Integral Representations over Isotropic Submanifolds and Equations of Zero Curvature

✍ Scribed by Mikhail Karasev; Yuri Vorobjev

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 656 KB
• Volume: 135
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1703

✦ Synopsis

In the phase space over a Riemann manifold we consider a submanifold 4 invariant with respect to a Hamilton flow, isotropic (i.e., the form pdq| 4 is closed), and stable with respect to the first variation equation. For semiclassical wavefunctions of the quantum Hamiltonian we propose a very simple global ansatz with an oscillation front on 4. This ansatz has a form of an integral along 4 from Gaussian packets framed by an ``amplitude.'' The amplitude is a parallel section of a bundle of polynomials. The connection on this bundle is generated by a symplectic connection with zero curvature on the normal symplectic bundle over 4.

The coefficients of such a connection are calculated explicitly in terms of infinitesimal symmetries, and also in adiabatic approximation. We investigate topological and geometrical objects arising as corrections to the Poincare Cartan invariant in quantization rule on 4 and calculate spectral series for the quantum Hamiltonian.