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\def \d{\partial}
\def \G{\Gamma}
\def \g{\gamma}
\def \a{\alpha}
\def \b{\beta}
\def \de{\delta}
\def \De{\Delta}
\def \ep{\varepsilon}
\def \kappa{\varkappa}
\def \la{\lambda}
\def \La{\Lambda}
\def \r{\rho}
\def \t{\theta}
\def \z{\zeta}
\def \Sg{\Sigma}
\def \sg{\sigma}
\def \Om{\Omega}
\def \U{\Cal Q}
\def \om{\omega}
\def \Var{\text{\rm Var}\;}
\def \tr{\,\text{\rm tr}\,}
\def \sign{\,\text{\rm sign}\,}
\def \diag{\,\text{\rm diag}\,}
\def \ind{\,\text{\rm ind}\,}
\def \Area{\,\text{\rm Area}\,}
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\def \rf{\root 4 \of}
\def \Ai{\,\text{\rm Ai}\,}
\def \iff{\quad\text{\rm if}\quad}
\def \ON{O(N^{-2})}
\def \iz{\int_{z_1}^{z_2}}
\def \ix{\int_{x_1}^{x_2}}
\def \A{\text{\rm A}}
\def \Re{\,\text{\rm Re}\,}
\def \Im{\,\text{\rm Im}\,}
\def \lacr{\la_{\text {\rm cr}}}
\def \WKB{\text{\rm EBK}}
\def \TP{\text{\rm TP}}
\def \Vol{\,\text{\rm Vol}\,}
\def \const{\,\text{\rm const}\,}
\def \sgn{\,\text{\rm sgn}\,}
{\nopagenumbers
\null
{}
\vskip 3cm
\ce {\bbrm Trace Formula for Quantum Integrable}
\vskip 4mm
\ce {\bbrm Systems, Lattice--Point Problem,}
\vskip 4mm
\ce{\bbrm and Small Divisors}
\vskip 1cm
\ce {\bbrm Pavel Bleher}
\vskip 4mm
\ce{\brm Indiana University -- Purdue University}
\vskip 2mm
\ce{\brm at Indianapolis}
\vskip 2cm
\ce {To be published in the Proceedings of the Summer Program on}
\ce{
``Emerging Applications of Number Theory'', July 15--26, 1996}
\vskip 1cm
\ce {Institute for Mathematics and Its Applications,}
\ce{University of
Minnesota, Minneapolis}
\vskip 1cm
\ce{ 1996}
\vskip 2cm
Address:
Department of Mathematical Sciences
Indiana University -- Purdue University at Indianapolis
402 N. Blackford Street
Indianapolis, IN 46202, USA
\vskip 5mm
E-mail: bleher\@math.iupui.edu
\vfill\eject}
\pageno = 1
\null${}$
\vskip 5cm
{\bf Abstract.} We review rigorous results concerning almost
periodicity by
Besicovitch of the spectral function of quantum integrable systems,
lattice point-problem in convex and nonconvex domains,
trace formula, and a related
problem of small divisors. This includes the following subjects:
(i) Semiclassical quantization by Einstein-Brillouin-Keller and the
problem of completeness of semiclassical eigenfunctions.
(ii) The lattice-point problem.
(iii) The Gutzwiller-Berry-Tabor trace formula for quantum integrable
systems.
(iv) The problem of small divisors.
(v) The saturation phenomenon.
\vskip 3cm
The work is supported in part by the NSF Grant No. DMS--9623214.
\vfill\eject
${}$
\vskip 2cm
\ce{\bbrm Contents}
\vskip 0.5cm
1. The Weyl Law.
2. General Lattice-Point Problem.
3. Surfaces of Revolution.
4. Liouville Surfaces.
5. Saturation Phenomenon.
6. Lattice-Point Problem in Dimension Greater Than 2.
7. Quantum Linear Oscillators.
References
\vskip 1cm
\beginsection 1. The Weyl Law \par
Let $M^d$ be a smooth closed compact Riemannian manifold, and let
$$
0=E_00$.
In this paper we will be interested in the opposite case of completely
integrable geodesic flows. Let us begin with a simple example of
a two--dimensional torus $\T^2=(2\pi)\R^2/\Z^2$ with the flat metric
$$
dq^2=dq_1^2+dq_2^2.\eqno (1.3)
$$
Then the eigenvalues are
$$
E_{n_1n_2}=n_1^2+ n_2^2,\qquad n_1,n_2\in\Z,
\eqno (1.4)
$$
so that
$$
N(E)=\#\{(n_1,n_2)\in\Z^2\: n_1^2+ n_2^2\le E\}
$$
is just the number of lattice points inside the circle $x_1^2+ x_2^2=E$,
and (1.1) reads
$$
N(E)=\pi E +n(E).
$$
In this case the
problem of evaluating $n(E)$ is the classical circle problem
which goes back to Gauss. Gauss proves the estimate
$$
n(E)=O(E^{1/2}),
$$
which is equivalent in this particular case to the H\"ormander
estimate (1.2). The Gauss estimate is rather obvious since $n(E)$
cannot grow faster than the length of the boundary. First nontrivial
estimate,
$$
n(E)=O(E^{1/3}),
$$
is derived by Sierpinski [Sie]. Then the exponent 1/3 in this estimate
has been improved due to the works by Hardy, Landau, Vinogradov,
Walfisz, Titchmarsh, Hua, Kolesnik, Iwaniec and Mozzochi, Huxley, and
others. Huxley [Hux] proves the estimate
$$
n(E)=O\left(E^{23/73}(\log E)^{315/146}\right),
\eqno (1.5)
$$
which is probably the best estimate at the present time.
Hardy's careful conjecture [Har1]: ``it is not unlikely that
$$
n(E)=O\left( E^{(1/4)+\ep}\right)\eqno (1.6)
$$
for all positive $\ep$,'' remains open. On the other hand, Hardy [Har2]
proves that
$$\eqalign{
\limsup_{E\to\infty} {n(E)\over E^{1/4}}>0,\cr
\liminf_{E\to\infty} {n(E)\over E^{1/4}(\log E)^{1/4}}>0,\cr}
\eqno (1.7)
$$
which shows that the exponent in (1.6) cannot be smaller than
$(1/4)+\ep$.
The Hardy conjecture (1.6) combined with (1.7) suggest that ``typical''
values of $n(E)$ are of the order of $E^{1/4}$, and it is confirmed by
the classical theorem of Cram\'er [Cra]:
$$
\lim_{T\to\infty} {1\over T^{3/2}}\int_0^T |n(E)|^2dE=C>0.
\eqno (1.8)
$$
Fig. 1 shows the error function $n(E)$ as a function of the radius
$R=\sqrt E$. It is easy to see that $n(R^2)$ is irregularly oscillating
around 0, and it looks like a random function. So it is quite natural
to study probabilistic characteristics of $n(R^2)$, like average value,
mean square deviation, limit distribution, correlations, etc. Since
there is nothing random in the problem, the probabilistic characteristics
of $n(R^2)$ are understood as ergodic averages. Let us introduce
the normalized error function as
$$
F(R)={n(R^2)\over \sqrt R}\,, \qquad R\ge 1,\eqno (1.9)
$$
and $F(R)=0$ when $R<1$.
The result of Cram\'er (1.8) implies that the second moment of $F(R)$ exists
and it is positive. Heath-Brown [H-B] proves the existence of a limit
distribution of $F(R)$.
{\bf Theorem 1.1 } (see [H-B]). {\it There exists a probability
density $p(x)$ such that for every bounded continuous
function $g(x)$,
$$
\lim_{T\to\infty}{1\over T} \int_0^T g(F(R))dR=\int_{-\infty}^\infty
g(x)p(x)dx.\eqno (1.10)
$$
In addition, the density $p(x)$ decays as $x\to\infty$ faster than
polynomially, and it can be extended to an entire function on a
complex plane.
}
\beginsection 2. General Lattice--Point Problem \par
{\it Convex Domains}.
