Radiation condition bounds on manifolds with ends

K. Ito; E. Skibsted

doi:10.1016/j.jfa.2019.108449

Radiation condition bounds on manifolds with ends

^*Corresponding author for this work

Department of Mathematics

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Abstract

We study spectral theory for the Schrödinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Certain exterior domains for possibly unbounded obstacles are included. We prove Rellich's theorem, the limiting absorption principle, radiation condition bounds and the Sommerfeld uniqueness result, striving to extending and refining previously known spectral results on manifolds. The proofs are given by an extensive use of commutator arguments. These arguments have a classical spirit (essentially) not involving energy cutoffs or microlocal analysis and require, presumably, minimum regularity and decay properties of perturbations. This paper has interest of its own right, but it also serves as a basis for the stationary scattering theory developed fully in the sequel [19].

Original language	English
Article number	108449
Journal	Journal of Functional Analysis
Volume	278
Issue	9
ISSN	0022-1236
DOIs	https://doi.org/10.1016/j.jfa.2019.108449
Publication status	Published - 15 May 2020

Keywords

Commutator argument
Riemannian manifold
Schrödinger operator
Spectral theory

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10.1016/j.jfa.2019.108449

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title = "Radiation condition bounds on manifolds with ends",

abstract = "We study spectral theory for the Schr{\"o}dinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Certain exterior domains for possibly unbounded obstacles are included. We prove Rellich's theorem, the limiting absorption principle, radiation condition bounds and the Sommerfeld uniqueness result, striving to extending and refining previously known spectral results on manifolds. The proofs are given by an extensive use of commutator arguments. These arguments have a classical spirit (essentially) not involving energy cutoffs or microlocal analysis and require, presumably, minimum regularity and decay properties of perturbations. This paper has interest of its own right, but it also serves as a basis for the stationary scattering theory developed fully in the sequel [19].",

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AU - Skibsted, E.

PY - 2020/5/15

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N2 - We study spectral theory for the Schrödinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Certain exterior domains for possibly unbounded obstacles are included. We prove Rellich's theorem, the limiting absorption principle, radiation condition bounds and the Sommerfeld uniqueness result, striving to extending and refining previously known spectral results on manifolds. The proofs are given by an extensive use of commutator arguments. These arguments have a classical spirit (essentially) not involving energy cutoffs or microlocal analysis and require, presumably, minimum regularity and decay properties of perturbations. This paper has interest of its own right, but it also serves as a basis for the stationary scattering theory developed fully in the sequel [19].

AB - We study spectral theory for the Schrödinger operator on manifolds possessing an escape function. A particular class of examples are manifolds with Euclidean and/or hyperbolic ends. Certain exterior domains for possibly unbounded obstacles are included. We prove Rellich's theorem, the limiting absorption principle, radiation condition bounds and the Sommerfeld uniqueness result, striving to extending and refining previously known spectral results on manifolds. The proofs are given by an extensive use of commutator arguments. These arguments have a classical spirit (essentially) not involving energy cutoffs or microlocal analysis and require, presumably, minimum regularity and decay properties of perturbations. This paper has interest of its own right, but it also serves as a basis for the stationary scattering theory developed fully in the sequel [19].

KW - Commutator argument

KW - Riemannian manifold

KW - Schrödinger operator

KW - Spectral theory

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VL - 278

JO - Journal of Functional Analysis

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ER -

Radiation condition bounds on manifolds with ends

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Keywords

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