Meiji Institute for Advanced Study of Mathematical Sciences

AN OVERVIEW OF SUNADA’S WORK
ATSUSHI KATSUDA AND POLLY WEE SY

We are proud of publishing this special paper introducing
excellent achievements by Prof. Sunada, a MIMS fellow.
(The authors are not MIMS members.)

MIMS Technical Report No.00013 (200807091)

Received 9 July 2008

AN OVERVIEW OF SUNADA’S WORK
ATSUSHI KATSUDA AND POLLY WEE SY

1. Brief Proﬁle of Professor Toshikazu Sunada
Professor Toshikazu Sunada was born in Tokyo, Japan, on September 7, 1948,
three years after the end of World War II. He dwelled and grew up in the suburbof Tokyo until the age of twenty-ﬁve. Sunada described himself in his childhood as
an ordinary boy, somewhat introverted and showing no particular interest in any
subjects taught in primary and junior high schools. According to his reminiscence,
he sat absentmindedly all day long during class hours. He even confessed thatarithmetic was then his instinctive dislike.
His zest for mathematics arose when he was a high school student and hada chance to read “History of Modern Mathematics” written by Takagi Teiji, aJapanese luminary who established the class ﬁeld theory, a culmination of algebraic
number theory. The book, including a vivid description of the lives of Gauss, Abel,and Galois together with the development of the theory of elliptic functions, wasso fascinating that it led him to the ambition of becoming a mathematician. Since
he had thought of himself as a literature-oriented person at that time, this was amajor turning point in his life.
He thus decided to study mathematics and entered Tokyo Institute of Technology(TIT), which had a department of mathematics of moderate size. However, soonafter his admission to the university (1968), his study was disrupted by student
riot, a movement sweeping over universities around the world. During this period,classes were cancelled and the campus was locked out. Interested students of themathematics department voluntarily requested their teachers to organize seminars
outside the campus. The subjects they took up then were vector bundles andcomplex multiplications; topics which were not covered in the regular lectures forundergraduate courses. The enthusiasm for mathematics that the teachers demonstrated as well as their selﬂess eﬀort to impart knowledge even in this extraordinary
period has left a lasting impression on the mind of the young Sunada. This experience made him more ambitious to become a professional mathematician. When
Sunada was in his senior year, his supervisor was Prof. Koji Shiga, who conducted
one of the seminars mentioned above and is now Sunada’s lifelong friend.
After his undergraduate studies, he was admitted to the graduate school of the
University of Tokyo (UT) and soon began his research under the supervision ofProf. Mikio Ise. The most decisive moment for his future career came when hedefended his master’s thesis which consists of three diﬀerent subjects in front of an
examination committee. Since the time allocated for presentation to each student
was short, he had to choose one out of the three subjects. Prof. Kunihiko Kodaira,a Fields Medal laureate, asked Sunada to explain in detail the other two subjects
as well, even though his time was already up. Moreover, Kodaira made valuable1

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ATSUSHI KATSUDA AND POLLY WEE SY

