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\title{\Large \bf Notes on the nonlinear nonlocal\\ "Schr\"odinger equation"
\cite{Fil}}
\author{{\large \bf Alex A. Samoletov}\thanks{E-mail:
samolet@host.dipt.donetsk.ua}\\
{\small \sl Institute for Physics and Technology, Natl. Acad. Sci. Ukraine,
340114 Donetsk, Ukraine}}
\date{}
\maketitle
\vspace{10mm}
\hrule
\vspace{3mm}
\noindent{\bf Abstract}\\
It is shown that the nonlinear nonlocal equation \cite{Fil} recently
numerically studied in the context of quantum mechanics by Filippov
is in fact a modification of the quantum-mechanical variational
principle. It is shown that the result of Letter \cite{Fil}
admits clear analytical reading.
\vspace{3mm}
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\vfill
\newpage
{\bf 0.} In a recent Letter \cite{Fil} Filippov has presented a numerical
study of nonlinear nonlocal "Schr\"odinger equation" in the context of
quantum mechanics. In the present letter I intend to point out first, that
in fact this equation is the result of a modification (generalization) of the
quantum-mechanical variational principle, and, second, that there is clear
analytical reading of the results of Letter \cite{Fil}.
I wish to express gratitude to A.Filippov for the possibility to read the
manuscript of the Letter \cite{Fil} before a publication as well as for
demonstration of the results of numerical simulations.
The present letter to be considered as a continuation of Letter \cite{Fil}.
\vspace{3mm}
{\bf 1.} To begin with, I wish to make some preliminary remarks about the
known facts \cite{Gar}, which are needed to the following.
Let us consider the Fokker-Planck equation for a scalar field $\psi(x)$
(generally, $x$ take on values in ${\bf R}^d$, but, for the sake of brevity,
the one-dimensional notation is taken)
\begin{equation}
\label{(1)}
\frac{\partial}{\partial t} P[\psi] = {\int} dx \frac{\delta}{\delta \psi(x)}
\Biggl[ \Biggl( \frac{\delta F[\psi]}{\delta \psi(x)} +
\frac{\delta}{\delta \psi(x)} \Biggr) P[\psi] \Biggr],
\end{equation}
where $t \in [0,\infty)$ is a (fictial) time. This equation has equilibrium
solution of the form
\begin{equation}
\label{(2)}
P_{\infty} \sim \exp (-F[\psi]).
\end{equation}
The functional $P_{\infty}$ reaches its maximum value at a set of function
which is associated with minimum of the functional $F[\psi]$.
Stochastic equation, which is stochastically equivalent to the Eq.(1), has
the form
\begin{equation}
\label{(3)}
\frac{\partial}{\partial t}\psi(x,t) =
-\frac{\delta F[\psi]}{\delta \psi} + \xi(x,t),
\end{equation}
where $\xi(x,t)$ is the standard $\delta$-correlated Gaussian field with
the following averages
$$
<\xi(x,t)> = 0, \qquad <\xi(x,t)\xi(x^\prime,t^\prime)> =
2\delta(x-x^\prime) \delta(t-t^\prime).
$$
Eq.(1) with equilibrium solution (2) to be obtained from stochastic Eq.(3)
as a result of averaging over all realizations of the random field $\xi(x,t)$.
If we have a unique realization of the field $\xi(x,t)$, {\it as this
happens to be the case for any numerical simulation}, and under a restriction
on the functional $F$, then the system to be relaxed to the set of function
on which a local minimum of the functional $F$ has been realized. And then
a formation of the equilibrium solution $P_\infty$ to be started as the
result of averaging in time. Next, after a sufficiently large time $t$
and on account of unlimited variation of the random field $\xi(x,t)$,
the field $\psi(x,t)$ and a value of the functional $F$ to be hustled away
the minimum of $F$. Next, once again, the system (field $\psi$) to be
relaxed, and so on, and so forth. Solution $P_\infty$ will be formed at
$t \to \infty$.
Thus, it is clear what the qualitative behavior of Eq.(3) solution
to be expected. And we are now in a position to state and to solve the
basic problem of the letter: Construction of a functional $F$ which is needed
to solution of Schr\"odinger equation.
Further we will consider only those systems whose evolution is given by the
stochastic equation of the form (3).
\vspace{3mm}
{\bf 2.}Let $H$ be a selfadjoint operator (Hamiltonian) and $H$ is defined
in Hilbert space $\cal H$, and let $\cal H$ is realized as
${\cal L}^2 ({\bf R}^d); \quad \psi \in {\cal L}^2 ({\bf R}^d)$.
