Singular Bohr–Sommerfeld rules for 2D integrable systems

Abstract

ABSTRACT. -This article gives Bohr-Sommerfeld rules for semiclassical completely integrable systems with two degrees of freedom with non-degenerate singularities (Morse-Bott singularities) under the assumption that the energy level of the first Hamiltonian is non-singular. The more singular case of focusfocus singularities was treated in previous works by San Vũ Ngo . c. The case of one degree of freedom has been studied by Colin de Verdière and Parisse.The results are applied to some famous examples: the geodesics of the ellipsoid, the 1 : 2-resonance, and Schrödinger operators on the sphere S 2 . A numerical test shows that the semiclassical Bohr-Sommerfeld rules match very accurately the "purely quantum" computations. Les résultats sont appliqués à quelques exemples célèbres : les géodésiques de l'ellipsoïde, la résonance 1 : 2 et les opérateurs de Schrödinger sur la sphère S 2 . Un test numérique montre que les règles de BohrSommerfeld semi-classiques sont en parfait accord avec les calculs "purement quantiques". 2003 Éditions scientifiques et médicales Elsevier SAS

Critical Loci for Higgs Bundles

“…The four umbilical points consist of two pairs of points ±u, ±v ∈ R 3 on the ellipse y = 0. A singular fibre of the integrable system is defined by (f 2 = 0, f 1 = const) and (see [25]) consists of the orbits of the geodesic flow on T * S 2 which project to geodesics passing through the umbilical points. These are closed geodesics and are given by two tori (one for the geodesics through ±u, one for ±v) and they intersect in two circles corresponding to the closed geodesic y = 0 joining u and v with the two orientations.…”

Qualitative Analysis of the Classical and Quantum Manakov Top

“…Both these problems have one-dimensional singular stratum in the image of the momentum map which is formed by points with inverse image being 'curled torus' [6,23]. This stratum is not passable in quantum version by an 'elementary cell' but it is passable by 'double cell'.…”

Integrable systems and group actions

“…Partial results in that direction (with dim L = 1 in a manifold of dimension 4) were obtained by Currás-Bosch and the author in [14] and independently by Colin de Verdière and Vũ Ngo . c in [12]. The final result in any dimension was obtained by Nguyen Tien Zung and the author in [56].…”