comments to each subject. This happening was a big boost to Sunada’s conﬁdencein pursuing his dream.
Just after receiving his master’s degree from UT in 1974, he was appointed as a
research associate at Nagoya University (NU) where he was to stay for the next 15
years. This stay has made his dream to become a mathematician comes true. In
1977, he received his doctorate degree by submitting a dissertation to UT.
In 1979-80, he was invited as a guest researcher at Bonn University. He says thatthe two-years stay in Bonn was the most fruitful time in his life. During this period,
he made the aqcuaintance of many active young mathematicians, and published aseries of excellent papers ([S7], [S8], [S9], [S10]). And it was also during this periodthat his geometric model of number theory was conceived (see Subsection 2.4).
After his return to NU, he was promoted to associate professor in 1982. In 1985,
he gave a beautiful construction of isospectral manifolds based on his geometricmodel of number theory. For this important contribution, he was subsequently
awarded the Iyanaga Prize by the Mathematical Society of Japan in 1987.
Sunada became a full professor at NU in 1988. Three years after, he was appointed professor at UT (1991-1993) and thereafter, at Tohoku University (TU,
1993-2003) before he has ﬁnally settled down at Meiji University in 2003. Currently
he is also professor emeritus of TU, a position held since 2003, and is aﬃliated with
the newly-established Meiji Institute for Advanced Study of Mathematical Sciencesin Tokyo. It is a rare case in Japan that a full professor transfers frequently fromone university to another since there is almost no diﬀerence in the ﬁnancial status.
The motivation for his frequent movement was to seek better research environment.
He frankly says “UT, one of the most prestigious universities in Japan, was worst
in my experience as far as the human relation is concerned”.
In the meanwhile, Sunada stayed for six months (1988) in Institut Hautes Études
Scientiﬁques (IHES) as a guest professor, for a few months in Isaac Newton Institute at Cambridge as an organizer of a special project (2007), and for seven months
in Max Planck Institute in Bonn (2008) as a visiting professor. In 2008, he heldan Andrejewski Lecturership at Humboldt University in Berlin under the auspicesof the Walter and Eva Andrejewski Foundation as a distinguished scholar. He
also stayed in Mathematical Sciences Research Institute (MSRI) in Berkeley, Johns
Hopkins University, Augsburg University, Institut Henri Poincaré (IHP), Tata Institute of Fundamental Research, Institut Mittag-Leﬄer, the Academy of Sciencein Beijing, National University of Singapore, and the University of the Philippines
for short periods. His ﬁrst stay in the Philippines (1986), which was the most
exciting moment in all of his travels (where he has witnessed the peaceful People
Power Revolution in Manila), was the beginning of his active involvement in theSoutheast Asia regional mathematical activities.
Sunada gave an invited lecture at the International Congress of Mathematicians(ICM) in Kyoto in 1990, at the Third Asian Mathematical Conference (AMC) in
Manila in 2000, and at the LMS South West and South Wales Regional Meeting in
Cardiﬀ, UK in 2007, to name a few. He was invited to numerous other international
conferences and symposia as a keynote speaker.
His activities are not limited to teaching and research. He was chosen a memberof the Kyoto Prize Selection Committee for three terms (1989, 1994, 2002) in thepast 20 years. In 2008, he was appointed a panel member of the European ResearchCouncil, an organization set up to promote outstanding, frontier research in all areas

AN OVERVIEW OF SUNADA’S WORK3

of science and humanities throughout Europe. His other services to the mathematics
community include his two-term board membership of the Mathematical Societyof Japan and the membership of the IMU-CDE committee where he served fortwo consecutive terms. Moreover, he helped in the organization of several major
conferences, including the celebrated Taniguchi Symposia, held in Asia as a memberof steering, scientiﬁc or advisory committee.
Besides his many research publications, Sunada has written a number of mathematics books for the general public as well as textbooks for undergraduate andgraduate students (most of which are in Japanese) ([SB1] - [SB9]), and enlightening essays which appeared in Sugaku Seminar (Mathematics Seminar) and othermathematical magazines. He has also been involved in the publication of several
series of mathematical books, journals, and proceedings as an editor. Sunada isat present a member of the Editorial Board of a popular Japanese mathematicalmagazine, Have Fun with Mathematics, published by Kame-Shobo.
Although Sunada usually portrays himself as a geometer, we realize from his
list of publications, that it is diﬃcult to single out his specialization. In fact,
Sunada’s work covers complex analytic geometry, spectral geometry, dynamicalsystems, probability, and graph theory, some of which we shall explain in detail in
the next section. Through his work, we would describe him as an extraordinary
and talented man with enormous insight and technical power who is constantly
generating new ideas and methods to form exciting and remarkable mathematicalresults.

2. Sunada’s work
This section gives an overview of Sunada’s achievement up to the age of 60.
Because of the wide range of subjects Sunada has been involved in and his research’s
style of moving back and forth among subjects, this narration of Sunada’s work shallbe done in accordance with the subject matters and not in chronological order.
The organization of the subsections is as follows:
2.1.
2.2.
2.3.
2.4.

