Bohr-Sommerfeld conditions for Integrable Systems with critical manifolds of focus-focus type

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dc.creatorSan, Vu Ngoc
dc.date1998-10-22
dc.date.accessioned2026-07-07T05:26:34Z
dc.date.available2026-07-07T05:26:34Z
dc.descriptionWe present a detailed study, in the semi-classical regime $h \to 0$, of microlocal properties of systems of two commuting h-PDO s $P_1(h)$, $P_2(h)$ such that the joint principal symbol $p=(p_1,p_2)$ has a special kind of singularity called a "focus-focus" singularity. Typical examples include the quantum spherical pendulum or the quantum Champagne bottle. In the spirit of Colin de Verdière and Parisse, we show that such systems have a universal behavior described by singular quantization conditions of Bohr-Sommerfeld type. These conditions are used to give a precise description of the joint spectrum of such systems, including the phenomenon of quantum monodromy and different formulations of the counting function for the joint eigenvalues close to the singularity, in which a logarithm of the semi-classical constant $h$ appears. Thanks to numerical computations done by M.S. Child for the case of the Champagne bottle, we are able to accurately illustrate our statements.
dc.description70 pages, 12 figures (prefer the .ps file) \usepackage{amsfonts,amssymb,euscript,a4,epsfig} preprint Institut Fourier/Utrecht Univ
dc.identifierhttps://arxiv.org/abs/math/9810137
dc.identifierhttp://arxiv.org/abs/math/9810137
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/77598
dc.subjectAnalysis of PDEs
dc.subjectMathematical Physics
dc.subjectSpectral Theory
dc.subject34C20; 34E20; 35P20; 57R70; 58F07; 81Q20
dc.titleBohr-Sommerfeld conditions for Integrable Systems with critical manifolds of focus-focus type
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