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Frontiers in Number Theory, Physics, and Geometry I
Pierre Cartier Bernard Julia
Pierre Moussa Pierre Vanhove (Eds.)
Frontiers in Number Theory,
Physics, and Geometry I
On Random Matrices, Zeta Functions,
and Dynamical Systems
ABC
Pierre Cartier
I.H.E.S.
35 route de Chartres
F-91440 Bures-sur-Yvette
France
e-mail:
Bernard Julia
LPTENS
24 rue Lhomond
75005 Paris
France
e-mail:
Pierre Moussa
Service de Physique Théorique
CEA/Saclay
F-91191 Gif-sur-Yvette
France
e-mail:
Pierre Vanhove
Service de Physique Théorique
CEA/Saclay
F-91191 Gif-sur-Yvette

France
e-mail:
Cover photos:
G. Pólya (courtesy of G.L. Alexanderson); Eugene P. Wigner (courtesy of M. Wigner).
Library of Congress Control Number: 2005936349
Mathematics Subject Classification (2000): 11A55, 11K50, 11M41, 15A52, 37C27,
37C30, 58B34, 81Q50, 81R60
ISBN-10 3-540-23189-7 Springer Berlin Heidelberg New York
ISBN-13 978-3-540-23189-9 Springer Berlin Heidelberg New York
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Preface
The present book contains fifteen contributions on various topics related to
Number Theory, Physics and Geometry. It presents, together with a forthcom-
ing second volume, most of the courses and seminars delivered at the meeting
entitled “Frontiers in Number Theory, Physics and Geometry”, which took
place at the Centre de Physique des Houches in the french Alps March 9-21,
2003.
The relation between mathematics and physics has a long history. Let us
mention only ordinary differential equations and mechanics, partial differential
equations in solid and fluid mechanics or electrodynamics, group theory is
essential in crystallography, elasticity or quantum mechanics. . .
The role of number theory and of more abstract parts of mathematics
such as topological, differential and algebraic geometry in physics has become
prominent more recently. Diverse instances of this trend appear in the works
of such scientists as V. Arnold, M. Atiyah, M. Berry, F. Dyson, L. Faddeev,
D.Hejhal,C.Itzykson,V.Kac,Y.Manin,J.Moser,W.Nahm,A.Polyakov,
D. Ruelle, A. Selberg, C. Siegel, S. Smale, E. Witten and many others.
In 1989 a first meeting took place at the Centre de Physique des Houches.
The triggering idea was due at that time to the late Claude Itzykson (1938-
1995). The meeting gathered physicists and mathematicians, and was the
occasion of long and passionate discussions.
The seminars were published in a book entitled “Number Theory and
Physics”, J M. Luck, P. Moussa, and M. Waldschmidt editors, Springer Pro-
ceedings in Physics, Vol. 47, 1990. The lectures were published as a second
book entitled “From Number Theory to Physics”, with C. Itzykson joining
the editorial team, Springer (2nd edition 1995).
Ten years later the evolution of the interface between theoretical physics
and mathematics prompted M. Waldschmidt, P. Cartier and B. Julia to re-
new the experience. However the emphasis was somewhat shifted to include

in particular selected chapters at the interface of physics and geometry, ran-
dom matrices or various zeta- and L- functions. Once the project of the new
meeting entitled “Frontiers in Number Theory, Physics and Geometry” re-
ceived support from the European Union the High level scientific conference
was organized in Les Houches.
VI Preface
The Scientific Committee for the meeting “Frontiers in Number The-
ory, Physics and Geometry”, was composed of the following scientists: Frits
Beukers, Jean-Benoˆıt Bost, Pierre Cartier, Predrag Cvitanovic, Michel Duflo,
Giovanni Gallavotti, Patricio Leboeuf, Werner Nahm, Ivan Todorov, Claire
Voisin, Michel Waldschmidt, Jean-Christophe Yoccoz, and Jean-Bernard Zu-
ber. The Organizing Committee included:
Bernard Julia (LPTENS, Paris scientific coordinator),
Pierre Moussa (SPhT CEA-Saclay), and
Pierre Vanhove (CERN and SPhT CEA-Saclay).
During two weeks, five lectures or seminars were given every day to about
seventy-five participants. The topics belonged to three main domains:
1. Dynamical Systems, Number theory, and Random matrices,
with lectures by E. Bogomolny on Quantum and arithmetical chaos, J. Conrey
on L-functions and random matrix theory, J C. Yoccoz on Interval exchange
maps, and A. Zorich on Flat surfaces;
2. Polylogarithms and Perturbative Physics,
with lectures by P. Cartier on Polylogarithms and motivic aspects, W. Nahm
on Physics and dilogarithms, and D. Zagier on Polylogarithms;
3. Symmetries and Non-pertubative Physics, with lectures by
A. Connes on Galoisian symmetries, zeta function and renormalization,
R. Dijkgraaf on String duality and automorphic forms,
P. Di Vecchia on Gauge theory and D-branes,
E. Frenkel on Vertex algebras, algebraic curves and Langlands program,
G. Moore on String theory and number theory,

