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\begin{document}
\title{One-dimensional Anderson Localization: distribution of wavefunction amplitude
and phase at the band center.}
\author{V. E. Kravtsov}{address={The Abdus Salam International Centre for
Theoretical Physics, P.O.B. 586, 34100 Trieste, Italy.\\
$^{2}$Landau Institute for Theoretical Physics, 2 Kosygina
st.,117940 Moscow, Russia}} 
\author{V. I. Yudson}{address={Institute for Spectroscopy,
Russian Academy of Sciences, 142190 Troitsk, Moscow reg., Russia.}}
\classification{72.15.Rn, 72.70.+m, 72.20.Ht, 73.23.-b}
\keywords{localization, mesoscopic fluctuations}
%\date{today}
\begin{abstract}
The statistics of normalized wavefunctions in the one-dimensional
(1d) Anderson model of localization is considered. It is shown that
at any energy that corresponds to a rational filling factor
$f=\frac{p}{q}$ there is a statistical anomaly which is seen in
expansion of the generating function (GF) to the order $q-2$ in the
disorder parameter. We study in detail the principle anomaly at
$f=\frac{1}{2}$ that appears in the leading order. The
transfer-matrix equation of the Fokker-Planck type with a
two-dimensional internal space is derived for GF. It is shown that
the zero-mode variant of this equation is integrable and a solution
for the generating function is found in the thermodynamic limit.
\end{abstract}

 \maketitle
\section{Introduction} Anderson localization (AL) enjoys an unusual
fate of being a subject of advanced research during a half of
century. The seminal paper by P.W.Anderson \cite{Anderson} opened up
a direction of research on the interplay of quantum mechanics and
disorder which is of fundamental interest up to now
\cite{Mirlin2008}. The one-dimensional tight-binding model with
diagonal disorder --the Anderson model (AM)-- which is the simplest
and the most studied model of this type, became a paradigm of AL:
\begin{equation}
\label{Ham}
\hat{H}=\sum_{i}\varepsilon_{i}\,c^{\dagger}_{i}c_{i}-\sum_{i}t_{i}\left(\,c^{\dagger}_{i}c_{i+1}+
c^{\dagger}_{i+1}c_{i}\right).
\end{equation}
In this model the hopping integral is deterministic $t_{i}=t$ and
the on-site energy $\varepsilon_{i}$ is a random Gaussian variable
uncorrelated at different sites and characterized by the variance
$\langle(\delta\varepsilon_{i})^{2}\rangle=w$.
%distribution function:
%\begin{equation}
%\label{Gauss} {\cal P}(\varepsilon_{i})=\frac{1}{\sqrt{2\pi
%w}}\,{\rm exp}\left[{-\frac{\varepsilon_{i}^{2}}{2w}}\right].
%\end{equation}
The dimensionless parameter $\alpha^{2}=w/t^{2}$ describes the
strength of disorder.

The best studied is the continuous limit of this model in which the
lattice constant $a\rightarrow 0$ at $ta^{2}$ remaining finite
\cite{Ber, AR, Mel, Kolok}. There was also a great deal of activity
\cite{1dRev, Pastur} aimed at a rigorous mathematical description of
1d AL. However, despite considerable efforts invested, some subtle
issues concerning 1d AM still remain unsolved. One of them is the
effects of commensurability between the de-Broglie wavelength
$\lambda_{E}$ (which depends on the energy $E$) and the lattice
constant $a$. The parameter that controls the commensurability
effects is the filling factor $f=\frac{2a}{\lambda_{E}}$ (fraction
of states below the energy $E$).