Let $\La$ be an open convex
domain on a plane $\R^2$, and let $\G=\partial\La$
be its boundary. We will assume that
(i) $\G$ is $C^7$-smooth;
(ii) $0\in\La$;
(iii) $\G$ is strictly convex, i.e., the curvature $\kappa(x)>0$
for all $x\in\G$.
Let $\a\in\R^2$ be an arbitrary fixed point on the plane.
For $R>0$, define the set
$$
R\La +\a=\{x\in\R^2\: (x-\a)/R\in\La\},
\eqno (2.1)
$$
which is a dilation of $\La$ with the coefficient $R$ and its shift in
$\a$.
Let
$$
N_\a(R)=\# \, \Z^2\cap (R\La+\a),
$$
the number of lattice points in $R\La+\a$, and
$$
F_\a(R)={N_\a(R)-AR^2\over \sqrt R}\,,
\qquad R\ge 1,
\eqno (2.2)
$$
where
$$
A=\,\text{\rm Area}\,\La,
$$
and $F_\a(R)=0$ for $R<1$.
{\bf Theorem 2.1.} (see [Ble1]) {\it If $\La$ satisfies the conditions
(i)--(iii), then for all $\a\in\R^2$, $F_\a(R)$ is an almost periodic
function in $R$ from the Besicovitch space $B^2$, and its Fourier
series in $B^2$ is
$$
F_\a(R)=\pi^{-1}\sum_{n\in \Z^2,\; n\not=0}|n|^{-3/2}|\kappa(x(n))|^{-1/2}
\cos\bigl(\om(n)R-\phi(n;\a)\bigr),
\eqno (2.3)
$$
where $x(n)\in\G$ is the unique point on $\G$ where the outer normal vector
to $\G$ coincides with $n/|n|$, $\kappa(x)$ is the curvature, and
$$
\om(n)=2\pi (n,x(n))>0;\qquad \phi(n;\a)= 2\pi (\a,n)+(3\pi/4).
\eqno (2.4)
$$}
The meaning of the $B^2$-almost periodicity and (2.3) is that
$$\eqalign{
\lim_{N\to\infty}\limsup_{T\to\infty}&{1\over T}
\int_0^T\left|F_\a(R)-\pi^{-1}\sum_{n\in \Z^2,\; 0<|n|0$,
$$
\lim_{|x|\to\infty}{\log p_\a(x)\over |x|^{4+\ep}}=0,
\eqno (2.10)
$$
and
$$
\lim_{x\to\infty}{\log P_\a^\pm (x)\over |x|^{4-\ep}}=\infty,
\eqno (2.11)
$$
where
$$
P_\a^\pm=\left|\int_{\pm x}^{\pm\infty} p_\a(x)\,dx\right|.
$$}
Roughly (2.10) and (2.11) mean that $p_\a(x)$ decays
at infinity as $\exp(-cx^4)$. The variance of $p_\a(t)$
turns out to be sensitive to arithmetical properties of $\a$. Bleher
and Dyson [BD1] proves the following result.
{\bf Theorem 2.4} (see [BD1]). {\it Let
$$
D(\a)=\lim_{T\to\infty}{1\over T}\int_0^T|F_\a(R)|^2dR=\int_{-\infty}
^\infty x^2p_\a(x)\,dx.
$$
Then $D(\a)$ is a continuous function of $\a$, and for
all rational $\a\in\Q^2$,
$$
\lim_{\b\to\a}{D(\a)-D(\b)\over |\a-\b|\bigl|\log|\a-\b|\bigr|}
=C(\a)>0.
\eqno (2.12)
$$}
The equation (2.12) implies that $D(\a)$ has a sharp local maximum
at every rational point.
Properties of $\nu_\a(dx)$ for a generic $\G$ are studied in [Ble2]
and [BKS].
Let
$$
Z=\{ n=(n_1,n_2)\in\Z^2\: n_1,n_2\not=0\; \text{are relatively prime}\}
\cup (1,0)\cup (0,1).
\eqno (2.13)
$$
{\bf Theorem 2.5} (see [Ble2] and [BKS]). {\it Assume that
the numbers $\{\om(n),\;
n\in Z\}$ are linearly independent over $\Z$, i.e., if $k_1\om_1+\dots
+k_m\om_m=0$ for some $k_i\in\Z$ then all $k_i=0$. Then for all $\a\in\R^2$,
the distribution $\nu_\a(dx)$ has a density $p_\a(x)=\nu_\a(dx)/dx$
which is an entire function in $x$, and
$$
\lim_{x\to\pm\infty} -{\log p_\a(x)\over x^4}=c_\pm(\a)>0.
$$}
If $\G$ possesses a symmetry then $\om(n)$ coincides for symmetric $n$.
In this case Theorem 2.5 remains valid if $\om(n_1),\dots,\om(n_m)$
are linearly
independent over $\Z$ for all $n_1,\dots,n_m\in Z$ such
that $n_i$ and $n_j$ are not symmetric. For an ellipse $x_1^2+\mu^{-2}
x_2^2=1$
the condition of linear independence of $\om(n)$ over $\Z$ holds if
$\mu^{-2}$ is a transcendental number (see [Ble2]), hence in this case
we have Theorem 2.5. For the ellipse $x_1^2+\mu^{-2}
x_2^2=1$ the normalized error function is
$$
F_\a(R)={\#\,\{ (n_1,n_2)\in\Z^2\: n_1^2+\mu^{-2} n_2^2\le R^2\}
-\pi \mu\over R^{1/2}}.
\eqno (2.14)
$$
Figs. 2--10 (taken from [Ble2]) show the density
$p_\a(x)$ of the limit distribution of $F_\a(R)$ for
ellipses with different value of $\mu$ and different $\a$. It
obviously varies and can be bimodal.
How to explain the non-Gaussian nature of $p_\a(x)$? Let us consider,
as an example, an
ellipse $x_1^2+\mu^{-2} x_2^2=1$
with some transcendental $\mu^{-2}$, and let us take
$\a=0$. We rewrite the Fourier series (2.3) in this case as
$$
F_0(R)=(\mu/\pi)\sum_{n\in\Z^2\: n\not=0}|n|_\mu^{-3/2}
\cos\bigl(2\pi|n|_\mu R-\phi\bigr),
$$
where $|n|_\mu^2=n_1^2+\mu^2n_2^2$ (see [Ble2]). Now we can take some
$m$ from the set
$$
Z_+=\{ m=(m_1,m_2)\in\Z^2\: m_1,m_2>0\; \text{are relatively prime}\}
\cup (1,0)\cup (0,1),
\eqno (2.15)
$$
and first make a summation over all $n=km,\; k\in \N,$ and
symmetric points, and
then over $m\in Z_+$.
This gives
$$
F_0(R)=\sum_{m\in Z_+}|m|^{-3/2}_\mu f_m(|m|_\mu R),
\eqno (2.16)
$$
where
$$
f_m(t)=(\mu/\pi)r(m)\sum_{k=1}^\infty k^{-3/2}\cos(2\pi kt-\phi),
\eqno (2.17)
$$
where $r(m)$ is the symmetry factor: $r(m)=4$ if $m_1m_2\not=0$,
and $r(m)=2$ otherwise. The function $f_m(t)$ is periodic
of period 1. The number $\mu$ is transcendental, hence
the frequencies $|m|_\mu,\; m\in Z_+,$ are linearly independent
over $\Z$ (see [Ble2]). Therefore the numbers $|m|_\mu R \mod 1$ behave
like independent random variables $\t_m$ uniformly distributed
on $[0,1]$. Thus the limit distribution of $F(R)$ is
the distribution of the random series
$$
\sum_{m\in Z_+} |m|_\mu^{-3/2} f_m(\t_m).
\eqno (2.18)
$$
This series is only conditionally convergent because $Z_+$ is
a two-dimensional set and the exponent $3/2$ is less than 2.