C. Soul´e on Arithmetic groups.
In addition seminars were given by participants many of whom could have
given full sets of lectures had time been available. They were: Z. Bern, A.
Bondal, P. Candelas, J. Conway, P. Cvitanovic, H. Gangl, G. Gentile, D.
Kreimer, J. Lagarias, M. Marcolli, J. Marklof, S. Marmi, J. McKay, B. Pioline,
M. Pollicott, H. Then, E. Vasserot, A. Vershik, D. Voiculescu, A. Voros, S.
Weinzierl, K. Wendland, A. Zabrodin.
We have chosen to reorganize the written contributions in two parts ac-
cording to their subject. These naturally lead to two different volumes. The
present volume is the first one, let us now briefly describe its contents.
This volume is itself composed of three parts including each lectures and
seminars covering one theme. In the first part, we present the contributions
on the theme “Random matrices : from Physics to Number Theory”. It begins
with lectures by E. Bogomolny, which review three selected topics of quan-
tum chaos, namely trace formulas with or without chaos, the two-point spec-
tral correlation function of Riemann zeta function zeroes, and the two-point
spectral correlation functions of the Laplace-Beltrami operator for modular
Preface VI I
domains leading to arithmetic chaos. The lectures can serve as a non-formal
introduction to mathematical methods of quantum chaos. A general introduc-
tion to arithmetic groups will appear in the second volume. There are then
lectures by J. Conrey who examines relations between random-matrix theory
and families of arithmetic L-functions (mostly in characteristics zero), that is
Dirichlet series satisfying functional equations similar to those obeyed by the
Riemann zeta-function. The relevant L-functions are those associated with
cusp-forms. The moments of L-functions are related to correlation functions
of eigenvalues of random matrices.
Then follow a numb er of seminar presentations: by J. Marklof on some
energy level statistics in relation with almost modular functions; by H. Then
on arithmetic quantum chaos in a particular three-dimensional hyperbolic

domain, in relation to Maass waveforms. Next P. Wiegmann and A. Zabrodin
study the large N expansion for normal and complex matrix ensembles. D.
Voiculescu reviews symmetries of free probability models. Finally A. Vershik
presents some random (resp. universal) graphs and metric spaces.
In the second part “Zeta functions: a transverse tool”, the theme is zeta-
functions and their applications.
First the lectures by A. Connes were written up in collaboration with M.
Marcolli and have been divided into two parts.
The second one will appear in the second volume as it relates to renor-
malization of quantum field theories. In their fi rst chapter they introduce
the noncommutative space of commensurability classes of Q-lattices and the
arithmetic properties of KMS states in the corresponding quantum statistical
mechanical system. In the 1-dimensional case this space gives the spectral
realization of zeroes of zeta-functions. They give a description of the multiple
phase transitions and arithmetic spontaneous symmetry breaking in the case
of Q-lattices of dimension two. The system at zero temperature settles onto a
classical Shimura variety, which parametrizes the pure phases of the system.
The noncommutative space has an arithmetic structure provided by a ratio-
nal subalgebra closely related to the modular Hecke algebra. The action of
the symmetry group involves the formalism of superselection sectors and the
full noncommutative system at positive temperature. It acts on values of the
ground states at the rational elements via the Galois group of the modular
field.
Then we report seminars given by A. Voros on zeta functions built on
Riemann zeroes; by J. Lagarias on Hilbert spaces of entire functions and
Dirichlet L-functions; and by M. Pollicott on Dynamical zeta functions and
closed orbits for geodesic and hyperbolic flows.
In the third part “ Dynamical systems: interval exchanges, flat surfaces and
small divisors”, are gathered all the other contributions on dynamical systems.
The lectures by A. Zorich provide an extensive self-contained introduction to

