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On the Landau levels on the hyperbolic plane. (English) Zbl 0635.58034
The author considers the problem of the motion of a charged particle on a hyperbolic manifold in the presence of a constant uniform magnetic field. This problem is of relevance to both the study of geodesic motion and the relation between quantum and classical mechanical chaos. The configuration space is taken to be the Poincaré model of the hyperbolic plane. The presence of the symmetry group SL(2,\({\mathbb{R}})\) is fully exploited. A criterion for the trapping of particles in closed orbits is obtained. The quantization is done following De Witt’s method. The problem of a quantized magnetic field is dealt with in detail. The semiclassical limit is studied using the associated heat kernel. The ambiguity associated with operator ordering is resolved following Gutzwiller. The major tool in this part of the work is the BKW approximation. The resolvent operator yields the correct Landau levels in the limit of flat space. As the group SL(2,\({\mathbb{R}})\) is the symmetry group of the problem, many of the formulae appearing in the paper have been obtained by various investigators of automorphic forms.
Reviewer: Y.Prahalad

58Z05 Applications of global analysis to the sciences
58J35 Heat and other parabolic equation methods for PDEs on manifolds
53D50 Geometric quantization
Full Text: DOI
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