Symplectic Techniques for Semiclassical Completely Integrable Systems

This article is a survey of classical and quantum completely integrable systems from the viewpoint of local ``phase space'' analysis. It advocates the use of normal forms and shows how to get global information from glueing local pieces. Many crucial phenomena such as monodromy or eigenvalue concentration are shown to arise from the presence of non-degenerate critical points.Comment: 32 pages, 7 figures. Review articl

Moment polytopes for symplectic manifolds with monodromy

A natural way of generalising Hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map. For a class of such ``almost-toric 4-manifolds'' which admits a Hamiltonian $S^1$-action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application, we derive a Duistermaat-Heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system.Comment: finally a revision of the 2003 preprint. 29 pages, 8 figure

Spectral asymptotics via the semiclassical Birkhoff normal form

This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters (E,\h), and gives improvements and new proofs for many of the results in the field. In the completely resonant case we show that the pseudo-differential operator can be reduced to a Toeplitz operator on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved. In the case of polynomial differential operators, a combinatorial trace formula is obtained.Comment: 44 pages, 2 figure

Symplectic inverse spectral theory for pseudodifferential operators

23 pagesWe prove, under some generic assumptions, that the semiclassical spectrum modulo O(h²) of a one dimensional pseudodifferential operator completely determines the symplectic geometry of the underlying classical system. In particular, the spectrum determines the hamiltonian dynamics of the principal symbol

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

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