the geometry of Flat surfaces which allows a description of flows on compact
VIII Preface
Riemann surfaces of arbitrary genus. The course by J C. Yoccoz analyzes
Interval exchange maps such as the first return maps of these flows. Ergodic
properties of maps are connected with ergodic properties of flows. This leads
to a generalization to surfaces of higher genus of the irrational flows on the
two dimensional torus. The adaptation of a continued fraction like algorithm
to this situation is a prerequisite to extension of small divisors techniques to
higher genus cases.
Finally we conclude this volume with seminars given by G. Gentile on Br-
juno numbers and dynamical systems and by S. Marmi on Real and Complex
Brjuno functions. In both talks either perturbation of irrational rotations or
twist maps are considered, with fine details on arithmetic conditions (Brjuno
condition and Brjuno numbers) for stability of trajectories under perturba-
tions of parameters, and on the size of stability domains in the parametric
space (Brjuno functions).
The following institutions are most gratefully acknowledged for their gen-
erous financial support to the meeting:
D´epartement Sciences Physiques et Math´ematiques and the Service de
Formation permanente of the Centre National de la Recherche Scientifique;
´
Ecole Normale Sup´erieure de Paris; D´epartement des Sciences de la mati`ere du
Commissariat `al’
´
Energie Atomique; Institut des Hautes Etudes Scientifiques;
National Science Foundation; Minist`ere de la Recherche et de la Technolo-
gie and Minist`ere des Affaires
´
Etrang`eres; The International association of
mathematical physics and most especially the Commission of the European

Communities.
Three European excellence networks helped also in various ways. Let
us start with the most closely involved “Mathematical aspects of Quantum
chaos”, but the other two were “Superstrings” and “Quantum structure of
spacetime and the geometric nature of fundamental interactions”.
On the practical side we thank CERN Theory division for allowing us
to use their computers for the webpage and registration process. We are also
grateful to Marcelle Martin, Thierry Paul and the staff of les Houches for their
patient help. We had the privilege to have two distinguished participants:
C´ecile de Witt-Morette (founder of the Les Houches School) and the late
Bryce de Witt whose communicative and critical enthusiasm were greatly
appreciated.
Paris, July 2005 Bernard Julia
Pierre Cartier
Pierre Moussa
Pierre Vanhove
List of Contributors
List of Authors: (following the order of appearance of the contributions)
• E. Bogomolny, Laboratoire de Physique Th´eorique et Mod`eles Statistiques
Universit´e de Paris XI, Bˆat. 100, 91405 Orsay Cedex, France
• J. Brian Conrey, American Institute of Mathematics, Palo Alto, CA, USA
• Jens Marklof, School of Mathematics, University of Bristol, Bristol BS8
1TW, U.K.
• H. Then, Abteilung Theoretische Physik, Universit¨at Ulm, Albert-Einstein-
Allee 11, 89069 Ulm, Germany
• A. Zabrodin, Institute of Biochemical Physics, Kosygina str. 4, 119991
Moscow, Russia and ITEP, Bol. Cheremushkinskaya str. 25, 117259 Moscow,
Russia
P. Wiegmann, James Frank Institute and Enrico Fermi Institute of the
University of Chicago, 5640 S.Ellis Avenue, Chicago, IL 60637, USA

Landau Institute for Theoretical Physics, Moscow, Russia
• D. Voiculescu, Department of Mathematics University of California at
Berkeley Berkeley, CA 94720-3840, USA
• A.M. Vershik, St.Petersburg Mathematical Institute of Russian Academy
of Science Fontanka 27 St.Petersburg, 191011, Russia
• A. Connes, Coll`ege de France, 3, rue Ulm, F-75005 Paris, France
I.H.E.S. 35 route de Chartres F-91440 Bures-sur-Yvette, France
M. Marcolli, Max–Planck Institut f¨ur Mathematik, Vivatsgasse 7, D-53111
Bonn, Germany
• A. Voros, CEA, Service de Physique Th´eorique de Saclay (CNRS URA
2306) F-91191 Gif-sur-Yvette Cedex, France
• J.C. Lagarias, Department of Mathematics, University of Michigan, Ann
Arbor,MI 48109-1109 USA
• M. Pollicott, Department of Mathematics, Manchester University, Oxford
Road, Manchester M13 9PL UK
• J C. Yoccoz, Coll`ege de France, 3 Rue d’Ulm, F-75005 Paris, France
• A. Zorich, IRMAR, Universit´e de Rennes 1, Campus de Beaulieu, 35042
Rennes, France
• G. Gentile, Dipartimento di Matematica, Universit`a di Roma Tre, I-00146
Roma, Italy
• S.Marmi, Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa,
Italy
X List of Contributors
P. Moussa, Service de Physique Th´eorique, CEA/Saclay, F-91191 Gif-sur-
Yvette, France
J C. Yoccoz, Coll`ege de France, 3 Rue d’Ulm, F-75005 Paris, France
Editors:
• Bernard Julia, LPTENS, 24 rue Lhomond 75005 Paris, France, e-mail:

• Pierre Cartier, I.H.E.S. 35 route de Chartres F-91440 Bures-sur-Yvette,

France, e-mail:
• Pierre Moussa, Service de Physique Th´eorique, CEA/Saclay, F-91191 Gif-
sur-Yvette, France, e-mail:
• Pierre Vanhove, Service de Physique Th´eorique, CEA/Saclay, F-91191 Gif-
sur-Yvette, France
Contents
Part I Random Matrices: from Physics to Number Theory
Quantum and Arithmetical Chaos
Eugene Bogomolny 3
Notes on L-functions and Random Matrix Theory
J. Brian Conrey 107
Energy Level Statistics, Lattice Point Problems, and Almost
Modular Functions
Jens Marklof 163
Arithmetic Quantum Chaos of Maass Waveforms
H. Then 183
Large N Expansion for Normal and Complex Matrix
Ensembles
P. Wiegmann, A. Zabrodin 213
Symmetries Arising from Free Probability Theory
Dan Voiculescu 231
Universality and Randomness for the Graphs and Metric
Spaces
A. M. Vershik 245
Part II Zeta Functions
From Physics to Number Theory via Noncommutative
Geometry
Alain Connes, Matilde Marcolli 269
XI I Contents
More Zeta Functions for the Riemann Zeros

Andr´e Voros 351
Hilbert Spaces of Entire Functions and Dirichlet L-Functions
Jeffrey C. Lagarias 367
Dynamical Zeta Functions and Closed Orbits for Geodesic
and Hyperbolic Flows
Mark Pollicott 381
Part III Dynamical Systems: interval exchange, flat surfaces, and
small divisors
Continued Fraction Algorithms for Interval Exchange Maps:
an Introduction
Jean-Christophe Yoccoz 403
Flat Surfaces
Anton Zorich 439
Brjuno Numbers and Dynamical Systems
Guido Gentile 587
Some Properties of Real and Complex Brjuno Functions
Stefano Marmi, Pierre Moussa, Jean-Christophe Yoccoz 603
Part IV Appendices
List of Participants 629
Index 633
Part I
Random Matrices: from Physics to Number
Theory
Quantum and Arithmetical Chaos
Eugene Bogomolny
Laboratoire de Physique Th´eorique et Mod`eles Statistiques
Universit´edeParisXI,Bˆat. 100, 91405 Orsay Cedex, France

Summary. The lectures are centered around three selected topics of quantum
chaos: the Selberg trace formula, the two-point spectral correlation functions of

Riemann zeta function zeros, and the Laplace–Beltrami operator for the modular
group. The lectures cover a wide range of quantum chaos applications and can serve
as a non-formal introduction to mathematical methods of quantum chaos.
Introduction 5
I Trace Formulas 7
1 Plane Rectangular Billiard 7
2 Billiards on Constant Negative Curvature Surfaces 15
2.1 Hyperbolic Geometry 16
2.2 Discretegroups 18
2.3 Classical Mechanics 20
2.4 QuantumProblem 21
2.5 Constructionof the GreenFunction 22
2.6 Density ofState 23
2.7 Conjugated Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 SelbergTraceFormula 26
2.9 Density ofPeriodicOrbits 29
2.10 SelbergZetaFunction 30
2.11 ZerosoftheSelbergZetaFunction 32
2.12 Functional Equation 33
3 Trace Formulas for Integrable Dynamical Systems 33
3.1 Smooth PartoftheDensity 34
3.2 OscillatingPartoftheDensity 34
4 Trace Formula for Chaotic Systems 36
4.1 SemiclassicalGreenFunction 36
4 Eugene Bogomolny
4.2 Gutzwiller Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Riemann Zeta Function 41
5.1 FunctionalEquation 42
5.2 TraceFormulafortheRiemannZeros 43
5.3 Chaotic Systems andtheRiemannZetaFunction 46

6 Summary 46
II Statistical Distribution of Quantum Eigenvalues 49
1 Correlation Functions 52
1.1 Diagonal Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.2 Criterion of Applicability of Diagonal Approximation . . . . . . . . . . . . 55
2 Beyond the Diagonal Approximation 58
2.1 TheHardy–Littlewood Conjecture 59
2.2 Two-Point Correlation Function of Riemann Zeros . . . . . . . . . . . . . . . 64
3 Summary 65
III Arithmetic Systems 70
1 Modular group 72
2 Arithmetic Groups 73
2.1 Algebraic Fields 74
2.2 QuaternionAlgebras 76
2.3 Criterion ofArithmeticity 81
2.4 Multiplicities of Periodic Orbits for General Arithmetic Groups . . . 82
3 Diagonal Approximation for Arithmetic Systems 85
4 Exact Two-Point Correlation Function for the Modular
Group 87
4.1 BasicIdentities 87
4.2 Two-Point Correlation Function of Multiplicities . . . . . . . . . . . . . . . . 89
4.3 ExplicitFormulas 92
4.4 Two-PointFormFactor 93
5 Hecke Operators 94
6 Jacquet–Langlands Correspondence 98
7 Non-arithmetic Triangles 99
8 Summary 102
References 103
Quantum and Arithmetical Chaos 5
Introduction