It was known for quite a while \cite{Wegner, Derrida} that the
Lyapunov exponent takes anomalous values at the filling factors
equal to $\frac{1}{2}$ and $\frac{1}{3}$ (compared to those at
filling factors $f$ beyond the window of the size $\alpha^{2}\ll 1$
around $f=\frac{1}{2}$ and $f=\frac{1}{3}$).   At weak disorder the
Lyapunov exponent sharply {\it decreases} at $f=\frac{1}{2}$ (which
is usually associated with {\it increasing} the localization
length). However, near $f=\frac{1}{3}$ the Lyapunov exponent
exhibits a sharp {\it peak} if the third moment of the on-site
energy distribution is non-zero \cite{Derrida}.
 More recently \cite{Titov, AL}
it was found that the statistics of conductance in 1d AM is
anomalous at the center of the band that corresponds to the filling
factor $f=\frac{1}{2}$. We want to stress that all these anomalies
were observed for the AM Eq.(\ref{Ham}) in which the on-site energy
$\varepsilon_{i}$ is random. This Hamiltonian does not possess the
{\it chiral symmetry} \cite{Dyson, Mirlin2008} which is behind the
statistical anomalies at the center of the band $E=0$ in the {\it
Lifshitz model} described by Eq.(\ref{Ham}) with the deterministic
$\varepsilon_{i}=0$ and a random hopping integral $t_{i}$. Thus the
statistical anomaly at $f=\frac{1}{2}$ raises a question about a
{\it hidden symmetry} that do not merely reduce to the
two-sublattice division \cite{Dyson, AL, Mirlin2008}.

The above results point out to existence of the entire {\it devil's
staircase} of statistical anomalies at a rational filling factor $f$
in different physical quantities of 1d Anderson model. There are
numerous questions concerning physics behind the anomalies. One of
such puzzles is the sign of the variation of the Lyapunov exponent
which corresponds to suppression of localization in the vicinity of
$f=\frac{1}{2}$ and to enhancement of localization near
$f=\frac{1}{3}$ (provided that there is an asymmetry of the on-site
energy distribution). It is difficult to explain such behavior by a
conventional model invoking reflection off Bragg mirrors with
doubling or tripling the lattice period due to peculiar fluctuations
of the random landscape of $\varepsilon_{i}$ \cite{Shapiro}, as this
model seems to favor localization in all cases.

A possible resolution of this conflict between physical intuition
and mathematical results is that the Lyapunov exponent describes
only the tails of the localized wavefunction while the global
picture of localization is  better represented by the average {\it
inverse participation ratio} $I=\int dx\;\langle
|\psi(x)|^{4}\rangle$, where $\psi(x)$ is a random {\it normalized}
eigenfunction obeying the Shroedinger equation
$\hat{H}\psi=E_{n}\,\psi$ and the boundary conditions on both ends
of the chain. Here it is worth noting that computing of $I$ is much
more difficult than the problem of Lyapunov exponent, since the
latter does not require an {\it eigenfunction} of  the Shroedinger
operator obeying all the boundary conditions, while the inverse
participation ratio is defined only for {\it normalizible
eigenfunctions}.

In this paper we give a regular description of the phenomenon of the
rational-$f$ statistical anomalies in terms of the generalized
transfer-matrix equation (TME) which is a universal tool to describe
properties of a generic 1d or quasi-1d system. As the result we
obtain the {\it joint probability distribution} (JPD) ${\cal
P}(u,\phi)$ of the amplitude $u$ and phase $\phi$  of the random
eigenfunction $\psi\sim \sqrt{u}\,\cos\phi$ which can be used to
compute {\it any} local statistics of {\it normalized}
eigenfunctions of 1d AM.

We will show that the TME for $f=\frac{1}{2}$ has anomalous terms
which make it essentially two-dimensional second-order PDE depending
on the amplitude variable $u$ as well as on the phase variable
$\phi$. Yet, we show that this equation has an exact solution for
the zero mode and we find this solution explicitly in quadratures.
Similar anomalous terms are shown to appear at any other rational
filling factor $f=\frac{p}{q}$.










%{\it generating function} (GF) $\Phi(u,\phi;x)$ that allows to
%compute {\it all} local statistical properties of 1d AM. The
%simplest of them is the statistics of the wavefunction amplitude
%$|\psi(x)|^{2}$ characterized by the moments $I_{m}=\langle
%|\psi(x)|^{2m}\rangle\,\ell_{0}^{m}$:
%\begin{eqnarray}
%\label{moment1}
%I_{m}&=&\frac{2}{(m-2)!}\,\int_{0}^{\pi}\frac{d\phi}{\pi}\,
%\cos^{2m}\phi\,\\
%\nonumber&\times&\int_{0}^{\infty}du\,u^{m-2}\,\Phi(u,\phi;x)\,
%\Phi(u,-\phi-2\pi f;L-x),
%\end{eqnarray}
%where $L$ is the total length of the system and
%$\ell_{0}=a\frac{2t^{2}}{w}\,\sin^{2}(\pi f)$ is the "bare"
%localization length.

