Complex germ method in the Fock space.~II. Asymptotics, corresponding to finite-dimensional isotropic manifolds

V. P. Maslov ; O. Yu. Shvedov

Teoretičeskaâ i matematičeskaâ fizika, Tome 104 (1995) no. 3, pp. 479-506

Résumé

Approximate as $\varepsilon \to 0$ solutions to secondary-quantized equations $$i\varepsilon \frac {\partial \Phi }{\partial t}=H(\sqrt {\varepsilon }\widehat {\psi }^+,\sqrt {\varepsilon }\widehat {\psi }^-)\Phi$$ where $\Phi$ is an element of the Fock space, $\widehat {\psi }^{\pm }$ are creation and annihilation operators in this space, were considered in the previous paper by the authors. Construction of this solutions was based on the presentation of the creation and annihilation operators in the form $$\widehat {\psi }^{\pm }=\frac {Q\mp \varepsilon \delta /\delta Q}{\sqrt {2\varepsilon }}$$ and application of the complex germ approach at a point to arrising infinite-dimensional Schrödinger equation. This approach gives asymptotics in $Q$-representation, which are concentrated in the vicinity of a point at a fixed time. In this paper we concider and generalize to the infinite-dimensional case the complex germ method in a manifold, which gives us asymptotics in $Q$-representation in the vicinty of some surfaces, which are projections of isotropic manifolds in the phase space to $Q$-plane. We construct corresponding asymptotics in the Fock representation. Examples of these asymptotics are approximate solutions to $N$-particle Schrödinger and Liouville equations as $N\sim 1/\varepsilon$ and quantum field theory equations.