However the series of the variances,
$$
\sum_{m\in Z_+} |m|_\mu^{-3} \int_0^1 f_m(\t)\,d\t,
$$
converges, and hence, by the Kolmogorov theorem,
the random series (2.18) is well defined
and it is non-Gaussian. An analysis of the characteristic function
of the random series (2.18) gives the tail behavior of its density as
$\exp(-cx^4)$. The exponent 4 is related to the one (3/2) in (2.18).
As concerns uniform estimate of the error term $n(E)$, Huxley's
estimate (1.5) is proved for arbitrary smooth domain with nonvanishing
curvature and the estimate of the error function is uniform in
translation and rotation (see [Hux]).
{\it Nonconvex Domains}.
Let us consider a nonconvex domain $\La$. We will assume that $\La$
is a star-like domain such that
(i) $\G=\partial\La$ is $C^7$-smooth;
(ii) $(n(x),x)>0$ for all $x\in\G$, where $n(x)$ is the outer normal
vector to $\G$ at the point $x$.
In addition, we will assume the following
Diophantine hypothesis.
{\bf Hypothesis D.} {\it (i) The curvature $\kappa(x)\not=0$ everywhere
on $\G$ except, maybe, a finite set $W=\{ w_1,\dots, w_K\}$, and
$$
{d\kappa\over ds}(w_k)\not=0,\qquad k=1,\dots,K,
\eqno (2.19)
$$
where $s$ is the natural coordinate (the arc length) on $\G$. (ii)
For all $w_k\in W$ the outer normal vector $n(w_k)$ to $\G$ is
either rational, i.e., $(m,n(w_k))=0$ for some $m\in\Z^2,\; m\not=0$,
or Diophantine in the sense that there exist $0<\z<1$ and $C>0$ such
that for all $m\in\Z^2,\; m\not=0$,
$$
|(m,n(w_k))|>C\,|m|^{-1-\z}.
\eqno (2.20)
$$}
For $x\in\G$ we denote by $L(x)$ the line on a plane through 0 with
slope $n(x)$. By $\G_r$ we denote the set of all points on $\G$
where the vector $n(x)$ is rational. The set $\G_r$ is countable.
{\bf Theorem 2.6} (see [Ble3]). {\it Assume that $\La$ satisfies (i),
(ii), and that the Hypothesis D holds. Then for all $\a\in\R^2$,
$$
N_\a(R^2)=AR^2+R^{2/3}\sum_{k\: n(\om_k)\;\text{is rational}}
\Phi_k(R;\a)+R^{1/2}F_\a(R),
\eqno (2.21)
$$
where $\Phi_k(R;\a)$ are continuous periodic functions of $R$,
$$
\Phi_k(R)={\G(2/3)\over 2\, 3^{2/3}\pi^{4/3}}
\left| {d\kappa\over ds}(w_k)\right|^{-1/3}\sum_{n\in L(w_k),\;n\not=0}
\sin \bigl[ 2\pi (w_k,n(w_k))|n|R-2\pi (n,\a)\bigr],
\eqno (2.22)
$$
and $F_\a(R)$ is a $B^2$-almost periodic function of $R$. The $B^2$-Fourier
series of $F_\a(R)$ is
$$
F_\a(R)=\pi^{-1}\sum_{x\in \G_r} |\kappa(x)|^{-1/2}
\sum_{n\in L(x),\; n\not=0} |n|^{-3/2}\cos\bigl[
2\pi (x,n(x)) |n|R-\phi(x,n,\a)\bigr],
\eqno (2.23)
$$
where $\phi(x,n,\a)=(\pi/2)+(\pi/4)\,\text{\rm sgn}\, \kappa(x)+2\pi (n,\a)$.}
We can compare this result with the Colin de Verdi\`ere estimate [CdV2].
{\bf Theorem 2.7} (see [CdV2]). {\it Assume that $\La$ satisfies
(i) and (ii). Then
$$
\bigl| N_\a(R^2)-AR^2\bigr|\le CR^{2/3}
\eqno (2.24)
$$}
Comparison of (2.21) with (2.24) shows that the Colin de Verdi\`ere estimate
(2.24) is sharp when there is an inflection point $w_k$ on $\G$ with
rational normal vector $n(w_k)$. In a ``typical'' situation,
however, there is no
inflection point with rational $n(w_k)$, and the intermediate
term with $R^{2/3}$ is
missed in (2.21). The typical situation means that the normal
vectors $n(w_k)$ at the inflection points $w_k$ satisfy the Diophantine
condition (2.20). This is a problem of small divisors, which
can be seen from (2.23). Namely, the multiplier $|\kappa(x)|^{-1/2}$
approaches $\infty$ as $x\to w_k$, and the Diophantine condition is
needed to secure that the growth of this multiplier is compensated
by the decrease of the multiplier $|n|^{-3/2}$ in (2.23). Along
the lines of the proof of Theorem 2.6, it is possible to show
that if for some $w_k$, $n(w_k)$ is a Liouville number, i.e., it is
exceptionally well approximated by rationals, then the error
function $n_\a(R^2)$ behaves eratically, with no powerlike
asymptotics.
The importance of the inflection points for the trace formula
was emphasized by Berry and Tabor [BT1]:
``Physically,
the region [where the curvature] $K=0$ would rise to a large peak in
$n(E)$, a dense cluster of energy levels. The simplest region $K=0$ is
a point of inflexion on the energy contour for a two-dimensional case,
and the corresponding feature in $n(E)$ could be called a `bound state
rainbow' (see [BM], section 6.3). More complicated coalescences would
give rise to higher order `catastrophes' (Thom 1975; Connor 1975;
Duistermaat 1974). At present we know of no case involving smooth
potentials where the energy contours have inflexions $K=0$, and so we
do not consider these coalescences any further here (Balian and Bloch
[BaB] study an enclosure in the shape of a waisted Greek vase whose
mode show a rainbow).''
In the next section we will consider a geodesic flow on a smooth
surface of revolution, and we will see that the points of inflection
on the energy contour are indeed possible in a generic situation.
\beginsection 3. Surfaces of Revolution \par
Surface of revolution is a simple example of integrable geodesic
flow. Assume that $M^2$ is a smooth surface of revolution,
which is diffeomorphic to a sphere. Let $A$ be the axis of $M^2$,
and let $N$ and $S$ be its north and south poles, respectively
(see Fig. 11). The geodesic flow on $M^2$ is integrable due
to the Clairaut integral,
$$
I=r\sin\a=\const,
$$
where $r$ is the radial coordinate and $\a$ is the angle between
the velocity vector and the meridian. Let
$$
r=f(s),\qquad 0\le s\le L,
$$
be the equation of $M^2$, where $s$ is the normal coordinate
(the arc length) along meridian. Then
$$
\De=f(s)^{-1}{\partial f\over \partial s}\left(f(s)
{\partial f\over \partial s}\right)
+f(s)^{-2}{\partial f^2\over \partial \f^2},
$$
where $\f$ is the angular coordinate. We will assume that $f(s)$ has
a simple structure, so that
$$
f'(s)\not=0,\quad s\not= s_{\max};\qquad f''(s_{\max})<0,
$$
where
$$
f(s_{\max})=\max_{0\le s\le L}f(s).
$$
For normalization we put $f(s_{\max})=1$. We call the circle
$s=s_{\max}$ on $M^2$ the equator.
Another assumption on $M^2$ is the twist hypothesis.
Let $x_0$ be an arbitrary point on the equator. Consider
a geodesic $\g$ which goes out of $x_0$ at some angle
$-(\pi/2)< \a_0< (\pi/2)$ to the north direction on the meridian.
The Clairaut integral is $I=\sin\a_0$ and we can parametrize $\g$
by $-1< I< 1$: $\g=\g(I)$. It follows from the Clairaut
integral that $\g(I)$ oscillates between two parallels,
$s=s_-$ and $s=s_+$ where $f(s_-)=f(s_+)=I$. Hence
$\g(I)$ intersects the equator infinitely many times.