%We  derive the corresponding transfer-matrix equation (TME) for the
%GF
%\begin{equation}
%\label{TME}
%\partial_{x} \Phi(u,\phi;x)=[\hat{L}_{f}(u,\phi)-u]\,\Phi(u,\phi;x),
%\end{equation}
%which {\it zero-mode} variant ($\Phi(u,\phi;x)\equiv \Phi(u,\phi)$
%is independent of the space coordinate $x$) appears to be a partial
%differential equation (PDE) depending on {\it two} variables. One of
%them (denoted by $u$)is associated with the amplitude of the
%wavefunction $\psi\sim \sqrt{u} \cos\phi$ while the other (denoted
%by $\phi$)has a physical meaning of its phase. The well known
%\cite{Kolok} TME in the continuous limit $f\ll 1$ can be obtained by
%the averaging of this PDE over the phase variable $\phi$ thus
%reducing it to an ODE in the {\it single} variable $u$.

%We  show that there are statistical anomalies at {\it any} rational
%filling factor $f=\frac{p}{q}$. Namely, the operator
%$\hat{L}_{f}(u,\phi)$ in Eq.(\ref{TME}) expanded in the disorder
%parameter $\alpha^{2}$
%\begin{equation}
%\label{expan}
%\hat{L}_{f}(u,\phi)=\hat{L}_{f}^{(0)}(u,\phi)+\alpha^{2}\,\hat{L}_{f}^{(1)}(u,\phi)+
%\alpha^{4}\,\hat{L}_{f}^{(2)}(u,\phi)+...
%\end{equation}
%is such that
%\begin{equation}
%\label{deltaL} \hat{L}_{f}^{(n)}=\hat{L}_{f}^{(n,{\rm
%reg})}+\sum_{p=1}^{n+1}\Delta\hat{L}^{(n)}_{p}\,\delta\left(f,\frac{p}{n+2}\right)
%\end{equation}
%contains a regular part $\hat{L}_{f}^{(n,{\rm reg})}$ with a smooth
%dependence on $f$ and an anomalous part that appears only at
%$f=\frac{1}{2+n},\frac{2}{2+n},...\frac{n+1}{2+n}$. In the leading
%order $(n=0)$ in $\alpha^{2}$ the anomalous term appears only at
%$f=\frac{1}{2}$. In the next order one can observe anomalies at
%$f=\frac{1}{3}$ and $\frac{2}{3}$, etc. Though anomalous terms
%corresponding to the denominator $q>2$ are small at weak disorder,
%they have an abrupt dependence on $f$. This allows to speak about
%the "devil's staircase of anomalies".

%We study in detail the principal anomaly at $f=\frac{1}{2}$.
%Remarkably, the corresponding zero-mode TME appears to be {\it
%integrable}. We find a unique solution to this equation which
%describes any local statistics of wavefunctions in the center of the
%band.


\section{Derivation of the TM equation.} 
The starting point of our
analysis is the TM equation for the generating function (GF)
$\Phi_{j}(u,\phi)$ on the lattice site $j$:
\begin{equation}
\label{op}
\Phi_{j+1}(u,\phi)=\left(1+\frac{2a}{\ell_{0}}\,\left[{\cal L
}(u,\phi)-c_{1}(\phi)\,u\right]\right)\,\Phi_{j}(u,\phi-\pi f),
\end{equation}
where  $\ell_{0}=a\frac{2t^{2}}{w}\,\sin^{2}(\pi f)$ is the "bare"
localization length; in the limit of weak disorder ${\cal
L}(u,\phi)=c_{2}(\phi)\,u^{2}\partial^{2}_{u}+c_{3}(\phi)\,(u\partial_{u}-1)+c_{4}(\phi)\,
u\partial_{u}\partial_{\phi}+c_{5}(\phi)\,\partial_{\phi}+c_{6}(\phi)\,\partial^{2}_{\phi}$.
The coefficients $c_{i}(\phi)$ are all combinations of $\cos(2\phi)$
and $\sin(2\phi)$ which at first glance do not show any nice
structure: $c_{1}(\phi)=\frac{1}{2}(1+\cos(2\phi))$,
$c_{2}(\phi)=1-\cos^{2}(2\phi)$,
$c_{3}(\phi)=-(1-\cos(2\phi)-2\cos^{2}(2\phi))$,
$c_{4}(\phi)=\sin(2\phi)(1+\cos(2\phi))$,
$c_{5}(\phi)=-\frac{3}{2}\sin(2\phi)(1+\cos(2\phi))$,
$c_{6}(\phi)=\frac{1}{4}(1+\cos(2\phi))^{2}$. This equation has been
derived in Ref.\cite{OsK} by expansion to the first order in
$\alpha^{2}$ of the exact integral TME  obtained by the
super-symmetry method \cite{Efet-book}.

