General formulation for direct evaluation of the local field amplitude and transition amplitude based on the Fredholm determinant

Abstract

We develop a determinantal method in a quantum scattering system for direct evaluation of various quantities including the local field amplitude and transition amplitude ($S$-matrix element). This method is applicable to multichannel elastic and inelastic scattering systems without rearranging the particles. The underlying principle of our formulation is that requested information can be extracted from a wave operator without solving the wave equation. The wave operator is the master operator maintaining all of the information about a system; the determinant of the inverse wave operator is just the Fredholm determinant. In the 1960s, a similar determinantal method was developed for the $S$-matrix element or equivalently the far-field amplitude of the wave function. Until now, however, no determinantal method for near-field quantities existed, although de Broglie and electromagnetic near fields are observable using scanning probe microscopes. Additionally, we prove that our formula for $S$-matrix element covers the known formula (the Le Couteur-Newton formula).