Scattering theory for $C^2$ long-range potentials
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arXiv
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| Natura: | Preprint |
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2024
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| author | Ito, K. Skibsted, E. |
| author_facet | Ito, K. Skibsted, E. |
| contents | 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 [Is1, Is2, II, GY], which all require a higher degree of smoothness. In this sense the spirit of our paper is similar to [Hö2, Chapter XXX], which also develops 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, 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-Hörmander 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02979 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scattering theory for $C^2$ long-range potentials Ito, K. Skibsted, E. Mathematical Physics Analysis of PDEs Functional Analysis Spectral Theory 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 [Is1, Is2, II, GY], which all require a higher degree of smoothness. In this sense the spirit of our paper is similar to [Hö2, Chapter XXX], which also develops 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, 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-Hörmander 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. |
| title | Scattering theory for $C^2$ long-range potentials |
| topic | Mathematical Physics Analysis of PDEs Functional Analysis Spectral Theory |
| url | https://arxiv.org/abs/2408.02979 |