Scattering theory for $C^2$ long-range potentials

We develop a complete stationary scattering theory for Schr\"odinger operators on $\mathbb R^d$, $d\ge 2$, with $C^2$ long-range potentials. This extends former results in the literature, in particular [Is1, Is2, II, GY], which all require a higher degree of smoothness. In this sense the spirit of our paper is similar to [H\"o2, Chapter XXX], which also develops a scattering theory under the $C^2$ condition, however being very different from ours. While the Agmon-H\"ormander theory is based on the Fourier transform, our theory is not and may be seen as more related to our previous approach to scattering theory on manifolds [IS1,IS2,IS3]. The $C^2$ regularity is natural in the Agmon-H\"ormander theory as well as in our theory, in fact probably being `optimal' in the Euclidean setting. We prove equivalence of the stationary and time-dependent theories by giving stationary representations of associated time-dependent wave operators. Furthermore we develop a related stationary scattering theory at fixed energy in terms of asymptotics of generalized eigenfunctions of minimal growth. A basic ingredient of our approach is a solution to the eikonal equation constructed from the geometric variational scheme of [CS]. Another key ingredient is strong radiation condition bounds for the limiting resolvents originating in [HS]. They improve formerly known ones [Is1, Sa] and considerably simplify the stationary approach. We obtain the bounds by a new commutator scheme whose elementary form allows a small degree of smoothness.