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

Abstract

We develop a complete stationary scattering theory for Schrödinger operators on ${\mathbb{R}}^d$, $d\ge 2$, with $C^2$ long-range potentials. This extends former results in the literature, in particular Isozaki (1980) and (1982), Ikebe and Isozaki (1982), and Gâtal and Yafaev (1999), which all require a higher degree of smoothness. In this sense, the spirit of our paper is similar to Hörmander [The Analysis of Linear Partial Differential Operators IV (1985), Chapter XXX] and J. Dereziński and C. Gérard [Scattering Theory of Classical and Quantum $N$-Particle Systems (1997), Section 4.7], which also develop a scattering theory under the $C^2$ condition, however being very different from ours. While the Agmon–Hörmander theory is based on the Fourier transform and a momentum-space representation, our theory is entirely position-space based and may be seen as more related to our previous approach to scattering theory on manifolds, Ito and Skibsted (2013), (2019), and (2021). The $C^2$ regularity is natural in the Agmon–Hörmander theory as well as in our theory, in fact probably being “optimal” in the Euclidean setting. We prove equivalence of the stationary scattering theory and a developed position-space based time-dependent scattering theory. 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 Cruz-Sampedro and Skibsted (2013). Another key ingredient is strong radiation condition bounds for the limiting resolvents originating in Herbst and Skibsted (1991). They improve formerly known ones by Isozaki (1980) and Saitō (1979) and considerably simplify the stationary approach. We obtain the bounds by a new commutator scheme whose elementary form allows a small degree of smoothness.