Let $x_n$ be the $n$th intersection of $\g(I)$ with the equator,
$n\in\Z$. Define
$$
\tau(I)=|\g[x_0,x_2]|
$$
to be the length of $\g=\g(I)$ between $x_0$ and $x_2$, and
$$
\om(I)=(2\pi)^{-1}(\f(x_2)-\f(x_0))
$$
to be the phase of $\g(I)$ between $x_0$ and $x_2$ (see Fig. 11).
Observe that $\om(I)$ is defined mod 1. To define $\om(I)$
uniquely, we choose a continuous branch of $\om(I)$ starting
at $\om(0)=0$. Define $\tau(1)=\lim_{I\to 1-0}\tau(I)$ and
$\om(1)=\lim_{I\to 1-0}\om(I)$. A finite geodesic $\g(I)$ on $M^2$
with $0*0$ for the oblong
ellipsoid, $\om'(I)<0$ for the oblate ellipsoid, and
$\om(I)$ has a critical point for the bell-like surface.
Hence Theorem 3.1 is applicable in the cases (a) and (b),
and it is not applicable in the case (c). To extend Theorem
3.1 to the case (c), we introduce the following condition.
{\bf Diophantine Hypothesis}. {\it Assume that $\om(I)$ has at most
finitely many critical points $00$
such that
$$
\left| \om(I_k)-{p\over q}\right|\ge {C\over q^{2+\z}},\qquad
\forall\, {p\over q}\in \Q.
$$}
{\bf Theorem 3.2} (see [Ble3]). {\it Assume that $M^2$ is a
surface of revolution of simple structure and that $M^2$
satisfies the Diophantine Hypothesis. Then
$$
N(E)={\Vol M\over 4\pi}\, E+E^{1/3}\sum_{k\:\om(I_k)\in\Q}
\Phi_k(E^{1/2})+E^{1/4}F(E^{1/2}),
\eqno (3.4)
$$
where $\Phi_k(R)$ are bounded periodic functions and
$F(R)$ is $B^2$-almost periodic function. The Fourier
series of $\Phi_k(R)$ is
$$
\Phi_k(R)={\G(2/3)\tau(I_k)^{4/3}\over 2\pi^{4/3}}
\sum_{\g\: I(\g)=I_k} (-1)^{m(\g)}|\g|^{-4/3}\sin(|\g|R),
\eqno (3.5)
$$
and the Fourier series of $F(R)$ in $B^2$ is
$$
F(R)=\sum_{\g\: I(\g)\not= I_1,\dots, I_K}
A(\g)\cos(|\g|R-\phi(\g)),
\eqno (3.6)
$$
where $\phi(\g)=(\pi/2)+(\pi/4)\sgn \om'(I),\; I=I(\g)$
and $A(\g)$ is as in (3.2).}
The proof of Theorem 3.2 (observe that Theorem 3.1
reduces to Theorem 3.2) uses the semiclassical result
of Colin de Verdi\`ere.
{\bf Theorem 3.3} (see [CdV3]). {\it If $M^2$ is a surface of revolution
of simple structure, then
$$
\text{\rm Spectrum}(-\De)=
\{ E_{kl}=Z(k+(1/2),l);\quad k,l\in\Z,\quad |l|\le k\},
\eqno (3.7)
$$
with $Z(p)=Z(p_1,p_2)\in C^{\infty}(\R^2)$ such that
$$
Z(p)=Z_2(p)+Z_0(p)+O(p^{-1}),\qquad |p|\to\infty,
$$
where
$$
Z_2(p),\; Z_0(p)\in C^\infty (\R^2\backslash\{0\});\qquad
Z_2(p)>0,\quad p\not=0,
$$
and
$$
Z_j(\la p)=\la^j Z_j(p),\qquad \forall\, \la>0,\; p\in\R^2;
\qquad j=0,2.
$$
In addition, in the sector $\{ p_1\ge |p_2|$, $Z_2(p)$ satisfies
the equation
$$
\pi^{-1}\int_a^b\sqrt{Z_2(p)-p_2^2f(s)^{-2}}\,ds=p_1-|p_2|,
$$
where $a$ and $b$ are the turning points, i.e.,
$$
Z_2(p)-p_2^2f(s)^{-2}=0,\quad\text{for}\quad s=a,b.
$$}
We define semiclassical eigenvalues as
$$
E_{kl}^{\WKB}=Z_2(k+(1/2),l).
$$
This is the Einstein-Brillouin-Keller quantization formula, applied to
the surface of revolution.
The semiclassical counting function is then
$$
N^{\WKB}(E)=\#\, \{(k,l)\: Z_2(k+(1/2),l)\le E\}.
\eqno (3.8)
$$
The following estimate shows that we can replace $N(E)$
by $N^{\WKB}(E)$.
{\bf Lemma 3.4} (see [Ble3]). {\it
$$
\lim_{T\to\infty}\int_0^T {|N(R^2)-N^{\WKB}(R^2)|^2 dR\over R}=0.
$$}
Since $Z_2(p)$ is a homogeneous function of order 2, the problem
of finding asymptotics of $N^{\WKB}(E)$ reduces to the
lattice-point problem. The condition $|l|\le k$ in (3.7)
restricts lattice points to the sector between diagonals,
and the additive term (1/2) in (3.7) shifts the lattice in (1/2)
along the $x$-axis. Fig. 13 depicts the graph of the level
set $Z_2(p)=1$ for the three surfaces of revolution shown in
Fig. 12. Observe that the critical point of $\om(I)$ for
the bell-like surface leads to the inflection point on
the level set $Z_2(p)=1$.
The influence of the inflection points on the oscillatory
asymptotics of the error function was observed by Berry and Tabor
[BT1]. They called this phenomenon a `bound state rainbow', with a
reference to the paper of Balian and Bloch [BaB].
Theorem 2.6 allows an extension to the lattice
points in the sector between diagonals (see [Ble3]), and
this proves the $B^2$-almost periodicity for the semiclassical
error function. Lemma 3.4 proves it then for the error
function $F(R)$.
Theorem 3.1 can be used to get a limit distribution of the
normalized error function $F(R)$.
{\bf Theorem 3.5} (see [Ble3], [KMS], and [BKS]). {\it Assume that
$M^2$ is a surface of revolution of simple structure and
that $M^2$ satisfies the Twist Hypothesis. Then there exists
a probability distribution $\nu(dx)$ on a line such that
for all bounded continuous functions $g(x)$,
$$
\lim_{T\to\infty}{1\over T}\int_0^T g(F(R))\,dR=
\int_{-\infty}^\infty g(x)\,\nu(dx).
$$
If, in addition, the lengths of all primitive closed
geodesics on $M^2$ with non-negative Clairaut integral $I$
are linearly independent over $\Z$, then the probability
distribution $\nu(dx)$ is absolutely continuous, and
the density $p(x)=\nu(dx)/dx$ is an entire function
such that
$$
\lim_{x\to\pm\infty} -{\log p(x)\over x^4}=c_\pm>0.
$$}
Similar result holds as well for the error function $F(R)$
in the case
when $M^2$ has a simple structure and it satisfies the
Diophantine Hypothesis (see [BKS]). It is to be noted that the limit
distribution $\nu(dx)$ is not Gaussian. As we have discussed before,
it is related to the convergence of the series of squared amplitudes
in the trace formula. For chaotic systems,
Aurich, Bolte, and Steiner [ABS] have found universally
a Gaussian limit distribution of the error function.
This can be explained as follows.
In the chaotic case we can apply the
Gutzwiller trace formula and we can write it for $0U_2(q_2)$ for all $q_1,q_2$.
The Laplacian is then
$$
\De=[U_1(q_1)-U_2(q_2)]^{-1}\left({\partial^2 \over \partial
q_1^2}+{\partial^2 \over \partial
q_2^2}\right).
\eqno (4.2)
$$
The Hamiltonian of the geodesic flow is
$$
H(p,q)={1\over 2(U_1(q_1)-U_2(q_2))}\, (p_1^2
+p_2^2).
