Connections on Lagrangian submanifolds and some problems of quasi- classical approximation. I. (Russian) Zbl 0741.58056

Karasev, M. V.

Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 172, 41-54 (1989).

[For the entire collection see Zbl 0673.00016.]
The quasiclassical approximation in quantum mechanics (i.e. the limit \(\hbar\to 0\)) is investigated. First the author defines the operator \(K\) intertwining the representation of the Heisenberg submanifold \(\Lambda\) and the standard one in \(\mathbb{R}^ n\). The operator \(K\) is described as an integral operator with the Gaussian kernel realizing the representation of an \(U(1)\)-connection on \(\Lambda\) with vanishing curvature. The operator \(K\) is then used for the construction of Hamiltonian eigenfunctions in the quasiclassical approximation, provided that the generalized Bohr-Sommerfeld quantization conditions are satisfied on \(\Lambda\).
The proposed formalism extends the standard quasiclassical methods, e.g. the quantization of separatrics in the double-well potential was investigated, and the quasiclassical asymptotics of eigenfunctions in a strongly singular potential at the origin is found.

Reviewer: P.Prešnajder (Bratislava)

MSC:

58Z05

Applications of global analysis to the sciences

Keywords:

Lagrangian submanifold; Heisenberg algebra representation; zero-curvature connection; quantization condition

Citations:

Zbl 0673.00016