Complex analysis
Trace formulae
Density of states
Isospectral manifolds

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ATSUSHI KATSUDA AND POLLY WEE SY

2.5.
2.6.
2.7.
2.8.
2.9.

Twisted Laplacians
Ihara zeta functions
Quantum ergodicity
Discrete geometric analysis
Strongly isotropic crystals (a diamond twin)

2.1. Complex analysis. Under the guidance of his supervisor Professor Mikio Ise,
Sunada succeeded in completing a master’s thesis consisting of three diﬀerent topics; namely, “Holomorphic equivalence problem of bounded Reinhaldt domains”,“Implicit function theorem for non-linear elliptic operators”, and “Random walks
on a Riemannian manifold”. The ﬁrst one [S4] published in Mathematische Annalen in 1978 is considered his debut paper to the mathematical community. As
he recalled in recent times, this work has given him self-conﬁdence of his ability
as a mathematician. In this paper, he generalized Thullen’s classical result [50]asserting that a 2-dimensional bounded Reinhaldt domain 1 containing the origin isbiholomorphic to one of the following domains provided that the orbit of the origin
by the automophism group has positive dimension:(1) {(z, w) ∈ C2 ; |z| < 1, |w| < 1} (polydisc);(2) {(z, w) ∈ C2 ; |z|2 + |w|2 < 1} (unit ball);(3) {(z, w) ∈ C2 ; |z|2 + |w|2/p < 1} (p > 0, ̸= 1) (Thullen domain).
Here, the orbit for the Thullen domain is {(z, 0); |z| < 1}. An interesting aspectof the classiﬁcation above is that even in the non-homogeneous case, the shape ofthe domain is explicitly described.
The main theorems in [S4] are stated below.
Theorem 2.1. An n-dimensional bounded Reinhaldt domain D containing the origin is biholomorphic to the Reinhaldt domain D̃ in Cn1 ×· · ·×Cns ×Cm1 ×· · ·×Cmt
which has the following characteristics:(1) Let D̃0 be the orbit of the origin for the action of the identity component of theautomorphism group of D̃. Then D̃0 = {(z1 , . . . , zs , w1 , . . . , wt ); zi ∈ Cni , wj ∈Cmj , |z1 | < 1, . . . , |zs | < 1, w1 = · · · = wt = 0}.
(2) D̃1 = {(w1 , . . . , wt ); (0, . . . , 0, w1 , . . . , wt ) ∈ D̃} is a bounded Reinhaldt
domain.
(3) D̃ is described in terms of D̃0 and D̃1 asnD̃ =(z1 , . . . , zs , w1 , . . . , wt ); (z1 , . . . , zs ) ∈ D̃0 ,
³

w1

s
s
o
Y
Y
¡
¢−pti /2 ´¡¢−p1i /2
1 − |zi |2, . . . , wt
∈ D̃1 ,
1 − |zi |2
i=1

i=1

where pij are non-negative real numbers.
Theorem 2.2. Two n-dimensional bounded Reinhardt domains D1 and D2 aremutually equivalent if and only if there exists a transformation ϕ : Cn −→ Cn
given by zi 7→ ri zσ(i) (ri > 0 and σ being a permutation of the indices) such thatϕ(D1 ) = D2 .
Sunada’s idea in the proofs is to employ the torus action (z1 , . . . , zn ) 7→ (eiθ1 z1 ,. . . , eiθn zn ) on a bounded Reinhaldt domain to obtain an analogue of the Cartan
1
A Reinhaldt domain D is an open set in Cn invariant under the transformation (z1 , . . . , zn ) 7→
(eiθ1 z1 , . . . , eiθn zn ) (θi ∈ R).