Let the quadratic form $(\psi, H \psi)$, where $(\cdot, \cdot)$ denote the
scalar product in ${\cal H}$, be bounded below:
$(\psi, H \psi) \geq \gamma (\psi,\psi)$. Then \cite{Hal} the numerical
range $\Theta (H)$ of $H$, otherwise speaking, a range of real numbers
$(\psi, H \psi)/(\psi,\psi)$, be also bounded below. Closure of $\Theta (H)$
is the same as the convex hull of operator $H$ spectrum $\Lambda(H)$
\cite{Hal} and the greatest lower bound of $\Theta(H)$
coinsides with the greatest lower bound of $\Lambda(H)$
which is a ground state energy.
Consider now the eigenvalue problem: $H\psi = E\psi, \quad (\psi,\psi) =
||\psi||^2 = 1$. As is well known, this problem is the same as the variational
principle for functional
$$
J[\psi] = (\psi,H\psi) + E {||\psi||}^2.
$$
Thus, if we choose functional $F[\psi]$ of Eqs.(1)-(3) in the form $J[\psi]$,
then the functional $P_{\infty}[\psi]$ has a minimum at the ground state and
Eq.(3) has a unique attractor solution in case where $H$ has a nondegenerate
ground state.
In order that all eigenstates of $H$ be search out, it is
necessary to shift
energy (numerical range of $H$) in positive direction of real axis. It is
clear that this shift must be dependent on "time" $t$ and functionally on
$\psi$. Then, for each of $\psi(t)$ we shall have as a local attractive
solution of Eq.(3) a lowest eigenstate depending on the value $||\psi||^2$
at the "time" $t$.
In a set of admissible shifts of the form $H \to H + S[\psi] \cdot I$,
where $S[\psi] > 0$ for all nonzero $\psi \in \cal H$, it is necessary to
select a subset such that the functional $S[\psi]$ has unique global minimum.
It is obvious that simplest form of $S[\psi]$ is the following
$$
S[\psi] = \alpha ||\psi||^2 , \qquad \alpha > 0.
$$
In the case when $S[\psi]$ has more than one local minimum the evolution of
$\psi(t)$ to be assumed an additional structure: a competition between the
attracting character of local minima of the functional $||\psi||^2 S[\psi]$
must be arisen (see Refs. in \cite{Fil}). Discussion of these
interesting
cases must be down separately.
Thus, if our purpose is to search out all eigenstates of $H$ as local
attractors for Eq.(3) then it is reasonable to take functional $F[\psi]$
in its simplest form
\begin{equation}
\label{(4)}
F[\psi] = (\psi,H \psi) + \lambda ||\psi||^2 + \alpha ||\psi||^4.
\end{equation}
Since the field $\xi (x,t)$ has unlimited variation (in any sense)
it follows that $||\psi||^2$ can takes on arbitrary values in positive
semiaxis. Hence, all eigenstates of $H$ are attainable as local attractors.
It remains only to verify that $F[\psi]$ Eq.(4) is the functional which is
used in Letter \cite{Fil}. In fact, it is easily verified directly, when
$H$ is the Schr\"odinger operator: $H=-\frac{1}{2}\Delta + V(x)$. Further
discussion and simulation of solutions of Eq.(3) given in Letter \cite{Fil}
remain true.
\vspace{5mm}
{\bf 3.} As a result we have
\begin{enumerate}
\item {Method for a solution of eigenstate problems for Schr\"odinger
operators of
of \cite{Fil} is the generalization (modification) of the quantum-mechanical
variational principle.}
\item A fictitious time is used in this generalization, so there is no reason
for corresponding equation to be referred to as the Schr\"odinger equation
with
the real time.
\item In applications the method of \cite{Fil} can be of the same importance
as the variational principle.
\item The time-dependent Ginzburg-Landau equation of a particular form (with
a complicated $S[\psi]$ is the case of the particular importance.
\end{enumerate}
\vspace{3mm}
\begin{thebibliography}{3}
\bibitem{Fil} A.E. Filippov, Nonlinear nonlocal Schr\"odinger equation in the
context of quantum mechanics.- Phys.Lett.{\bf A} (to be published)
\bibitem{Gar} C.W. Gardiner, Handbook of stochastic methods (Springer,
Berlin, 1985).
\bibitem{Hal} P.R. Halmos, A Hilbert space problem book (Van Nostrand,
Princeton, 1976).
\end{thebibliography}
\end{document}