Quasi-energy spectral series and the Aharonov-Anandan phase for the nonlocal Gross--Pitaevsky equation

TL;DR

This paper constructs asymptotic solutions for a nonlocal Gross-Pitaevsky operator, analyzes its spectral properties, and calculates the Aharonov-Anandan phases for quasi-energy states, advancing understanding of quantum phase phenomena.

Contribution

It introduces a method to find asymptotic solutions and spectral series for the nonlocal Gross-Pitaevsky operator, including the calculation of geometric phases.

Findings

Constructed formal solutions up to O(ħ^{3/2})

Identified spectral series related to stable phase trajectories

Calculated Aharonov-Anandan phases for quasi-energy states

Abstract

For the nonlocal $T$-periodic Gross-Pitaevsky operator, formal solutions of the Floquet problem asymptotic in small parameter $\hbar$, $\hbar\to0$, up to $O(\hbar^{3/2})$ have been constructed. The quasi-energy spectral series found correspond to the closed phase trajectories of the Hamilton-Ehrenfest system which are stable in the linear approximation. The monodromy operator of this equation has been constructed to within $\hat O(\hbar^{3/2})$ in the class of trajectory-concentrated functions. The Aharonov-Anandan phases have been calculated for the quasi-energy states.