Bohr-Sommerfeld conditions for integrable systems with critical manifolds of focus-focus type
✍ Scribed by San Vũ Ngoc
- Publisher
- John Wiley and Sons
- Year
- 2000
- Tongue
- English
- Weight
- 684 KB
- Volume
- 53
- Category
- Article
- ISSN
- 0010-3640
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✦ Synopsis
We present a detailed study, in the semiclassical regime h → 0, of microlocal properties of systems of two commuting h-pseudodifferential operators P 1 (h) and 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 [11,12,13], we show that such systems have a universal behavior described by singular quantization conditions of the Bohr-Sommerfeld type, involving geometrical invariants of the associated singular Lagrangian foliation.
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 semiclassical constant h appears. Thanks to numerical computations done by M. S.