Spectral Analysis of N-Body Schrödinger Operators at Two-Cluster Thresholds (Mathematical Physics Studies) - Softcover

Skibsted, Erik; Wang, Xue Ping

Softcover

ISBN 10: 9819726263 ISBN 13: 9789819726264

Publisher: Springer, 2025

Synopsis

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. 

About this title

About the Author

Erik Skibsted is Associate Professor at Department of Mathematics of Aarhus University.

Xue Ping Wang is Professor at Université de Nantes.

From the Back Cover

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. 

Edition details

• Publisher: Springer
• Publication date: 2025
• Language: English
• ISBN 10: 9819726263
• ISBN 13: 9789819726264
• Binding: Paperback
• Number of pages: 267