AN OVERVIEW OF SUNADA’S WORK

5decomposition g = k + p of the Lie algebra g of the automorphism group and a root
decomposition of the complexiﬁcation gC of g, which allows him to determine thestructure of g. Note that gC is identiﬁed with a Lie algebra consisting of polynomial
vector ﬁelds on Cn . A key fact is that a system ξ of vector ﬁelds on a real subspaceof Cn which consists of the restriction of vector ﬁelds in p is integrable on an open
dense subset. Using this fact, he could characterize D by solving the system ofdiﬀerential equations derived from ξ. Later on, his idea has been generalized by S.
Shimizu, A. Kodama, D. E. Barret, T. Barton, N. G. Kruzhilin, J. P. Vigué andothers for various classes of complex domains.
When Sunada was a research associate of NU (1974-1975) and UT (1975-1977),
he focused his study on the family of holomorphic maps of a Kähler manifold into
a compact quotient of a symmetric bounded domain ([S2], [S6]), which formed partof his doctoral dissertation submitted to UT. At that time, he had been looking
for a suitable example related to the result in his MS thesis on non-linear elliptic
equations.
He came across a general result due to A. Douady [20] which says that the setof holomorphic maps Hol(X, Y ) of a compact complex manifold X into anothercomplex manifold Y carries the structure of a complex space, and the evaluation
map ϕx : Hol(X, Y ) −→ Y deﬁned by ϕx (f ) = f (x) is an analytic map. What
Sunada conceived was that Hol(X, Y ) should inherit some structures from the target
manifold Y , if not always for every Y . For example, if Y is compact and Kobayashi
hyperbolic, then so is each component of Hol(X, Y )2. His question was: what about
the case that Y is a compact quotient of a symmetric domain ? His answer isadequately provided in the following theorem.
Theorem 2.3. Let X be a compact Kähler manifold, and Y be a compact quotientof a symmetric bounded domain Γ\D. We assume that Γ is torsion free. Then wehave(1) Each connected component of Hol(X, Γ\D) is a compact quotient of a symmetric bounded domain;(2) Let Γ1 \D1 be a connected component of Hol(X, Γ\D). Then the evaluation
map ϕx is a totally geodesic immersion, and hence D1 is identiﬁed with a Hermitian symmetric subspace of D (in other words, the lifting of ϕx is a holomorphic
embedding in the sense of Kuga-Satake [41]);(3) If f ∈ Γ1 \D1 , then the group Γ1 is identiﬁed with the centralizer of the imageof the induced homomorphism f∗ : π1 (X) −→ Γ;(4) If f, g ∈ Hol(X, Γ\D) are homotopic, then f and g are contained in the same
component of Hol(X, Γ\D).
A real analogue was also established for the space Harm(M, N ) of harmonic
maps of a compact Riemannian manifold M into a compact quotient N of a symmetric space of non-positive curvature ([S6]). This being the case, each componentof Harm(M, N ) is also a compact quotient of a symmetric space of non-positive
curvature, and the evaluation map is totally geodesic. In his proof, he made useof the classical result due to Eells and Sampson [21] and the fact that the energy
2This is easy to see if one uses Brody’s criterion on Kobayashi hyperbolicity “A compact
complex manifold Y is Kobayashi hyperbolic if and only if there is no non-constant holomorphic
map of C into Y ”.