The  GF $\Phi(u,\phi)$ determines the JPD ${\cal P}(u,\phi)$ of
eigenfunction amplitude and phase {\it in the bulk} in a long chain
${\cal P}(u,\phi)$. However the relationship between them is
non-trivial:
\begin{equation}
\label{JPD} {\cal P}(u,\phi)=-\frac{1}{2\pi
i\,u}\,\partial_{u}\int_{-i\infty+0}^{+i\infty+0}
\frac{dy}{y}\,e^{y}\, \Phi^{2}(uy,\phi).
\end{equation}
The main feature of Eq.(\ref{JPD}) is that it is {\it quadratic} in
$\Phi(u,\phi)$. This reflects the identical boundary conditions at
the {\it two} ends of a chain \cite{Mirlin2000} and the fact that
the point of observation is in the bulk. In contrast to that in the
problem of Lyapunov exponent one considers essentially a {\it
semi-infinite} chain with points of observation close to its end. In
this case the GF  itself plays a role of the distribution function.

  By construction \cite{OsK} the
function $\Phi_{j}(u,\phi)$ must be periodic in $\phi$ with the
period of $\pi$ which corresponds to the phase factor $\cos\phi$ of
the wave function sweeping all possible values in the interval
$[0,\pi]$. However, the shift in the argument $\phi$ in the r.h.s.
of Eq.(\ref{op}) is by a {\it fraction} $f$ of $\pi$. For a rational
$f=\frac{p}{q}$ one has to make $q$ iterations in Eq.(\ref{op}) in
order to get a closed equation for the GF. In the leading order in
$\alpha$ we obtain:
\begin{eqnarray}
\label{lin} &&\Phi_{j+q}(u,\phi)-\Phi_{j}(u,\phi)
=\frac{2a}{\ell_{0}}\\ &\times&\left[\sum_{r=0}^{q-1}{\cal
L}(\phi-r\,\pi p/q)-u\sum_{r=0}^{q-1}c_{1}(\phi-r\,\pi p/q)
\right]\,\Phi_{j}(u,\phi).\nonumber
\end{eqnarray}
%A remarkable identity that gives rise to the anomaly is the
%following:
The reason for the principle anomaly at $f=\frac{1}{2}$  is the
following identity that shows a jump at $q=2$:
\begin{equation}
\label{id1} \sum_{r=0}^{q-1}e^{2i\phi-2i r\,\pi p/q}=0,\;\;
\sum_{r=0}^{q-1}e^{4i\phi-4i r\,\pi p/q }=\left\{\begin{matrix}0, &
q>2\cr 2e^{4i\phi},& q=2\cr
\end{matrix} \right.
\end{equation}
%One can see from this identity that for $q>2$ the summation in
%Eq.(\ref{lin}) is the same as averaging over $\phi$: all the
%$\phi$-dependent terms vanish in both cases.
Assuming $q\ll
\ell_{0}/a$, expanding the l.h.s. of Eq.(\ref{lin}) and introducing
the dimensionless  coordinate $x=ja/\ell_{0}$ we obtain:
\begin{equation}
\label{ord}
\partial_{x}\Phi=\left[u^{2}\partial^{2}_{u}-u+\frac{3}{4}\partial^{2}_{\phi} \right]
\,\Phi + \delta_{f,\frac{1}{2}}\;\Delta{\cal L}(u,\phi)\,\Phi.
\end{equation}
The second term in Eq.(\ref{ord}) is the anomaly that is present
only for the filling factor $f=\frac{1}{2}$. The corresponding
operator takes the form:
\begin{eqnarray}
\label{del-2} \Delta {\cal
L}&=&\cos(4\phi)\,\left[-u^{2}\partial^{2}_{u}+2u\partial_{u}
+\frac{1}{4}\partial^{2}_{\phi}-2 \right]\nonumber \\
&+&\sin(4\phi)\,\left[u\partial_{u}\partial_{\phi}-\frac{3}{2}\partial_{\phi}
\right].
\end{eqnarray}
Without this part, the variables $u$ and $\phi$ are separated and
one can immediately find the independent of $x$ solution
 $\Phi(u)=\sqrt{u}\,K_{1}(2\sqrt{u})$. This {\it zero mode}
 solution describes the limit of a long chain with
the length $L\gg \ell_{0}$. It has been earlier obtained
\cite{Kolok} in the continuous limit $f\ll 1$. It also arises in the
theory of a multi-channel disordered wire \cite{Efet-book,
Mirlin2000}.  Plugging this solution into Eq.(\ref{JPD}) we obtain
the following probability distribution of $|\psi|^{2}$ in a long
{\it strictly} one-dimensional system (amazingly, this result was
not known before):
\begin{equation}
\label{1d-dist} P(|\psi|^{2})=\frac{\ell_{0}}{L}\,\frac{{\rm
exp}\left(-|\psi|^{2}\ell_{0}\right)}{|\psi|^{2}}.
\end{equation}
This distribution is valid for $|\psi|^{2}\ell_{0}\gg
e^{-L/\ell_{0}}$ and should be cut off at very small $|\psi|^{2}$ to
ensure normalizability \cite{rem1}.


