New methods in spectral theory of N -body Schrödinger operators

T. Adachi; K. Itakura; K. Ito; E. Skibsted

doi:10.1142/S0129055X2150015X

New methods in spectral theory of N -body Schrödinger operators

T. Adachi, K. Itakura, K. Ito, E. Skibsted

Institut for Matematik

Publikation: Bidrag til tidsskrift/Konferencebidrag i tidsskrift /Bidrag til avis › Tidsskriftartikel › Forskning › peer review

Abstract

We develop a new scheme of proofs for spectral theory of the N-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich's theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for L2-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with 'zeroth order' operators. In particular, they do not rely on Mourre's differential inequality technique.

Originalsprog	Engelsk
Artikelnummer	2150015
Tidsskrift	Reviews in Mathematical Physics
Vol/bind	33
Nummer	5
ISSN	0129-055X
DOI	https://doi.org/10.1142/S0129055X2150015X
Status	Udgivet - jun. 2021

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10.1142/S0129055X2150015X

https://arxiv.org/pdf/1804.07874.pdf

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abstract = "We develop a new scheme of proofs for spectral theory of the N-body Schr{\"o}dinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich's theorem, limiting absorption principle bounds, microlocal resolvent bounds, H{\"o}lder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for L2-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with 'zeroth order' operators. In particular, they do not rely on Mourre's differential inequality technique.",

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AU - Adachi, T.

AU - Itakura, K.

AU - Ito, K.

AU - Skibsted, E.

PY - 2021/6

Y1 - 2021/6

N2 - We develop a new scheme of proofs for spectral theory of the N-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich's theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for L2-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with 'zeroth order' operators. In particular, they do not rely on Mourre's differential inequality technique.

AB - We develop a new scheme of proofs for spectral theory of the N-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich's theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich's theorem which is unified with exponential decay estimates studied previously only for L2-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with 'zeroth order' operators. In particular, they do not rely on Mourre's differential inequality technique.

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New methods in spectral theory of N -body Schrödinger operators

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