived from the other without any reference to the phase-equation. Results proven for the one, such as convergence and bound theorems, therefore a r e valid for the other. This can prove useful when one of the forms is suitable for analysis and the other for computation. If a(~) and b(x) are two linearly independent solutions of the free-particle partialwave SchrSdinger equation then, in deriving the phase-equation, one takes for the solut ion of the full equation u ( x ) = C l ( x ) a ( x ) + (7~($)b(x). The phase-function S(x) is defined by S(x) = O2(x)/Cl(x). S(oo) is a function of the phase-shift depending on the choice of a(x) and b(x). The variational expression S,(oo) for S(cr is given by (1):