6

ATSUSHI KATSUDA AND POLLY WEE SY

functional on the mapping space Map(M, N ) is convex (see also R. Schoen and S.
T. Yau [43]).
Claims (3) and (4) in the theorem above tells, in particular, that, if the imageof f ∈ Hol(X, Γ\D) is “topologically big”, then f is rigid in the sense that, if
g ∈ Hol(X, Γ\D) is homotopic to f , then g = f .
In connection with this rigidity property, it is worthwhile to quote the following
result [S2] which is a generalization of de Franchis’ theorem on ﬁniteness of nonconstant holomorphic maps of an algebraic curve into an algebraic curve with genus
greater than 1: Let ℓ(D) be the maximum value of dimension of proper boundary
components of D ([52]). Then there are only a ﬁnite number of holomorphic mappings of M into Γ\D, each of rank greater than ℓ(D). In particular, the set ofsurjective holomorphic mappings is ﬁnite.
The de Franchis theorem has several generalizations. For instance, Kobayashi
and Ochiai [29] proved that there are only a ﬁnite number of dominant meromorphic
maps onto a complex space of general type. Moreover, in his collaborative work
[S11] with Junjiro Noguchi, a good friend of Sunada since their undergraduate years
in TIT, they established the following theorem, a more algebraic-geometric result.
Theorem 2.4. Let M be an algebraic variety and N be a smooth complete algebraic
variety. We denote by Ratµ (M, N ) the family of rational maps f : M −→ N withrank f ≥ µ. If the µ-th exterior power ∧µ T N is negative, then Ratµ (M, N ) is
ﬁnite.
We note that ∧µ T (Γ\D) is negative if µ > ℓ(D). Hence this theorem is a
generalization of the result above.
After these work on complex analysis, Sunada’s interest has shifted to geometricanalysis, especially spectral geometry.
2.2. Trace formulae. Among Sunada’s scientiﬁc papers, there are four papers
whose titles include the term “trace formula” ([S7], [S10], [S12], [S33]). Two otherpapers [S5] and [S17] are also closely related to trace formulae. His intention inthese papers was to use the trace formulae for the spectral study of Laplacians on
general Riemannian manifolds.
Needless to say, the “trace formula philosophy” has its origin in the famouswork by A. Selberg [42] who established a non-commutative version of the Poissonsummation formulaXX
fb(n),
f (m) =
(2.1)
m∈Z

n∈Z

where f is a rapidly decreasing function, and fb(ξ) =Z ∞−∞f (x)e−2π√−1xξdx. TheSelberg trace formula, which works eﬀectively for a closed surface of constant negative curvature, gives rise to a precise relation between eigenvalues of the Laplacianand closed geodesics. The Poisson summation formula also leads to a relation between eigenvalues and closed geodesics; that is, eigenvalues of −d2 /dx2 and closed
geodesics x 7→ kx (k ∈ Z) on S 1 = R/Z. This relation is more clearly understood
if we rewrite (2.1) as
(2.2)

∞
X
k=0

X
p
f (k),
fbc ( λk ) =k∈ZAN OVERVIEW OF SUNADA’S WORK

where
fbc (s) =

Z ∞
−∞

7

f (t) cos st dt,

and 0 = λ0 < λ1 ≤ λ2 ≤ . . . are eigenvalues of −d2 /dx2 ; that is, λ2k−1 = λ2k =4π 2 k 2 (k ≥ 1). In the expression on the right-hand side of (2.2), the whole number|k| (k ∈ Z) is thought of as representing the length of a closed geodesic in S 1 . If
one deﬁnes the distribution Θ ∈ D′ (R) by(2.3)〈Θ, f 〉 =∞Xp
fbc ( λk ),k=0then the formula above is written asXδk(δk is the Dirac delta with support {k}),
(2.4)
Θ=
k∈Z

which implies that the singular support of Θ coincides with the set of ± lengths ofclosed geodesics in S 1 . This view was generalized by Chazarain [13] to a general
Riemannian manifold M , who showed that the singular support of the distribution
Θ deﬁned in the same way as (2.3) is contained in the set{±ℓ; ℓ is the length of a closed geodesic in M }.
Sunada’s view of the summation formula is a bit diﬀerent, much akin to theoriginal trace formula, and is explained in the following way. The integers k in (2.4)
which are thought of as elements in the fundamental group π1 (S 1 ) = Z parameterize
the connected components of the mapping space Map(S 1 , S 1 ), and hence the righthand side of (2.4) may be regarded as the sum of distributions over free homotopy
classes of closed paths in S 1 . In the case of a general M , the connected componentsof Map(S 1 , M ) are parameterized by conjugacy classes [σ] ∈ [π1 (M )]. Thus it isnatural to expect that Θ is expressed as a sum of certain distributions Θ[σ] closely
related to homotopy class [σ] of closed paths. He proved that this is actually thecase, and presented his result below at the ICM Kyoto in 1990 ([S33]).
Theorem 2.5. (A generalized trace formula) Let ρ : π1 (M ) −→ U (N ) be a unitary
representation, and λ0 (ρ) ≤ λ1 (ρ) ≤ · · · be the eigenvalues of the Laplacian ∆ρacting on sections of the ﬂat vector bundle associated with ρ. Deﬁne Θ(ρ) ∈ D′ (R)by∞Xpfbc ( λk (ρ)).
〈Θ(ρ), f 〉 =k=0

