A Second Minimum Principle For Schrödinger’s Equation.
James H. Rose (Iowa State University and Laboratoire Ondes et Acoustique, ESPCI)

Jeudi 7 mai 1998

Consider Schrödinger’s equation for the scattering of spinless particles from a local real potential. Further suppose that the complex scattering amplitude is known. The inverse scattering problem is to determine the potential given the scattering amplitude. The solution to this problem is usually given in terms of linear integral equations : the Marchenko equation in 1D, and the Newton-Marchenko equation in 3D.

It will be shown that considering this problem starting from the Principle of Least Action leads to a second global minimum principle for Schrödinger’s equation. That is, I exhibit a functional of the scattering amplitude and a variational wavefield and show that the minimum of this functional with respect to variations of the wavefield yields a simple function from which the potential can be immediately extracted.