Phase shifts and the modified and Møller wave operatos (Q788211)

This is an announcement of the author's results which have appeared subsequently in Trans. Am. Math. Soc. 275, 641-677 (1983). The results concern spectral and scattering theory for Schrödinger operators, \(H=- \Delta +V\) with long range central potentials V. Typical of the potentials considered are \((*)\quad V(x)=(1+r)^{- \delta}\sin(r^{\alpha})\) where \(\alpha\geq 0\), \(\delta>\min(0,1-\alpha)\) and \(r=| x|\). The spectral results for H are: essential self adjointness on \(C_ 0^{\infty}({\mathbb{R}}^ n)\), absence of singularly continuous spectrum, unitary equivalence of the continuous part of H to - \(\Delta\), and an eigenfunction expansion for H. The possibility of positive eigenvalues is also considered. We state the scattering theory results only for V as in (*): the (Dollard) modified wave operators exist and are complete if \(\delta>1/2\) or \((\delta>1/4\) and \(\delta +\alpha>1)\) or \((\delta +\alpha>5/4)\); the usual wave operators exist (and are complete) if and only if \((\delta +\alpha>1\) and \(\delta>1/2)\) or \((\delta +\alpha)>3/2\). The role of the eigenfunction expansion for H in establishing the scattering theoretical results is briefly discussed.