Theoretical Physics Department,

Tomsk State University,

36 Lenin ave.,

Tomsk,

634050, RUSSIA

and

Mathematical Physics Laboratory of the High Mathematics

and Mathematical Physics Department,

Tomsk Polytechnic University,

30 Lenin ave.,

Tomsk,

634050, RUSSIA

E-mail: shpv@phys.tsu.ru, trifonov@mph.phtd.tpu.edu.ru

**Exact Solutions and Symmetry Operators for the Nonlocal
Gross-Pitaevskii Equation with Quadratic Potential**

**Abstract:**

The nonlocal Gross-Pitaevskii equation (GPE) with quadratic potential
in $n$-dimensional space is written as

\begin{equation}

\left( -i\hbar\partial_{t} +\hat{\cal{H}}(\vec x,\hat\vec p,t)+\int_{\mathbb{R}^{n}}V(t,\vec
x,\vec y)|\Psi(\vec y, t)|^{2}d\vec y\right) \Psi(\vec x,t) =0,

\end{equation}

where $\vec x,\vec y\in\mathbb{R}^{n}$, $\hat{\vec p}=-i\hbar\nabla_{\vec
x}$, a linear operator $\hat{\cal H}$ and the nonlocal potential $V(t,\vec
x,\vec y)$ are Weyl ordered smooth functions of $\hat{\vec p}, \vec x$,
and $\vec y$. In the theory of Bose-Einstein condensate (BEC) Eq. (1) describes
the condensate states $\Psi$ taking into account the nonlocal interaction
between atoms. The nonlocal BEC models are more realistic, and the
nonlocal property can play the role of compensating factor to the collapse
effect of the BEC states. In [1] we develop a method of semiclassical solutions
for the Eq. (1) asymptotical in $\hbar$, $\hbar\to 0$, based on the WKB-Maslov
theory of the complex germ. In the present work we construct the {\it exact}
solution of the Cauchy problem for the Eq. (1) in the class of trajectory
concentrated functions (TCF) when the linear operator $\hat{\cal H}$ and
the nonlocal potential $V(t,\vec x,\vec y)$ are quadratic in $\hat{\vec
p}$, $\vec x$, and $\vec y$. The nonlinear evolution operator is obtained
in explicit form for the Eq. (1) in the class of TCF. With the help of
symmetry operators, families of exact solutions of the equation are 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.

[1] Belov V.V., Trifonov A.Yu. and Shapovalov A.V., The Trajectory-Coherent Approximation and the System of Moments for the Hartree Type Equation, IJMMS (USA), 2002, V.32, No 6, P.325-370.

The work has been supported in part by the Grant for Support of Russian Scientific Schools 1743.2003.2, and the Grant of the President of the Russian Federation YD-246.2003.02.

See abstract in PDF.