At $f=\frac{1}{2}$ the quantities $u$ and $\phi$ are no longer
independent. Furthermore, the integrability of Eq.(\ref{ord}) --
even in its zero-mode variant-- is not guaranteed. Yet, with a
suitable choice of co-ordinates the variables are separated in the
zero-mode TM equation.

\section{Separation of variables.} 
The integrability of the zero mode
TME Eq.(\ref{ord}) is shown in three steps. The step one is to
introduce new set of variables $u$ and $v=u\cos(2\phi)$ instead of
$(u,\phi)$ and a new function
$\tilde{\Phi}(u,v)=u^{-1}\,\Phi(u,\frac{1}{2}\arccos(v/u))$. In
these variables the zero-mode TME Eq.(\ref{ord}) takes a very
symmetric form:
\begin{eqnarray}
\label{stat-xy}
%&&\sqrt{u^{2}-v^{2}}\,\left\{\partial_{u}\,\,\sqrt{u^{2}-v^{2}}\,\,\partial_{u}
%+\partial_{v}\,\,\sqrt{u^{2}-v^{2}}\,\,\partial_{v}\right\}\,\tilde{\Phi}=\nonumber
[D_{1}^{2}+D_{3}^{2}]\,\tilde{\Phi}= \frac{u}{2}\,\,\tilde{\Phi},
\end{eqnarray}
where the operators $D_{1}$ and $D_{3}$ belong to the family of
three operators from the representation of the $sl_{2}$ algebra:
\begin{equation}
\label{A-xy} D_{1(3)}=\pm \sqrt{u^{2}-v^{2}}\,\,\partial_{u(v)},\;
D_{2}=u\,\partial_{v}+v\,\partial_{u},
%\;D_{3}=-\sqrt{u^{2}-v^{2}}\,\,\partial_{v}
\end{equation}
obeying the commutation relations:
\begin{eqnarray}
\label{algebra} [D_{1},D_{2}]=-D_{3},\;
 [D_{3},D_{1}]=D_{2},\;
 [D_{2},D_{3}]=D_{1}.
\end{eqnarray}
Now it is clear that there is a hidden order in a set of
$\phi$-dependent terms in Eq.(\ref{del-2}) and the way they match
the regular part  in r.h.s. of Eq.(\ref{ord}).

One can further extend the algebra including also the operator $u$
in the r.h.s. of Eq.(\ref{stat-xy}). To this end we define:
\begin{equation}
\label{B} B_{1}=v,\;\;\;B_{2}=\sqrt{u^{2}-v^{2}},\;\;\;B_{3}=u.
\end{equation}
One can easily check that
\begin{equation}
\label{all-com} [B_{i},B_{j}]=0,\;\;\;[D_{i},B_{j}]=e_{ijk}\,B_{k}.
\end{equation}
The 6-dimensional algebra defined by
Eqs.(\ref{algebra}),(\ref{all-com}) constitutes the closed set of
operators sufficient to formulate all the symmetries of
Eq.(\ref{stat-xy}). Establishing the symmetries and the
corresponding operators commuting with the "Hamiltonian"
$D_{1}^{2}+D_{3}^{2}-\frac{1}{2}\,B_{3}$ is an important task which
was not accomplished so far. When achieved, it would probably help
to construct the new co-ordinates (very much in the same way as the
Kramers symmetry for the 3D Coulomb problem helps to identify the
set of parabolic co-ordinates) which would allow for a complete
solution to the problem. However, for the time being we proceed with
guessing the coordinates to separate variables in the zero-mode
problem.