Quantum chaos is a nickname for the investigation of quantum systems which
do not permit exact solutions. The absence of explicit formulas means that
underlying problems are so complicated that they cannot be expressed in
terms of known ( simple) functions. The class of non-soluble systems is very
large and practically any model (except a small set of completely integrable
systems) belongs to it. An extreme case of quantum non-soluble problems
appears naturally when one considers the quantization of classically chaotic
systems which explains the word ‘chaos’ in the title.
As, by definition, for complex systems exact solutions are not possible,
new analytical approaches were developed within quantum chaos. First, one
may find relations between different non-integrable models, hoping that for
certain questions a problem will be more tractable than another. Second,
one considers, instead of exact quantities, the calculation of their smoothed
values. In many cases such coarse graining appears naturally in experimental
settings and, usually, it is more easy to treat. Third, one tries to understand
statistical properties of quantum quantities by organizing them in suitable
ensembles. An advantage of such an approach is that many different models
may statistically be indistinguishable which leads to the notion of statistical
universality.
The ideas and methods of quantum chaos are not restricted only to quan-
tum models. They can equally well be applied to any problem whose analytical
solution either is not possible or is very complicated. One of the most spec-
tacular examples of such interrelations is the application of quantum chaos
to number theory, in particular, to the zeros of the Riemann zeta function.
Though a hypothetical quantum-like system whose eigenvalues coincide with
the imaginary part of Riemann zeta function zeros has not (yet!) been found,
the Riemann zeta function is, in many aspects, similar to dynamical zeta func-
tions and the investigation of such relations already mutually enriched both
quantum chaos and number theory (see e.g. the calculation by Keating and
Snaith of moments of the Riemann zeta function using random matrix theory

[43]).
The topics of these lectures were chosen specially to emphasize the inter-
play between physics and mathematics which is typical in quantum chaos.
In Chapter I different types of trace formulas are discussed. The main at-
tention is given to the derivation of the Selberg trace formula which relates
the spectral density of automorphic Laplacian on hyperbolic surfaces gener-
ated by discrete groups with classical periodic orbits for the free motion on
these surfaces. This question is rarely discussed in the Physics literature but
is of general interest because it is the only case where the trace formula is
exact and not only a leading semiclassical contribution as for general dynam-
ical systems. Short derivations of trace formulas for dynamical systems and
for the Riemann zeta function zeros are also presented in this Chapter.
6 Eugene Bogomolny
According to the well-known conjecture [17] statistical properties of eigen-
values of energies of quantum chaotic systems are described by standard ran-
dom matrix ensembles depending only on system symmetries. In Chapter II
we discuss analytical methods of confirmation of this conjecture. The largest
part of this Chapter is devoted to a heuristic derivation of the ‘exact’ two-
point correlation function for the Riemann zeros. The derivation is based on
the Hardy–Littlewood conjecture about the distribution of prime pairs which
is also reviewed. The resulting formula agrees very well with numerical calcu-
lations of Odlyzko.
In Chapter III a special class of dynamical systems is considered, namely,
hyperb olic surfaces generated by arithmetic groups. Though from the view-
point of classical mechanics these models are the best known examples of
classical chaos, their spectral statistics are close to the Poisson statistics typ-
ical for integrable models. The reason for this unexpected behavior is found
to be related with exponential degeneracies of periodic orbit lengths charac-
teristic for arithmetical systems. The case of the modular group is considered
in details and the exact expression for the two-point correlation function for

this problem is derived.
To be accessible for physics students the lectures are written in a non-
formal manner. In many cases analogies are used instead of theorems and
complicated mathematical notions are illustrated by simple examples.
Quantum and Arithmetical Chaos 7
I. Trace Formulas
Different types of trace formulas are the cornerstone of quantum chaos.
Trace formulas relate quantum properties of a system with their classical
counterparts. In the simplest and widely used case the trace formula expresses
the quantum density of states through a sum over periodic orbits and each
term in this sum can be calculated from pure classical mechanics.
In general, dynamical trace formulas represent only the leading term of
the semiclassical expansion in powers of . The computation of other terms is
possible though quite tedious [1]. The noticeable exception is the free motion
on constant negative curvature surfaces generated by discrete groups where
the trace formula (called the Selberg trace formula) is exact. The derivation
of this formula is the main goal of this Section.
For clarity, in Sect. 1 the simplest case of the rectangular billiard is briefly
considered and the trace formula for this system is derived. The derivation
is presented in a manner which permits to generalize it to the Selberg case
of constant negative curvature surfaces generated by discrete groups which
is considered in details in Sect. 2. In Sects. 3 and 4 the derivations of the
trace formula for, respectively, classically integrable and chaotic systems are
presented. In Sect. 5 it is demonstrated that the density of Riemann zeta
function zeros can be written as a sort of trace formula where the role of
periodic orbits is played by prime numbers. Section 6 is a summary of this
Chapter.
1 Plane Rectangular Billiard
To clarify the derivation of trace formulas let us consider in details a very
simple example, namely, the computation of the energy spectrum for the plane

