The evolution operator of the Hartree-type equation with a quadratic potential

TL;DR

This paper develops an explicit method to solve the Hartree-type equation with quadratic potential using semiclassical functions, deriving the evolution operator, symmetry operators, and geometric phases.

Contribution

It introduces a new approach based on Maslov's complex germ theory to find exact solutions and symmetry operators for the nonlinear Hartree-type equation.

Findings

Explicit form of the nonlinear evolution operator obtained.

Families of exact solutions constructed using symmetry operators.

Explicit expressions for quasi-energies and geometric phases derived.

Abstract

Based on the ideology of the Maslov's complex germ theory, a method has been developed for finding an exact solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically concentrated functions. The nonlinear evolution operator has been obtained in explicit form in the class of semiclassically concentrated functions. Parametric families of symmetry operators have been found for the Hartree-type equation. With the help of symmetry operators, families of exact solutions of the equation have been constructed. Exact expressions are obtained for the quasi-energies and their respective states. The Aharonov-Anandan geometric phases are found in explicit form for the quasi-energy states.