Spectral Analysis of N-Body Schrödinger Operators at Two-Cluster Thresholds

Erik Skibsted, Xue Ping Wang

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English

Springer Nature
04 July 2024

Functional analysis & transforms; Mathematical physics

Series: Mathematical Physics Studies

This book provides a systematic study of spectral and scattering theory for many-body Schrödinger operators at two-cluster thresholds. While the two-body problem (reduced after separation of the centre of mass motion to a one-body problem at zero energy) is a well-studied subject, the literature on many-body threshold problems is sparse. However, the authors’ analysis covers for example the system of three particles interacting by Coulomb potentials and restricted to a small energy region to the right of a fixed nonzero two-body eigenvalue. In general, the authors address the question: How do scattering quantities for the many-body atomic and molecular models behave within the limit when the total energy approaches a fixed two-cluster threshold? This includes mapping properties and singularities of the limiting scattering matrix, asymptotics of the total scattering cross section, and absence of transmission from one channel to another in the small inter-cluster kinetic energy region. The authors’ principal tools are the Feshbach–Grushin dimension reduction method and spectral analysis based on a certain Mourre estimate. Additional topics of independent interest are the limiting absorption principle, micro-local resolvent estimates, Rellich- and Sommerfeld-type theorems and asymptotics of the limiting resolvents at thresholds. The mathematical physics field under study is very rich, and there are many open problems, several of them stated explicitly in the book for the interested reader.

By: Erik Skibsted, Xue Ping Wang
Imprint: Springer Nature
Country of Publication: Singapore
Edition: 2024 ed.
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9789819726233
ISBN 10: 9819726239
Series: Mathematical Physics Studies
Pages: 258
Publication Date: 04 July 2024
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Introduction.- Many-Body Schrödinger Operators, Conditions and Notation.- Reduction to a One-Body Problem.- Spectral Analysis of H0 near 0.- Rellich-Type Theorems.- Resolvent Asymptotics near a Two-Cluster Threshold.- Elastic Scattering at a Threshold.- Non-Transmission at a Threshold for Physical Models.- Threshold Behaviour of Cross-Sections in Atom–Ion Scattering.

Erik Skibsted is Associate Professor at Department of Mathematics of Aarhus University. Xue Ping Wang is Professor at Université de Nantes.