The next step is to transform Eq.(\ref{stat-xy}) to the
Schroedinger-like  equation $H\Psi\equiv
-(\partial_{u}^{2}+\partial_{v}^{2})\,\Psi+U(u,v)\,\Psi=0$ for the
function $\Psi(u,v)=(u^{2}-v^{2})^{\frac{1}{4}}\,\tilde{\Phi}$,
where
\begin{eqnarray}
\label{Schroedinger}
 %\hat{H}\Psi&\equiv&
%-(\partial_{u}^{2}+\partial_{v}^{2})\,\Psi+U(u,v)\,\Psi=0,\\
U=-\frac{3}{4}\,\frac{u^{2}+v^{2}}{(u^{2}-v^{2})^{2}}+\frac{1}{2}\,
\frac{u}{u^{2}-v^{2}}.
\end{eqnarray}
Finally we introduce the variables
\begin{equation}
\label{xi-eta} \xi=\frac{u+v}{2}=u\,\cos^{2}\phi,\;\;\;\;
\eta=\frac{u-v}{2}=u\,\sin^{2}\phi.
\end{equation}
It is easy to see that in these  variables the operator in
Eq.(\ref{Schroedinger}) becomes a sum of two identical {\it
one-dimensional} Hamiltonians $H=\hat{H}_{\xi}+\hat{H}_{\eta}$ where
$\hat{H}_{\xi}$ is given by:
\begin{equation}
\label{H-1d}
\hat{H}_{\xi}=-\partial_{\xi}^{2}-\frac{3}{16}\,\frac{1}{\xi^{2}}+\frac{1}{4\xi}.
\end{equation}
Thus in new variables Eq.(\ref{xi-eta}) the TME Eq.(\ref{ord}) is
separable also at $f=\frac{1}{2}$ and can be reduced to the two
ODE's of the Schredinger type
$\hat{H}_{\xi}\varphi_{\lambda}(\xi)=\lambda\varphi_{\lambda}(\xi)$
and
$\hat{H}_{\eta}\varphi_{-\lambda}(\eta)=-\lambda\varphi_{-\lambda}(\eta)$
which have a well-known solution in terms  of the hypergeometric
functions (Whittaker functions) \cite{GR}.

Remarkably, $\xi$ and $\eta$ play a role of the co-ordinate and the
momentum in the equivalent classical model of kicked oscillator
\cite{Izrail}.


\section{Uniqueness of the solution.} 
The general solution to the
"Schroedinger equation" $H\Psi=0$ is given by the integral over the
parameter $\lambda$:
\begin{equation}
\label{gen} \Psi=\int d\lambda
d\bar{\lambda}\;c(\lambda,\bar{\lambda})\;\varphi_{\lambda}(\xi)\,\varphi_{-\lambda}(\eta),
\end{equation}
where integration is generically over the complex plane of $\lambda$
and $c(\lambda,\bar{\lambda})$ is an arbitrary function \cite{rem2}.
How does this huge degeneracy comply with the intuitive expectation
that the statistics of wavefunctions in an infinite disordered chain
should be unique and independent of the boundary conditions? Below
we show that the natural physical requirements on $\Phi(u,\phi)$
help to determine GF up to a constant factor which can be further
fixed using the wave function normalization $\langle
|\psi|^{2}\rangle=\frac{1}{L}$.

