Complex germ method in fock space. II. Asymptotic solutions corresponding to finite-dimensional isotropic manifolds

Abstract

In an earlier paper [1], the authors obtained approximate solutions of second-quantized equations of the form

$$i\varepsilon \frac{{\partial \Phi }}{{\partial t}} = H\left( {\sqrt \varepsilon \hat \psi ^ + ,\sqrt \varepsilon \hat \psi ^ - } \right)\Phi$$

(φ is an element of a Fock space, and φ^± are creation and annihilation operators) in the limitɛ→0. The construction of these solutions was based on the expression of the operators φ^± in the form

$$\hat \psi ^ \pm = \frac{{Q \mp \varepsilon \delta /\delta Q}}{{\sqrt {2\varepsilon } }}$$

and on the application to the obtained infinite-dimensional analog of the Schrödinger equation of the complex germ method at a point. This gives asymptotic solutions in theQ representation that are concentrated at each fixed instant of time in the neighborhood of a point. In this paper, we consider and generalize to the infinite-dimensional case the complex germ method on a manifold. This gives asymptotic solutions in theQ representation that are concentrated in the neighborhood of certain surfaces that are the projections of isotropic manifolds in the phase space onto theQ plane. The corresponding asymptotic solutions in the Fock representation are constructed. Examples of constructed asymptotic solutions are approximate solutions of theN-particle Schrödinger and Liouville equations (N∼1/ɛ), and also quantum-field equations.