rectangular billiard with periodic boundary conditions.
This problem consists of solving the equation
(∆ + E
n
)Ψ
n
(x, y)=0 (1)
where ∆ = ∂
2
/∂x
2
+ ∂
2
/∂y
2
is the usual two-dimensional Laplacian with
periodic boundary conditions
Ψ
n
(x + a, y)=Ψ
n
(x, y + b)=Ψ
n
(x, y)(2)
where a and b are sizes of the rectangle.
The plane wave
Ψ
n
(x, y)=e
ik

1
x+ik
2
y
is an admissible solution of (1). Boundary conditions (2) determine the allowed
values of the momentum k
8 Eugene Bogomolny
k
1
=
2π
a
n
1
,k
2
=
2π
b
n
2
,
with n
1
,n
2
=0, ±1, ±2, , and, consequently, energy eigenvalues are
E
n
1

n
2
=

2π
a
n
1

2
+

2π
b
n
2

2
. (3)
The first step of construction of trace formulas is to consider instead of indi-
vidual eigenvalues their density defined as the sum over all eigenvalues which
explains the word ‘trace’
d(E) ≡
+∞

n
1
,n
2
=−∞

δ(E −E
n
1
n
2
) . (4)
To transforms this and similar expressions into a convenient form one often
uses the Poisson summation formula
+∞

n=−∞
f(n)=
+∞

m=−∞

+∞
−∞
e
2πimn
f(n)dn. (5)
An informal proof of this identity can, for example, be done as follows.
First
+∞

n=−∞
f(n)=

+∞
−∞

f(x)g(x)dx
where g(x)istheperiodicδ-function
g(x)=
+∞

n=−∞
δ(x −n) .
As any periodic function with period 1, g(x) can be expanded into the Fourier
series
g(x)=
+∞

m=−∞
e
2πimx
c
m
.
Coefficients c
m
are obtained by the integration of g(x)overoneperiod
c
m
=

+1/2
−1/2
g(y)e
−2πimy
dy =1

which gives (5).
By applying the Poisson summation formula (5) to the density of states
(4) one gets
Quantum and Arithmetical Chaos 9
d(E)=
+∞

m
1
,m
2
=−∞

e
2πi(m
1
n
1
+m
2
n
2
)
×
× δ

E −

2π
a

n
1

2
−

2π
b
n
2

2

dn
1
dn
2
.
Perform the following substitutions: E = k
2
, n
1
= ar cos ϕ/2π,andn
2
=
br sin ϕ/2π.Thendn
1
dn
2
= abrdrdϕ/(2π)

2
and
d(E)=
µ(D)
(2π)
2
+∞

m
1
,m
2
=−∞

e
i(m
1
a cos ϕ+m
2
b sin ϕ)r
δ(k
2
− r
2
)rdrdϕ
=
µ(D)
2(2π)
2
+∞


m
1
,m
2
=−∞

2π
0
e
ik
√
(m
1
a)
2
+(m
2
b)
2
cos ϕ
dϕ
=
µ(D)
4π
+∞

m
1
,m

2
=−∞
J
0
(kL
p
) ,
where µ(D)=ab is the area of the rectangle,
J
0
(x)=
1
2π

2π
0
e
ix cos ϕ
dϕ
is the Bessel function of order zero (see e.g. [32], Vol. 2, Sect. 7), and
L
p
=

(m
1
a)
2
+(m
2

b)
2
is (as it is easy to check) the length of a periodic orbit in the rectangle with
perio dic boundary conditions.
Separating the term with m
1
= m
2
= 0 one concludes that the eigenvalue
density of the rectangle with periodic boundary conditions can be written as
the sum of two terms
d(E)=
¯
d(E)+d
(osc)
(E) ,
where
¯
d(E)=
µ(D)
4π
(6)
is the smooth part of the density and
d
(osc)
(E)=
µ(D)
4π

p.o.

