Spectral Representation for Schrödinger Operators with Long-Range Potentials, II — Perturbation by Short-Range Potentials

In our previous work (Ikebe [1]) we have obtained a spectral representation for the Schrodinger operator — A + V(x) with a purely long-range, real-valued potential F(x) acting in the Euclidean three-space U. That is, we have assumed that V(x) = O(\x\~~), grad V(x) = O(\x\-l~) and AV(ra)) = 0(r~) for |x| = r->oo, 0, where A is the negative Laplace-Beltrami operator acting on the aungular variable CDEQ = {xeR\ |x| = l}. In [1] we have pointed out that the introduction of a short-range perturbation Vs(x) = O(\x\~~ ~) is not trivial, and cursorily indicated how to handle the matter. The purpose of the present paper is to show that this is actually possible. What we have done in [1] is roughly as follows. Let H be the self-adjoint realization in the Hilbert space H=L2(R ) (see below) of the Schrodinger operator T= —A + V(x) with a long-range potential V(x) in R, and let E be the associated spectral measure. (Although we have treated in [1] the case n = 3, the dimension of the underlying space is no essential restriction.) For y e f f = I2 and GczH" let