\eqno (4.3)
$$
The geodesic flow on the Liouville surface is integrable due to the
integral
$$
S(p,q)={1\over 2(U_1(q_1)-U_2(q_2))}\, [U_2(q_2)p_1^2
+U_1(q_1)p_2^2].
\eqno (4.4)
$$
We associate with $S(p,q)$ the quantum operator
$$
\sg=[U_1(q_1)-U_2(q_2)]^{-1}\left(U_2(q_2){\partial^2 \over \partial
q_1^2}+U_1(q_1){\partial^2 \over \partial
q_2^2}\right).
$$
A direct computation shows that $\De$ and $\sg$ commute.
Consider the counting function $N(E)$ of $-\De$ and the normalized
error function
$$
F(R)={N(R^2)-CR^2\over R^{1/2}},\qquad C={\Area M^2\over 4\pi}.
\eqno (4.5)
$$
The asymptotic behavior of $F(R)$ is studied in the paper of
Kosygin, Minasov, and Sinai [KMS]. It is shown in [KMS] that under
some condition of non-degeneracy of the Liouville surface,
the function $F(R)$ is $B^1$-almost periodic. If, in addition,
the Liouville surface is ``generic'' then the distribution density of
$F(R)$ is an entire function, and it decays at infinity at least
as $\exp(-|x|^{(16/9)-\ep})$.
These results are further extended by Bleher, Kosygin, and Sinai
(see [BKS]). To describe the results we have to discuss some
properties of the geodesic flow. Define
$$\eqalign{
c_1=\max_{0\le x\le 1}U_1(x)=U_1(M_1),\qquad
c_2=\min_{0\le x\le 1}U_1(x)=U_1(m_1),\cr
c_3=\max_{0\le x\le 1}U_2(x)=U_2(M_2),\qquad
c_4=\min_{0\le x\le 1}U_2(x)=U_2(m_2). \cr}
\eqno (4.6)
$$
Consider the energy surface
$$
M_E=\{ (p,q)\: H(p,q)=E\},\qquad E>0,
$$
and partition $M$ into the invariant sets
$$
M_{E,c}=\{ (p,q)\in M_E\: S(p,q)=cE\}.
$$
By (4.3) and (4.4),
$$
{S(p,q)\over H(p,q)}={U_2(q_2)p_1^2+U_1(q_1)p_2^2\over p_1^2+p_2^2},
\eqno (4.7)
$$
hence on $M_E$ the value $c=S(p,q)/H(p,q)$ varies from $c_4$ to
$c_1$. If
$c_41$ and $C>0$ such that for all nonzero
pairs of integers $(k_1,k_2)$,
$$
|k_1\a_1+k_2\a_2|\ge {C\over |k|(\log |k|)^\tau}\,,
\qquad |k|=(k_1^2+k_2^2)^{1/2}.
$$
A number $\a$ is called Diophantine if the pair $(1,\a)$ is Diophantine.
The complement to the set of Diophantine numbers has zero Lebesgue measure.
Introduce the following condition.
{\bf Diophantine Hypothesis.}
{\it (i) Assume that $a(c)$ has
at most finitely many zeros, $a(c_1),a(c_4)\not=0$,
and all these zeros are simple.
(ii) If $c$ is a zero of $a(c)$ then the pair $(f_1'(c),f_2'(c))$
is Diophantine.
(iii) The pairs $(f_1(c_2),f_2(c_2))$ and $(f_1(c_3),f_2(c_3))$ are
Diophantine.
(iv) The pairs $(f_1'(c_1),f_2'(c_1))$ and $(f_1'(c_4),f_2'(c_4))$
are Diophantine.}
We will identify all closed geodesics which belong to the
same invariant torus.
{\bf Theorem 4.1} (see [BKS]). {\it Assume that a Liouville surface $M^2$
satisfies the Diophantine Hypothesis. Then the error function
$F(R)$ (see (4.5)) is a $B^1$-almost periodic function. The
$B^1$-Fourier series of $F(R)$ is
$$
F(R)=(2\pi^3)^{-1/2}\sum_{\text{\rm closed geodesics}\;\g}
|\g|^{-3/2}\kappa(\g)^{-1/2}
\sin\left( |\g|R-{\pi\ind\g\over 2}-{\pi\sg(\g)\over 4}\right),
$$
where
$$
\kappa(\g)=\left|\det\left( {\partial^2 H\over \partial
I_k\partial I_l}\right)(c)\right|,
\qquad \sg(\g)=\sgn\det\left( {\partial^2 H\over \partial
I_k\partial I_l}\right)(c),
\qquad \g\in M_{E,c},
$$
and $\ind \g$ is the Maslov index of $\g$.}
The function $F(R)$ is probably a $B^2$-almost periodic function.
Still in the proof of Theorem 4.1 there is a problem of
estimating the difference between the semiclassical counting
function $N^{\WKB}(E)$ and $N(E)$ (cf. Lemma 3.4 above).
The usual Einstein--Brillouin--Keller semiclassical quantization formula
for the eigenvalues,
$$
E^{\text{\rm EBK}}(n_1,n_2)=
H\left(\pi\bigl(n_1+{\a_1\over 4}\bigr),\pi\bigl(n_2
+{\a_2\over 4}\bigr)\right),
$$
where $H(I_1,I_2)$ is the Hamiltonian in action variables and
$\a_1,\a_2$ are Maslov's indices, turns out to be insufficient
for this purpose because it does not work near unstable periodic
orbits. An improved formula can be derived which gives
all semiclassical eigenvalues, but it is difficult to obtain
a good estimate of the error function in this formula (see [KMS]
and [Ble4]).
Theorem 4.1 is used to get a limit probability distribution of
$F(R)$.
{\bf Theorem 4.2} (see [BKS]). {\it Assume that a Liouville surface $M^2$
satisfies the Diophantine Hypothesis. Assume, in addition,
that the lengths of all primitive geodesics on $M^2$ are
linearly independent over $\Z$. Then the error function
$F(R)$ has a limit distribution $p(x)\,dx$ where $p(x)$ is an entire
function such that
$$
\lim_{x\to\pm\infty}-{\log p(x)\over x^4}=c_\pm>0.
$$}
\beginsection 5. Saturation Phenomenon \par
Let $\La$ be a convex domain on a plane, such that (i) $\G=\partial
\La$ is $C^7$-smooth, (ii) $0\in\La$, and (iii) the curvature
$\kappa(x)>0$ for all $x\in\G$. For normalization we will assume
that
$$
\Area\La=1.
\eqno (5.1)
$$
Let
$$
N_\a(E)=\# \, \Z^2\cap (E^{1/2}\La+\a),\qquad \a\in\R^2.
$$
We are interested in the statistical properties of
$N_\a(E+S)-N_\a(E)$ as $E\to\infty$. As was discovered by Casati,
Chirikov, and Guarneri [CCG] and explained by Berry [Ber1], the answer
depends on the relation between $E$ and $S$. We should distinguish
the four cases:
(i) $S\gg E^{1/2}$; (ii) $S\sim E^{1/2}$; (iii)
$E^{1/2}\gg S\gg 1$, and (iv) $S\sim 1$.
We present some rigorous
results in this direction by Bleher and Lebowitz [BL1, BL2].
{\bf Theorem 5.1} (see [BL1]). {\it Let
$$
F_\a(E,S)={N_\a(E+S)-N_\a(E)-S\over E^{1/4}}\,.