J
0
(kL
p
) , (7)
is the oscillating part equal to a sum over all periodic orbits in the rectangle.
As
J
0
(z)
z→∞
−→

2
πz
cos

z −
π
4

10 Eugene Bogomolny
the oscillating part of the level density in the semiclassical limit k →∞takes
the form
d
(osc)
(E)=
µ(D)
√
8πk


p.o.
1

L
p
cos

kL
p
−
π
4

. (8)
Let us repeat the main steps which lead to this trace formula. One starts
with an explicit formula (like (3)) which expresses eigenvalues as a function
of integers. Using the Poisson summation formula (5) the density of states (4)
is transformed into a sum over periodic orbits. In Sect. 3 it will be demon-
strated that exactly this method can be applied for any integrable system in
the semiclassical limit where eigenvalues can be approximated by the WKB
formulas.
More Refined Approach
The above method of deriving the trace formula for the rectangular billiard
can be applied only if one knows an explicit expression for eigenvalues. For
chaotic systems this is not possible and another method has to be used.
Assume that one has to solve the equation
(E
n
−

ˆ
H)Ψ
n
(x)=0
for a certain problem with a Hamiltonian
ˆ
H. Under quite general conditions
eigenfunctions Ψ
n
(x) can be chosen orthogonal

Ψ
n
(x)Ψ
∗
m
(x)dx = δ
nm
and they form a complete system of functions

n
Ψ
n
(x)Ψ
∗
n
(y)=δ(x −y) .
The Green function of the problem, by definition, obeys the equation
(E −
ˆ

H)G
E
(x, y)=δ(x −y)
and the same boundary conditions as the original eigenfunctions. Its explicit
form can formally be written through exact eigenfunctions and eigenvalues as
follows
G
E
(x, y)=

n
Ψ
n
(x)Ψ
∗
n
(y)
E −E
n
+i
. (9)
The +i prescription determines the so-called retarded Green function.
Quantum and Arithmetical Chaos 11
Example
To get used to Green functions let us consider in details the calculation of the
Green function for the free motion in f-dimensional Euclidean space. This
Green function obeys the free equation
(E + 
2
∆)G

(0)
E
(x, y)=δ(x −y) . (10)
Let us look for the solution of the above equation in the form G
(0)
E
(x, y)=G(r)
where r = |x −y| is the distance between two points.
Simple calculations shows that for r =0G(r) obeys the equation
d
2
G
dr
2
+
f − 1
r
dG
dr
+
k
2

2
G =0
where E = k
2
.
After the substitution
G(r)=r

1−f/2
g

k

r

one gets for g(z) the Bessel equation (see e.g. [32], Vol. 2, Sect. 7)
d
2
g
dz
2
+
1
z
dg
dz
+

1 −
ν
2
z
2

g = 0 (11)
with ν = |f/2 −1|.
There are many solutions of this equation. The above +i prescription
means that when k → k + i with a positive  the Green function has to

decrease at large distances. It is easy to see that G(r) is proportional to
e
±ikr/
at large r.The+i prescription selects a solution which behaves at
infinity like e
+ikr/
with positive k. The required solution of (11) is the first
Hankel function (see [32], Vol. 2, Sect. 7)
g(z)=C
f
H
(1)
ν
(z) (12)
where C
f
is a constant and H
(1)
ν
(z) has the following asymptotics for large
and small z
H
(1)
ν
(z)
z→∞
−→

2
πz

e
i(z−πν/2−π/4)
and
H
(1)
ν
(z)
z→0
−→

−i2
ν
Γ (ν)z
−ν
/π , ν =2
2i ln z/π , ν =2
.
The overall factor in (12) has to be computed from the requirement that the
Green function will give the correct δ-function contribution in the right hand
side of (10). This term can appear only in the result of differentiation of the
Green function at small r where it has the following behaviour
12 Eugene Bogomolny
G(r)
r→0
−→ G
0
(r)=A
f
r
2−f

with
A
f
= C
f
2
ν

ν
Γ (ν)
iπk
ν
.
One should have

2
∆G
0
(r)=δ(r) . (13)
Multiplying this equality by a suitable test function f(r) quickly decreasing
at infinity one has

2

f(r)∆G
0
(r)dr = f(0) .
Integrating by parts one obtains

2


∂
∂x
µ
f(r)
∂
∂x
µ
G
0
(r)dr = −f(0) .
As both functions f(r)andG
0
(r) depend only on the modulus of r one finally
finds

2

∞
0
df(r)
dr
dG
0
(r)
dr
r
f−1
drS
f−1

= −f(0)
where S
f−1
is the volume of the (f −1)-dimensional sphere x
2
1
+ + x
2
f
=1.
Using (13) one concludes that in order to give the δ-function term A
f
has to
obey

2
A
f
(f − 2)S
f−1
= −1 .
One of the simplest method of calculation of S
f−1
is the following identity

∞
−∞
e
−x
2

1
dx
1

∞
−∞
e
−x
2
2
dx
2


∞
−∞
e
−x
2
f
dx
f
= π
f/2
.
By changing Cartesian coordinates in the left hand side to hyper-spherical
ones we obtain

