Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms
type: Item
instance of: scholarly article
title: Invariant tori, effective stability, and quasimodes with exponentially small error terms. II: Quantum Birkhoff normal forms
zbMATH Open document ID: 1002.37028
DOI: 10.1007/PL00001005
author: R. Smith
published in: Annales Henri Poincaré
publication date: 2000-12-04T00:00:00Z
review text: In the first part of this paper [cf. ibid. 1, No. 2, 223-248 (2000; Zbl 0970.37050)], the effective stability of quasi periodic orbits in nearly integrable classical Hamiltonian systems has been shown for a set \(\Lambda\) of invariant tori whose frequencies obey a certain Diophantine condition. Using this result, the authors obtain quasimodes for a Schrödinger type operator \(P_h\) with exponentially small error terms in the semiclassical limit by constructing a quantum Birkhoff normal form for this operator around \(\Lambda\). Applying this result, sharp lower bounds for the number of resonances of the operator are found. The theory is also applied to the case of discrete spectrum of \(P_h\).
reviewed by: Simos Ichtiaroglou
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 81Q15
Mathematics Subject Classification ID: 70H33
Mathematics Subject Classification ID: 70H08
zbMATH DE Number: 1540533
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: Schrödinger type operators
zbMATH Keywords: quantum Birkhoff normal forms
zbMATH Keywords: quasimodes
zbMATH Keywords: discrete spectrum
MaRDI profile type: Publication
full work available at URL: pl00001005
OpenAlex ID: W2082872605
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description: scientific article; zbMATH DE number 1540533