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SPECTRAL ANALYSIS OF N-BODY SCHRöDINGER OPERATORS AT TWO-CLU IBD

ERIK SKIBSTED / XUE PING WANG
SPRINGER
10 / 2024
9789819726233
Inglés

Sinopsis

This book provides a systematic study of spectral and scattering theory áfor many-body Schrödinger operators at two-cluster thresholds. While the átwo-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 áliterature on ámany-body threshold problems áis sparse. However, the authorsâÇÖ analysis covers for example the system of three particles áinteracting by Coulomb potentials and restricted to a small energy áregion to the right of a fixed nonzero two-body eigenvalue. In general, áthe authors address the question: How do scattering quantities for the ámany-body atomic and molecular models behave within the limit when the átotal energy approaches a fixed two-cluster threshold? This includes ámapping properties and singularities of the limiting scattering matrix, áasymptotics of the total scattering cross section, and absence of átransmission from one channel to another in the small inter-cluster ákinetic energy region. The authorsâÇÖ principal tools are the áFeshbach-Grushin dimension reduction method and spectral analysis based áon a certain Mourre estimate. Additional topics of independent interest áare the limiting absorption principle, micro-local resolvent estimates, áRellich- and Sommerfeld-type theorems and asymptotics of the limiting áresolvents at thresholds. The mathematical physics field under study is ávery rich, and there are many open problems, several of them stated áexplicitly in the book for the interested reader.á