\eqno (5.2)
$$
Then for all continuous bounded functions $g(x)$ on
a line,
$$
\lim_{S,T\to\infty\: (S/T^{1/2})\to\infty,\; (S/T)\to 0}\;
{1\over T}\int_T^{2T} g\left( F_\a(E,S)\right) \, dE=
\int_{-\infty}^\infty g(x)\mu_\a(dx),
\eqno (5.3)
$$
where $\mu_a(dx)$ is the probability distribution of the difference
$\xi_1-\xi_2$ of two independent identically distributed random
varibles $\xi_1,\xi_2$ whose distribution is the limit distribution
$\nu_\a(dx)$ of
$$
F_\a(E)={N_\a(E)-E\over E^{1/4}}
$$
(see the formula (2.9) above).}
Theorem 5.1 can be explained as follows. Actually the claim is that
if $S,T\to\infty$ in such a way that $S/T^{1/2}\to\infty$ and
$S/T\to 0$,
then the normalized error functions $[N_\a(E+S) -(E+S)]/E^{1/4}$ and
$[N(E)-E]/ E^{1/4}$ are asymptotically independent. The proof uses
ergodic averaging at two scales, and the large scale averaging secures
the existence of a limit distribution of the two normalized error
functions, while the small scale averaging secures their
independence. This gives the theorem.
{\bf Theorem 5.2} (see [BL1]). {\it Let $F_\a(E,S)$ be as in
(5.2). Then there exists a one-parameter family of probability
measures $\{\mu_\a(dx;z),\; 00$ and all continuous bounded functions $g(x)$,
$$
\lim_{S,T\to\infty\: (S/T^{1/2})\to z}
{1\over T}\int_T^{2T} g\left( F_\a(E,S)\right) \, dE=
\int_1^2 \int_{-\infty}^\infty g(x)\mu_\a(dx;z/c)\,dc.
\eqno (5.4)
$$
If, in addition, the Fourier frequencies $\{ \om(n),\; n\in Z\}$ (see
(2.4) and (2.13)) are linearly
independent over $\Z$ then for all $z>0$,
the distribution $\mu_a(dx;z)$ possesses a density
$p_\a(x;z)=\mu_a(dx;z)/dx$ which is an entire function in $x$ such that
$$
\lim_{x\to\pm\infty}-{\log p_\a(x;z)\over x^4}=c_\pm(\a;z)>0.
$$
Also in this case,
if $\xi_{\a,z}$ is a random variable with the distribution
$\mu_\a(dx;z)$, then as $z\to 0$ the distribution of the random
variable ${\xi_{\a,z}/ (E\xi_{\a,z}^2})^{1/2}$ approaches the standard
Gaussian distribution, with zero mean and variance 1.}
If $\G$ has a symmetry, Theorem 5.2 is valid if the numbers
$\om(n_1),\dots,\om(n_m)$ are linearly independent for all $n_1,\dots,
n_m$ such that all $n_i$ and $n_j$ are not symmetric. In particular,
this
holds for all ellipses $x_1^2+\mu^{-2}x_2^2=1$ with transcendental
$\mu$. A special consideration shows that an analog of Theorem 5.2
holds for the circle as well (see [BL1]). The second moment of the
distribution $\mu_\a(dx;z)$,
$$
D_\a(z)=\int_{-\infty}^\infty x^2\mu_\a(dx;z)
$$
is the number variance, so that
$$
\lim_{S,T\to\infty\: (S/T^{1/2})\to z}
{1\over T}\int_T^{2T} F_\a(E,S)^2 dE=
\int_1^2 D_\a(z/c)\,dc.
\eqno (5.5)
$$
Figs. 14-16, taken from [BL1], show the number variance $D_\a(z)$ for
the circle and $\a=0$,
for the ellipse with $\mu=\pi$ and $\a=0$, and for the same ellipse
and $\a=(0.1,0.1)$. The number variance approaches
an almost periodic function at infinity, and it has different
asymptotics at the origin, which depends on the symmetry of the
problem (see [BL1]). For the circle $D_0(z)\sim Cz\,|\log z|$,
and for the ellipse $D_0(z)\sim 4z$ and $D_\a(z)\sim z$ if
$\a_1\a_2\not=0$ (see Figs. 14--16).
Let us consider now the limit when $1\ll S\ll T^{1/2}$. In this limit
we have results only for the number variance and only for the
ellipse. We will call a number $\mu$ Diophantine if there exist
$M,C>0$ such that for all rational numbers $p/q$,
$$
\left| \mu-{p\over q}\right|\ge {C\over q^M}.
$$
$\mu$ is a Liouville number if there exist a sequence of rationals
$p_i/q_i$ and a sequence of positive numbers $c_i$, $i=1,2,\dots$,
such that $\lim_{i\to\infty} c_i=\infty$ and
$$
\left| \mu-{p_i\over q_i}\right|\le e^{-c_iq_i},\qquad i=1,2,\dots.
$$
{\bf Theorem 5.3} (see [BL2]). {\it Let
$$
\G=\left\{\, (x_1,x_2)\: x_1^2 + \mu^{-2}x_2^2={1\over
\pi\mu}\,\right\}
$$
be an ellipse of area 1, and let $S=T^\g$ where $\g$ is an arbitrary
fixed number such that
$0<\g<1/2$. Then if $\mu$ is a Diophantine number then
$$
\lim_{T\to\infty} {1\over T}\int_T^{2T}{|N(E+S)-N(E)-S|^2\over S
}\,dE=4
\eqno (5.6)
$$
(with the factor 4 coming from symmetry considerations). Contrariwise,
if $\mu$ is rational then
$$
\lim_{T\to\infty} {1\over T\log T}\int_T^{2T}{|N(E+S)-N(E)-S|^2\over S
}\,dE=\left({1\over 2}-\g\right) c(\mu),
\eqno (5.7)
$$
with some $c(\mu)>0$, and if $\mu$ is a Liouville number then
$$
\limsup _{T\to\infty} {1\over T}\int_T^{2T}{|N(E+S)-N(E)-S|^2\over S
}\,dE=\infty.
\eqno (5.8)
$$}
The limit (5.6) is consistent with the conjecture that for every
Diophantine $\mu$, the numbers
$$
E_{n_1,n_2}=n_1^2+\mu^{-2}n_2^2,\qquad n_1,n_2\in\Z,\qquad
n_1,n_2\ge 0,
$$
are locally distributed as a Poisson random process. The limits (5.6)
and (5.7) are not consistent with the Poisson distribution, and this
is related to the
fact that the Poisson
conjecture violates for both rational and Liouville $\mu$.
The last part of Theorem 5.2 concerning the Gaussian limit of
$\xi_{\a,z}$ as $z\to\infty$, supports the following conjecture.
{\bf The Gaussian Conjecture.} {\it Let
$$
F_\a(E,S)={N(E+S)-N(E)-S\over D(T,S)^{1/2}},
$$
where
$$
D(T,S)={1\over T}\int_T^{2T}|N(E+S)-N(E)-S|^2dE,
$$
and $S=T^\g$ where $0<\g<1/2$ is fixed.
Then for every bounded continuous function $g(x)$,
$$
\lim_{T\to\infty} {1\over T}\int_T^{2T} g(F_\a(E,S))\,dE
=(2\pi)^{-1/2}\int_{-\infty}^\infty g(x)e^{-x^2/2}dx.
$$ }
Bleher and, independently, B\"acker have checked this conjecture
numerically. This is not easy because the rate of convergence is very
slow. Fig. 17 shows the distribution density found by Bleher
for a
circular annulus with the exponent $\g=0.25$. The value of $T$ is
4,000,000. The graph is close to a Gaussian density,
with a small bias to negative values.
Let us consider now the case of fixed $S>0$. In this case it is
conjectured that in a ``generic'' situation, $N(E+S)-N(E)$ has a limit
Poisson distribution with the parameter $S$. For an ellipse the
conjecture can be formulated as follows.
{\bf The Poissonian Conjecture.} {\it Let
$$
N(E)=\#\,\{(n_1,n_2)\: n_1^2+\mu^{-2}n_2^2\le {E\over \pi\mu}\},
$$
and $N(E,S)=N(E+S)-N(E)$.
Assume that $\mu$ is a Diophantine number in the sense that for all
$\ep>0$ there exists $C(\ep)>0$ such that
$$
\left|\mu-{p\over q}\right|\ge {C(\ep)\over q^{2+\ep}}\,.
