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

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

1998-10-22

arxiv.org/abs/math/9810137v1

Mathematics

Analysis of PDEs

70 pages, 12 figures (prefer the .ps file) \usepackage{amsfonts,amssymb,euscript,a4,epsfig} preprint Institut Fourier/Utrech

Scientific paper

We 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 Verdi\`ere 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.

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