∞
0

e
−r
2
r
f−1
drS
f−1
= π
f/2
which gives
S
f−1
=
2π
f/2
Γ (f/2)
where Γ (x) is the usual gamma-function (see e.g. [32], Vol. 1, Sect. 1).
Combining together all terms and using the relation xΓ (x)=Γ(x+1) one
gets the explicit expression for the free Green function in f dimensions
G
(0)
E
(x, y)=
k
ν
4i
2
(2πr)
ν
H

(1)
ν

k

|x −y|

(14)
where ν = |f/2 −1|. In particular, in the two-dimensional Euclidean space
Quantum and Arithmetical Chaos 13
G
(0)
E
(x, y)=
1
4i
2
H
(1)
0

k

|x −y|

. (15)
Another method of calculation of the free Green function is based on (9) which
for the free motion is equivalent to the Fourier expansion
G
(0)

E
(x, y)=

dp
(2π)
f
e
ip(x−y)/
E −p
2
+i
. (16)
Performing angular integration one obtains the same formulas as above.
The knowledge of the Green function permits to calculate practically all
quantum mechanical quantities. In particular, using
Im
1
x +iε
ε→0
−→ −πδ(x)
one gets that the eigenvalue density is expressed through the exact Green
function as follows
d(E)=−
1
π
Im

D
G
E

(x, x)dx . (17)
This general expression is the starting point of all trace formulas.
For the above model of the rectangle with periodic boundary conditions
the exact Green function has to obey
(
∂
2
∂x
2
+
∂
2
∂y
2
+ E)G
E
(x, y; x

,y

)=δ(x −x

)δ(y − y

) (18)
and the periodic boundary conditions
G
E
(x + na, y + mb; x


,y

)=G
E
(x, y; x

,y

) (19)
for all integer m and n.
The fact important for us later is that the rectangular billiard with periodic
boundary conditions can be considered as the result of the factorization of the
whole plane (x, y) with respect to the group of integer translations
x → x + na, y → y + mb (20)
with integer m and n.
The factorization of the plan (x, y) with respect to these transformations
means two things. First, any two points connected by a group transformation
is considered as one point. Hence (19) fulfilled. Second, inside the rectangle
there is no points which are connected by these transformations. In mathe-
matical language the rectangle with sizes (a, b) is the fundamental domain of
the group (20).
14 Eugene Bogomolny
Correspondingly, the exact Green function for the rectangular billiard with
periodic boundary conditions equals the sum of the free Green function over
all elements of the group of integer translations (20)
G
E
(x, y; x

,y


)=
∞

n,m=−∞
G
(0)
E
(x + na, y + mb; x

,y

) .
Here G
(0)
E
(x, x

) is the Green function corresponding to the free motion with-
out periodic boundary conditions. To prove formally that it is really the exact
Green function one has to note that (i) it obeys (18) because each term in the
sum obeys it, (ii) it obeys boundary conditions (19) by construction (provided
the sum converges), and (iii) inside the initial rectangle only identity term can
produce a δ-function contribution required in (18) because all other terms will
give δ-functions outside the rectangle.
The next steps are straightforward. The free Green function for the two-
dimensional Euclidean plane has the form (15). From (17) it follows that the
eigenvalue density for the rectangular billiard is
d(E)=−
1

π
Im

D
G
E
(x, x)dx
=
1
4π

mn

D
Im H
(1)
0

k

(ma)
2
+(nb)
2

dx
=
µ(D)
4π
+

µ(D)
4π


p.o.
J
0
(kL
p
) (21)
which coincides exactly with (6) and (7) obtained directly from the knowledge
of the eigenvalues.
The principal drawback of all trace formulas is that the sum over periodic
orbits does not converge. Even the sum of the squares diverges. The simplest
way to treat this problem is to multiply both sides of (21) by a suitable test
function h(E) and integrate them over E. In this manner one obtains

n
h(E
n
)=
µ(D)
4π

∞
0
h(E)dE +
µ(D)
4π


p.o.

∞
0
h(E)J
0
(
√
EL
p
)dE.
When the Fourier harmonics of h(E) decrease quickly the sum over periodic
orbits converges and this expression constitutes a mathematically well de-
fined trace formula. Nevertheless for approximate calculations of eigenvalues
of energies one can still use ‘naive’ trace formulas by introducing a cut-off on
periodic orbit sum. For example, in Fig. 1 the result of numerical application
of the above trace formula is presented. In performing this calculation one
uses the asymptotic form of the oscillating part of the density of state (8)
with only 250 first periodic orbits. Though additional oscillations are clearly
seen, one can read off this figure the positions of first energy levels for the
problem considered. In the literature many different methods of resummation
of trace formulas were discussed (see e.g. [19] and references therein).