\eqno (5.9)
$$
Then for all bounded continuous functions $g(x)$,
$$
\lim_{T\to\infty}{1\over T}\int_0^T g(N(E,S))\,dE
=\sum_{k=0}^\infty {g(k)S^ke^{-S}\over k!}\,.
$$}
We formulate the Diophantine condition (5.9) in the conservative way,
with the exponent $2+\ep$. Probably this condition can be weakened.
An analytical heuristic proof of the Poisson distribution in
integrable systems is done in the
paper [BT2] by Berry and Tabor (see also [Ber2] and [Tab]).
Sinai [Sin1,Sin2] and Major [Maj] prove a limit Poisson distribution
for $N(E,S)$ in the case of a random domain $\La$. Major
[Maj] also proves some other results in this direction, and
he shows that a typical oval from the probability space, used
in his and Sinai's papers, is not twice differentiable: the
derivative of the oval equation behaves roughly as a Brownian
trajectory. For smooth (say, analytic) ovals the Poisson
conjecture remains open. Sarnak [Sar2] studies the distribution of the
values at nonnegative integers of a positive binary quadratic form
$P(x,y)=\a x^2+\b xy+\g y^2$. He proves that there is a set $M$ in the
space of the coefficients $(\a,\b,\g)$ with $4\a\g>\b^2$
of full Lebesgue measure such that for all $(\a,\b.\g)\in
M$, the second correlation function of the values of $P(m,n)$,
$m,n\in\Z$,
$m,n\ge 0$, has the Poisson asymptotics. To prove the individual
asymptotics for the second correlation function, Sarnak actually
estimates the
fourth correlation function in the ensemble of the quadratic forms.
Cheng, Lebowitz, and Major [CLM] proves
the Poisson-type asymptotics for the second
order correlation function in the lattice-point problem with a random
shift.
The conjecture of local Poisson distribution for integrable quantum
systems is a part of the universality conjecture for
generic quantum systems: for integrable systems the local
statistics is Poissonian, while for hyperbolic systems it is
the Wigner--Dyson statistics of the ensemble of Gaussian
matrices. This conjecture is based on a number of analytical,
numerical and experimental results: see the papers and monographs
by
Bohigas, Giannoni, Schmidt [BGS], Bohigas [Boh], Casati, Chirikov,
Guarneri
[CCG], Berry and Tabor [BT], Berry [Ber2], Gutzwiller [Gut],
Ozorio de Almeida [OdA], Tabor [Tab], Steiner [Ste],
Gr\"af, Harney, Lengeler,
Lewenkopf, Rangacharyulu, Richter, Schardt, and Weidenm\"uller
[G-W], Stoffregen, Stein, St\"ockmann, Ku\'s, and Haake
[S-H], and many others.
\beginsection 6. Lattice-Point Problem in Dimension Greater Than 2 \par
Let $\a\in \R^d$ be a fixed point. Define
$$
N_\a(R)=\#\,\{\, n\in\Z^d\: |n-\a|\le R,
$$
which is the number of lattice points in a $d$-dimensional ball of
radius $R$ centered at $\a$, and
$$
F_\a(R)={N_\a(R)-\Om_d R^d\over R^{(d-1)/2}},
$$
where $\Om_d$ is the volume of a unit ball in $R^d$.
We will assume that $d\ge 3$. Let $\a=(\a_1,\dots, \a_d)$
satisfy the following Diophantine condition: for all $\ep>0$ there
exists $C(\ep)>0$ such that for all $m=(m_0,m_1,\dots,m_d)\in
Z^{d+1},\; m\not= 0$,
$$
\left(\prod_{j=1}^d|m_j|\right)^{1+\ep}\left|m_0+\sum_{j=1}^d
m_j\a_j\right| \ge C(\ep).
\eqno (6.1)
$$
This condition is fulfilled for almost all $\a$ with respect to the
Lebesgue measure. Bleher and Bourgain [BB] prove the following result.
{\bf Theorem 6.1 } (see [BB]. {\it
(1) Let $\a$ satisfy (6.1). Then $F_\a(R)$ has for $R\to\infty$ a
limit distribution $\nu_\a(dx)$ satifying for all $\ep>0$ the tail
estimate
$$
\nu_\a[|x'|>x]< C_1(\ep)\exp(-C_2(\ep)x^{4-\ep}).
$$
(2) For almost all $\a$, $\nu_\a(dx)$ is absolutely continuous with
respect to the Lebesgue measure, with a smooth density $p_\a(x)$
satisfying for all $\ep>0$ the tail estimate
$$
p_\a(x)< C_1(\ep)\exp(-C_2(\ep)x^{4-\ep}).
$$
(3) For $d=3$ and almost all $\a$, $p_\a(x)$ extends to an entire
function and, in addition, one has for all $\ep>0$ the lower estimate
$$
p_\a(x)>c_1(\ep) \exp(-c_2(\ep) x^{4+\ep}).
$$}
The proof of Theorem 6.1 uses the Poisson summation formula,
which yields essentially that
$$
F_\a(R)=C_d\sum_{n\in\Z^d,\; n\not=0}|n|^{-{(d+1)/2}}
\cos (2\pi|n| R-\phi)e^{2\pi i(n,\a)}.
\eqno (6.2)
$$
Writing $|n|=k\sqrt m$ where $m$ is square free one may rewrite
$$
F_\a(R)=\sum_{m\;\text{\rm square free}}f_m(R\sqrt m),
\eqno (6.3)
$$
where
$$
f_m(t)=C_dm^{-(d+1)/4}\sum_{k=1}^\infty k^{-(d+1)/2}
r_{k^2m}(\a)\cos (2\pi kt-\phi)
\eqno (6.4)
$$
and
$$
r_m(\a)=\sum_{n\in\Z^d\:|n|^2=m} e^{2\pi i (n,\a)}.
\eqno (6.5)
$$
The functions $f_m(t)$ are periodic functions of period 1. The crucial
role in the proof of Theorem 6.1 plays the following lemma.
{\bf Lemma 6.2} (see [BB]). {\it
$$\eqalign{
&\sum_{m\le M} |r_m(\a)|^2\ll M^{{d\over 2}+\ep},\cr
&\sum_{m\le M} |r_m(\a)|^2> cM^{{d\over 2}},\cr
&|r_m(\a)|\ll m^{{d-1\over 4}+\ep}\,.\cr}
$$}
With the help of Lemma 6.2, it is proved that for typical $\a$,
$F_\a(R)$ is a $B^1$-almost periodic function and Theorem 6.1 holds.
The Diophantine condition on $\a$ is essential. If $\a$ is rational
then $N_\a(R)$ has big jumps. To see it let us consider $\a=0$.
For $d>4$ the number of solutions of the equation
$$
n_1^2+\dots+n_d^2=m
$$
is of the order of $m^{(d-2)/2}$, which gives rise to the jumps of the
function $N_0(R)$.
For $d=3$ Jarnik [Jar] proves the following result. Let
$$
N_0(R)=\#\, \{ n\in\Z^3\: |n|\le R\},
$$
and
$$
F_0(R)={N_0(R)-{4\pi R^3\over 3}\over R},\qquad R\ge 1.
$$
Put $F_0(R)=0$ for $R<1$.
{\bf Theorem 6.3} (see [Jar]). {\it
$$
\lim_{T\to\infty}{1\over T\log T} \int_0^T|F_0(R)|^2dR=K>0.
$$}
Bleher and Dyson [BD3] find that
$$
K={32\z(2)\over 7\z(3)}\,,
$$
and they conjecture that the limit distribution of $F_0(R)$ is
Gaussian.
Theorem 6.3 can be extended to $F_\a(R)$ with rational $\a$.
