Berry phases for 3D Hartree type equations with a quadratic potential and a uniform magnetic field

TL;DR

This paper constructs localized solutions for 3D Hartree equations with quadratic potential and magnetic field, generalizes Berry phases for these nonlinear equations, and explicitly computes these phases.

Contribution

It introduces a method to find localized solutions and extends Berry phase concepts to nonlinear Hartree equations with explicit formulas.

Findings

Constructed asymptotic localized solutions using complex germ method

Generalized Berry phase concept for Hartree type equations

Explicit formulas for Berry phases in the solutions

Abstract

A countable set of asymptotic space -- localized solutions is constructed by the complex germ method in the adiabatic approximation for 3D Hartree type equations with a quadratic potential. The asymptotic parameter is 1/T, where $T\gg1$ is the adiabatic evolution time. A generalization of the Berry phase of the linear Schr\"odinger equation is formulated for the Hartree type equation. For the solutions constructed, the Berry phases are found in explicit form.