A limiting absorption principle for Schr�dinger operators with Von Neumann-Wigner type potentials

Abstract

In this paper we prove the main step in establishing a limiting absorption principle for von Neumann-Wigner type Schrodinger Hamiltonians of the form −Δ+csinb|x|/|x|+V(x), whereV(x) is a short range potential. The first fundamental step is to obtain a limiting absorption principal for the “free” operator −Δ+csinb.|x|/|x|. The free operator is unitarily equivalent to a direct sum of ordinary differential operators. We obtain uniform estimates for the resolvents of these ordinary differential operators. by obtaining uniform estimates for the Weyl-Green kernels of these resolvents. In turn, these latter estimates require uniform estimates on the Wronskians of certain generalized eigen-solutions of these differential operators.