An integral representation on the path space for scattering length
Suppose v is a potential function. Consider a solution \(\phi_ k\) of the equation \(\Delta \phi_ k-v\phi_ k=-k^ 2\phi_ k\) such that asymptotically as \(| x| \to \infty\), and with \(e\in R^ 3\) a unit vector, and \(e'=x/| x|\), \[ \phi_ k(x)\sim e^{ik<u,e>}+f_ k(e,e')e^{ik<u,e'>}/| x|. \] The unique constant \(f_ k(e,e')\) is the scattering amplitude. The scattering length \(\Gamma\) is the limit of \(f_ k(e,e')\) as \(k\to 0\). If \(\phi_ 0(x)\) is the solution of \[ \phi_ 0(x)=1-(2\pi)^{-1}\int_{R^ 3}v(y)\phi (y)| x-y|^{-1}dy, \] M. Kac showed \(\Gamma =(2\pi)^{-1}\int_{R^ 3}v(x)\phi_ 0(x)dx\). By using an integral representation of \(\Gamma\) (v) on the path space \(C((-\infty,\infty),R^ 3)\) the author gives proofs of the following two conjectures of M. Kac: 1) \(\Gamma\) (\(\alpha\) v) has a limit as \(\alpha \to +\infty\); 2) the limit \(\gamma_ u\) is independent of the potential v and depends only on the support \(U=\{x: v(x)>0\}\).
scientific article; zbMATH DE number 4172108
scientific article