First of all we note that
$F(\lambda;\xi,\eta)=\varphi_{\lambda}(\xi)\,\varphi_{-\lambda}(\eta)$
is a holomorphic function of $\lambda$, i.e. it depends only on
$\lambda=\rho e^{i\sigma}$ but not on $\bar{\lambda}=\rho
e^{-i\sigma}$. The idea is to represent the integral over the
complex plane as an integral over $\rho$ and $\sigma$ and then
rotate the contour of integration  $\rho\rightarrow t e^{-i\sigma}$
so that the dependence on $\sigma$ remains only in
$c(\lambda,\bar{\lambda})$ and in the integration measure but not in
$F(\lambda;\xi,\eta)$. Then performing integration over $\sigma$ one
obtains a new function $C(t)=t\int
d\sigma\,e^{-2i\sigma}\,c(t,te^{-2i\sigma})$ which stands for
$c(\lambda,\bar{\lambda})$ in an expression similar to
Eq.(\ref{gen}) but involving only a one-dimensional {\it contour
integral}. This contour can be further rotated to make the
expression more symmetric.  Thus without loss of generality we write
a solution to the zero-mode TM equation Eq.(\ref{stat-xy}) for
$f=\frac{1}{2}$:
\begin{eqnarray}
\label{canon}
&&\Phi(\xi,\eta)=\frac{\xi+\eta}{(\xi\eta)^{1/4}}\int_{0}^{\infty}d\lambda\,
C(\lambda)
\\
&\times&\left[W_{-\lambda\epsilon,\frac{1}{4}}\,\left(
\frac{\bar{\epsilon}\xi}{4\lambda}\right)\,W_{-\lambda\bar{\epsilon},\frac{1}{4}}\,\left(
\frac{\epsilon\eta}{4\lambda}\right)+ c.c\right].\nonumber
\end{eqnarray}
Here $W_{\kappa,\mu}(z)$ is the Whittaker function \cite{GR};
$\epsilon=e^{i\pi/4}$, $\bar{\epsilon}=e^{-i\pi/4}$, and
$C(\lambda)$ is a real function yet to be determined.

Before we proceed with determining this function it is important to
establish its properties as $\lambda\rightarrow 0$. To this end we
note that $\langle|\psi|^{2}\rangle=\int
d\phi\,du\,u\,\cos^{2}\phi\;{\cal P}(u,\phi)\propto \int
d\phi\;\cos^{2}\phi \;\Phi^{2}(0,\phi)$. This is immediately seen
upon integration by parts over $u$ in  Eq.(\ref{JPD}). Thus the GF
$\Phi(\xi,\eta)$ must tend to a finite limit as $\xi\rightarrow 0$
and $\eta\rightarrow 0$. Given the asymptotic behavior of Whittaker
functions this is equivalent to:
\begin{equation}
\label{tile-non}
C(\lambda)=\lambda^{-\frac{3}{2}}\;\tilde{C}(\lambda),\;\;\;\tilde{C}(0)={\rm
const}.
\end{equation}
GF defined by Eq.(\ref{canon}) is periodic in $\phi$ with the period
$\frac{\pi}{2}$ as it should be for $q=2$. This is guaranteed by the
adding of the {\it c.c} term in Eq.(\ref{canon}). What is not
automatically guaranteed is that $\Phi(\xi,\eta)$ is {\it smooth} as
a function of $\phi$ at $\phi=0$. We will see that it is the
requirement of {\it smoothness}  at $\phi=0$ which fixes (up to a
constant factor) the unknown function $\tilde{C}(\lambda)$.

Indeed, the discontinuity of derivatives at $\phi=0$ may arise from
the branching of the expression in Eq.(\ref{canon})  at a small
$\eta$. From the representation of the Whittaker function in terms
of the hypergeometric functions one concludes that the general
solution Eq.(\ref{canon}) is a sum of a part which is regular in the
vicinity of $\eta=0$ and a part which has a square-root singularity
$\sqrt{\eta}\approx \sqrt{u} |\phi|$. The condition that this latter
part cancels out in the solution Eq.(\ref{canon}) is the following
($t$ is real):
\begin{eqnarray}
\label{rot} &&\Im
\left[\frac{\tilde{C}(\bar{\epsilon}t)}{\Gamma\left(\frac{1}{4}-it\right)}\,
e^{-\frac{i\eta}{8t}}\,_{1}F_{1}\left(
\frac{3}{4}-it,\frac{3}{2},\frac{i\eta}{4t}\right)\right]=0.
%\\
%&-&\left.\frac{\tilde{C}(\epsilon t)}
%{\Gamma\left(\frac{1}{4}+it\right)}\,
%e^{+\frac{i\eta}{8t}}\,_{1}F_{1}\left(
%\frac{3}{4}+it,\frac{3}{2},\frac{-i\eta}{4t}\right)
%\right]=0.\nonumber
\end{eqnarray}
The crucial fact for the possibility to fulfil this condition is the
identity for the hypergeometric functions \cite{GR}:
%\begin{equation}
%\label{gr-iden} e^{-z/2}\,_{1}F_{1}\left(\alpha,\gamma,z
%\right)=e^{z/2}\,_{1}F_{1}\left(\gamma-\alpha,\gamma,-z \right).
%\end{equation}
%It leads to:
\begin{equation}
\label{gr-iden-part}
e^{-z/2}\,_{1}F_{1}\left(\frac{3}{4}-it,\frac{3}{2},z
\right)=e^{z/2}\,_{1}F_{1}\left(\frac{3}{4}+it,\frac{3}{2},-z
\right).
\end{equation}
Now one can immediately guess the solution for $\tilde{C}(\lambda)$:
\begin{equation}
\label{GG} C_{0}(\lambda)=\Gamma\left(\frac{1}{4}+\epsilon\lambda
\right)\,\Gamma\left(\frac{1}{4}+\bar{\epsilon}\lambda \right).
\end{equation}