{\bf Theorem 6.4.}
{\it If
$$
\a=\left({p_1\over q_1},{p_2\over q_2},{p_3\over q_3}\right)\in\Q^3,
\qquad
\text {\rm g.c.d.}\,(q_1,q_2,q_3)=1,
$$
then
$$
\lim_{T\to\infty}{1\over T\log T} \int_0^T|F_\a(R)|^2dR=K(\a)>0.
$$
If $q_1,q_2,q_3$ are odd then
$$
K(\a)={32\z(2)\over 7q^2\z(3)}\prod_{p|q}{p^2\over p^2+p+1}\,,
$$
where $q$ is the least common multiplier of $q_1,q_2,q_3$ and the
product is taken over all prime divisors $p$ of $q$. If all $q_i=2\mod
4$ then $K(\a)=K(\a/2)$. In all other cases
$$
K(\a)={8\z(2)\over q^2\z(3)}\prod_{p|q}{p^2\over p^2+p+1}\,.
$$}
The proof of Theorem 6.4 is an application of the formula (6.2) and
the following lemma which is based on the Hardy-Littlewood circle
method.
{\bf Lemma 6.5.} {\it Assume that $d=3,\;\a\in\R^3$, and $r_m(\a)$ is
defined as in (6.5). Then
$$
\lim_{M\to\infty} M^{-2}\sum_{m=1}^M|r_m(\a)|^2=L(\a)>0,
$$
with $L(\a)=(\pi^2/2)K(\a)$, where $K(\a)$ is defined in Theorem 6.4.}
\beginsection 7. Quantum Linear Oscillators \par
The spectrum of a system of $m$ linear oscillators is written as
$$
E(n_1,\dots,n_m)=E_0+n_1\om_1+\dots +n_m\om_m,\qquad n_j\in\Z,\quad
n_j\ge 0,
\eqno (7.1)
$$
where $E_0$ is the ground state energy and $\om_j$ are
frequencies. When $m=2$,
$$
E(n_1,n_2)=E_0+\om_1(n_1+\a n_2),\qquad \a={\om_2\over \om_1}\,,
$$
hence the problem of finding the asymptotics of the eigenvalues
$E(n_1,n_2)$ reduces to the same problem for the numbers
$$
\la(n_1,n_2)=n_1+\a n_2,\qquad n_1,n_2\in \Z,\quad n_1,n_2\ge 0.
\eqno (7.2)
$$
We will assume that $\a>0$ is irrational. A numerical and analytical
study by Berry
and Tabor [BT] indicates that the distribution of spacing
between neighboring $\la(n_1,n_2)$ depends on
the arithmetic properties of $\a$. The results for the golden mean
$\a=(\sqrt 5-1)/2$ were obtained by Pandey, Bohigas, and Giannoni
[PBG] and by Bleher [Ble5]. Let us order the numbers $\la(n_1,n_2)$ in
the increasing order and denote them by
$$
0=\la_0<\la_1<\la_2<\dots\to\infty.
$$
Consider the spacings
$$
\De_j=\la_{j+1}-\la_j,
$$
and the minimal spacing on $[0,\la]$,
$$
\De_{\min}(\la)=\min\{\De_j\: \la_j\le \la\}.
$$
Define normalized spacings $s_j$ on $[0,\la]$ as
$$
s_j={\De_j\over \De_{\min}},
$$
and the distribution function $P_\la(s)$ as
$$
P_\la(s)={\#\, \{ \la_j\le \la\: s_j\le s\}\over \#\,\{ \la_j\le
\la\}}.
$$
{\bf Theorem 7.1} (see [Ble5]). {\it Assume that $\a=(\sqrt
5-1)/2$. Then there exists a periodic
of period 1 family of distribution functions $P(s;\tau)$,
$$
P(s;\tau+1)=P(s;\tau),
$$
such that
$$
\lim_{\la\to\infty} \sup_{0\le s<\infty}|P_\la(s)- P(s;\log_\b\la)|=0,
$$
where $\b=\a^{-1}$. For every $\tau$ the probability distribution
$dP(s;\tau)$ is an atomic
measure, with atoms at the points $\b^k$ and with some weights $w_k(\tau)$,
$k=0,1,2,\dots$, which decay when $k\to\infty$ as $C\a^{2k}$.}
In other words, when $\la\to\infty$ the spacing distribution
function $P_\la(s)$
oscillates periodically as a function of $\log_\b\la$. In particular,
this implies that $P_\la(s)$ has no limit as $\la\to\infty$.
An explicit formula for the limiting periodic family $P(s;\tau)$ is
derived in [Ble5]. As a matter of fact, $dP_\la(s)$ is an atomic
measure with atoms at the points $\b^k$ with some weights $w_{k,\la}$,
$k=0,1,2,\dots$. In the proof of Theorem 7.1 it is verified that
the weights $w_{k,\la}$ approach $w_k(\log_\b\la)$ when
$\la\to\infty$. It is to be noted that the spectrum is very rigid, and
locally the spacing takes only three values (see, e.g., [AA], [Sta],
[PBG], and [Ble5]). This implies a
hard core repulsion of energy levels with $P_\la(s)=0$ for all $s<1$.
Along the same lines, a similar result can
be obtained for every quadratic irrational $\a$. The important point in
the proof is that the continued fraction of any quadratic
irrational is periodic.
The case of generic $\a$ is more complicated. In general, there
exists no limit of spacing distribution, and the problem is
to find, in what sense the limit can exist?
This problem is studied by Bleher [Ble6]. Consider the
continued fraction $[a_1,a_2,\dots]$ which represents $\a$ and
the partial fractions
$$
{p_k\over q_k}=[a_1,\dots,a_k],\qquad k\ge 1.
$$
{\bf Theorem 7.2} (see [Ble6]). {\it There exists a probability
distribution function $P(s)$ such that for almost all $\a$ on $[0,1]$
with respect to Lebesgue measure,
$$
\lim_{n\to\infty} n^{-1}\sum_{k=1}^n P_{p_k}(s)=P(s).
$$}
The limit distribution function $P(s)$ is expressed in terms of the
natural extension of the Gauss map $x\to\{1/x\}$ (see [Ble6]).
\vskip 1cm
{\it Acknowledgements.} The author thanks Peter Sarnak and Martin
Huxley for their comments to the preliminary version of this
paper. The paper is based on the talk presented at the IMA Summer
program on ``Emerging Applications of Number Theory'', July 15-20,
1996. The author is grateful to the organizers of the program for
the invitation to give a talk at the meeting and to participate in
this volume. He is also indebted to
the Institute for Mathematics and its Applications for hospitality.
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\vfill\eject
\beginsection Figures Captures \par
Fig. 1. The graph of the function $n(R^2)=\#\, \{ n\in\Z^2\: |n|\le
R\}-\pi R^2$.
Fig. 2. The density $p_\a(x)$ of the limit distribution of the
normalized error function $F_\a(R)$ given in (2.14) with $\mu=1$
(circle) and $\a=(0,0)$.
Fig. 3. $\mu=\pi/10$, $\a=(0,0)$.
Fig. 4. $\mu=\pi/50$, $\a=(0,0)$.
Fig. 5. $\mu=1$, $\a=(0.2,0.2)$.
Fig. 6. $\mu=\pi/10$, $\a=(0.2,0.2)$.
Fig. 7. $\mu=\pi/50$, $\a=(0.2,0.2)$.
Fig. 8. $\mu=1$, $\a=(0.3437,0.4304)$.
Fig. 9. $\mu=\pi/10$, $\a=(0.3437,0.4304)$.
Fig. 10. $\mu=\pi/50$, $\a=(0.3437,0.4304)$.
Fig. 11. Geodesic on a surface of revolution.
Fig. 12. The phase function $\om(I)$ for different surfaces of
revolution. Cross sections of the surfaces of revolution are shown in
the lower part of the figure.
Fig. 13. The energy level set for the surfaces of revolution shown on
Figure 12.
Fig. 14. The number variance for the circle problem.
Fig. 15. The number variance for an ellipse with $\mu=\pi$,
$\a=(0,0)$.
Fig. 16. The number variance for an ellipse with $\mu=\pi$,
$\a=(0.1,0.1)$.
Fig. 17. The density of distribution of the number of lattice points
in a circular annulus with the parameter $\g=0.25$.
\bye
*