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

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › 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.

Original language	English
Article number	2150015
Journal	Reviews in Mathematical Physics
Volume	33
Issue	5
ISSN	0129-055X
DOIs	https://doi.org/10.1142/S0129055X2150015X
Publication status	Published - Jun 2021

Keywords

minimal non-threshold generalized eigenfunctions
N -body Schrödinger operators

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10.1142/S0129055X2150015X

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

Cite this

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title = "New methods in spectral theory of N -body Schr{\"o}dinger operators",

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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T1 - New methods in spectral theory of N -body Schrödinger operators

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.

KW - minimal non-threshold generalized eigenfunctions

KW - N -body Schrödinger operators

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U2 - 10.1142/S0129055X2150015X

DO - 10.1142/S0129055X2150015X

M3 - Journal article

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VL - 33

JO - Reviews in Mathematical Physics

JF - Reviews in Mathematical Physics

IS - 5

M1 - 2150015

ER -

New methods in spectral theory of N -body Schrödinger operators

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Keywords

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