It is easy to see that the general solution to Eq.(\ref{rot})is
\begin{equation}
\label{gen-C}
\tilde{C}(\lambda)=C_{0}(\lambda)S(\lambda)=C_{0}(\lambda)\,\sum_{k=0}^{\infty}a_{k}\,\lambda^{4k},
\end{equation}
where the function $S(\lambda)$ must be regular in the entire
complex plane of $\lambda$. Now we apply the condition of
convergence of the integral over $\lambda$ in Eq.(\ref{canon}) at
large $\lambda$ to find the allowed asymptotic behavior of
$S(\lambda)$ at $\lambda\rightarrow\infty$. Substituting
Eq.(\ref{gen-C}) into Eq.(\ref{canon}) and using the asymptotics of
the Whittaker and $\Gamma$-functions we find that the integrand
behaves as $\lambda^{-3}S(\lambda)$ at $\lambda\rightarrow\infty$.
This means that $|S(\lambda)|$ should increase not faster than
$\lambda^{2}$. There is only one such entire function with the
structure of Eq.(\ref{gen-C}): this is a constant
$S(\lambda)=a_{0}={\rm const}$. This constant has to be determined
from the normalization condition $\langle|\psi|^{2} \rangle =L^{-1}$
using Eq.(\ref{JPD})

\section{Conclusion and discussion}
Eqs.(\ref{canon}),(\ref{tile-non}),(\ref{GG}) is the main result of
the paper. They give an exact and unique solution for the generating
function at  $f=\frac{1}{2}$ anomaly. The latter determines the JPD
of eigenfunction amplitude and phase, Eq.(\ref{JPD}) which can be
used to compute all local statistics of the one-dimensional Anderson
model in the bulk of a long chain $L\gg \ell_{0}$. The integrability
of TME Eq.(\ref{ord}) suggests that there is a hidden symmetry of
the problem at $f=\frac{1}{2}$. We make a conjecture that this
symmetry is naturally formulated in the three dimensional space
rather than in the two-dimensional space $(\xi,\eta)$ and that it
has to do with the symmetry of the 3d harmonic oscillator. This
conjecture is based on an analogy between our main result
Eq.(\ref{canon}) and the expression for the Green's function of the
3d harmonic oscillator problem \cite{BV}. This analogy concerns the
parameter ($\lambda$ in our problem and $k$ in Ref.\cite{BV})
entering both in the argument of the Whittaker functions and in its
first index in a mutually reciprocal way, as well as the second
index of the Whittaker functions being $\frac{1}{4}$ in both cases.
Establishing this symmetry would also be useful for studying the
anomalies at $f=\frac{p}{q}$ with $q>2$.

We have obtained \cite{rem4} the anomalous operator
$\Delta^{(3)}{\cal L}(u,\phi)$ which stands for $\Delta{\cal L}$ in
Eq.(\ref{ord}) at $f=\frac{1}{3}$ and shown that the mechanism
similar to Eq.(\ref{id1}) is also responsible for the anomaly at
$f=\frac{1}{3}$. The results of this study  will be published
elsewhere.

\begin{thecknowledgements} We appreciate stimulating discussions with A.Agrachev,
Y.V.Fyodorov, A.Kamenev and A.Ossipov and a support from RFBR grant
06-02-16744.
\end{thecknowledgements}
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\end{document}