Then with each conjugacy class [σ] ∈ [π1 (M )], a distribution Θ[σ] ∈ D′ (R) having
the following properties is associated.
Xtr(σ)Θ[σ] ;(1) Θ(ρ) =[σ](2) supp Θ[σ] ⊂ {t ∈ R; |t| ≥ ℓ[σ] }, where ℓ[σ] is the length of the shortest closed
geodesics in M whose homotopy class is [σ];(3) sing. supp Θ[σ] is contained in the set {±ℓ; ℓ is the length of a closed geodesicin M whose homotopy class is [σ]};8ATSUSHI KATSUDA AND POLLY WEE SY(4) Each Θ[σ] extends to a generalized function on a space of test functions
including the gaussian function³ t2 ´fτ (t) = (4πτ )−1/2 exp −(τ > 0).
4τ
P
Moreover, for such a test function f , the sum [σ] 〈Θ[σ] , f 〉 converges absolutely
and equals 〈Θ, f 〉.
The distribution Θ[σ] is explicitly deﬁned as
Z
Z
(2.5)
〈Θ[σ] , f 〉 =dx
f (t)U (t, σx, x)dt,
Γσ \M̃

R

where Γ = π1 (M ), Γσ is the centralizer of σ, π : M̃ −→ M is the universal
coveringp
map over M , and U (t, x, y) is the kernel function of the operator cos(t ∆M̃ ), whichis the fundamental solution of the wave equation on M̃³ ∂2´+ ∆ u = 0, u(0, x) = δ(x), ut (0, x) = 0.
2∂tClaim (1) is proven in much the same manner as the classical trace formula.
p
Indeed, if we denote by UM (t, x, y) the kernel function of the operator cos t ∆ρon M , then¡¢ X
UM t, π(x), π(y) =ρ(σ)U (t, σx, y),σ∈Γfrom which (1) follows immediately. The second claim (2) is a consequence of the
ﬁnite propagation property of the wave equation. As for (3), one just follows theway in [13]. A technical diﬃculty caused by non-compactness of Γσ \M̃ can beovercome again by using the ﬁnite propagation property.
If we take the gaussian function fτ as a test function, then we have, for thetrivial ρ,Z∞Xe−τ λk ,〈Θ, fτ 〉 =kM (τ, x, x)dx =Z(2.6)〈Θ[σ] , fτ 〉 =M

k=0Γσ \M̃

kM̃ (τ, σx, x)dx,

where kM and kM̃ are the heat kernel on M and M̃ , respectively.
The “path space feature” of trace formulae becomes more apparent if we use themodiﬁed Wiener measure µτ on the path spacea
Map(S 1 , M ) =Map[σ] (S 1 , M ),[σ]∈[Γ]which is characterized byZ¡¢
f c(t1 ), . . . , c(tN ) dµτ (c)Map(S 1 ,M )Z
kM (τ (t2 − t1 ), x1 , x2 ) × · · · × kM (τ (tN − tN −1 ), xN −1 , xN )=M ×···×M

×kM (τ (1 + t1 − tN ), xN , x1 )f (x1 , . . . , xN ) dx1 · · · dxN ,where f is an arbitrary continuous function on the N -tuple product M ×· · ·×M , and0 ≤ t1 < t2 < · · · < tN < 1 ([S12]). Indeed, one has µτ (Map(S 1 , M )) = 〈Θ, fτ 〉 andAN OVERVIEW OF SUNADA’S WORK

9µτ (Map[σ] (S 1 , M )) = 〈Θ[σ] , fτ 〉, and hence the generalized trace formula applied to
the gaussian function reduces to the additivity of the Wiener measure.
Also, if one takes the functional Fτ on the path space deﬁned by
Z
³
¡
¢ ´
Fτ (c) = exp − τ
q c(s) ds ,S1

where q is a smooth function on M , then, in view of the Feynmann-Kac formula,
the summation formula
Z
XZ
(2.7)
Fτ dµτ =Fτ dµτ
Map(S 1 ,M )Map[σ] (S 1 ,M )[σ]

is equivalent to the trace formulaZXZh(t, x, x)dx =M[σ]

Γσ \M̃h̃(t, σx̃, x̃)dx̃,where h (resp. h̃) is the fundamental solution of the equation³∂´+∆+q u=0∂t
on MZ(resp. on M̃ ). In general, one may not expect to have an exact shape of theterm

Map[σ] (S 1 ,M )Fτ dµτ . However one may give an asymptotic expression instead([S10]) provided that M is of non-positive curvature. More precisely, if, in addition,
the function x̃ → d(x̃, σx̃)2 is non-degenerate (this is the case for symmetric spaces),thenZ
2Fτ dµτ ∼ (4πτ )−dim M[σ] /2 e−ℓ[σ] /4τ (a0 + a1 τ + +a2 τ 2 + · · · )Map[σ] (S 1 ,M )as τ goes to zero. Here M[σ] denotes the set of closed geodesics in the homotopy
class [σ] (having the structure of a smooth connected manifold), and ℓ[σ] is thelength of c ∈ M[σ] .
When M̃ is a symmetric space of non-positive curvature, so is M[σ] which ismapped into M by a totally geodesic immersion, and π1 (M[σ] ) = Γσ as mentioned
in the previous subsection. By choosing a suitable ﬁbration map : Γσ \M̃ −→ M[σ] ,one can decompose the integration in (2.6) into integrations along the ﬁber andintegration over M[σ] . In a special case (for instance, M̃ is a hyperbolic space, andq ≡ 0), one may establish an exact trace formula in this way (see McKean [35]).
The following example is due to Sunada [S5].
Example 2.1. Let M = Γ\Rn be a compact ﬂat manifold. Then∞X
X
tr ρ(σ)α(σ)vol(M[σ] )(4πτ )−dim M[σ] /2 exp(−ℓ2[σ] /4τ ),e−λi (ρ)τ =i=0[σ]∈[Γ]where, for σ : x 7→ Ax + b (A ∈ O(n)), we put¯¡¢¯−1α(σ) = ¯ det A − I|Image(A − I) ¯ .
(Note that each M[σ] is also ﬂat. Using this exact formula, one can prove that there
are only ﬁnitely many isometry classes of ﬂat manifolds with a given spectrum. One
can also employ this formula to express the Ray-Singer zeta function in terms of
the Hurwits zeta function and to compute the Reidemeister-Franz torsion [S59].)

10

ATSUSHI KATSUDA AND POLLY WEE SY

Let us go back to the simplest case M = S 1 . We thus treat the eigenvalues {λk }of Hill’s operator −d2 /dx2 + q(x) (q(x + 1) = q(x)). The following proposition, a
special case of (2.7), yields a generalization of the Jacobi inversion formula.
Proposition 2.1. ([S7]) Let µ be the Wiener measure on the loop space Ω = {ω :[0, 1] −→ R; ω(0) = ω(1) = 0}. Then we have∞XZ
fn (q; τ ) =2e−n /4τ fn (q; τ ),

n=−∞

k=0

where

∞
X

e−τ λk = (4πτ )−1/2
Z

S1

dx

Ω

Z 1
³
¡
¢ ´
√
exp − τq x + nt + τ ω(t) dt dµ(ω).
0Using this integral expression, one can easily carry out the asymptotic expansion
fn (q; τ ) ∼ 1 + An1 (q)τ + An2 (q)τ 2 + · · ·(τ ↓ 0),and see that the coeﬃcients A0i (q) are expressed as
Z 1
ai (q(x), q ′ (x), . . .)dx.
A0i (q) =0

Here the ai ’s are universal polynomials of q, q ′ , q ′′ , . . ., which give us the well-known
KdV-invariants (see McKean and Van Moerbeke [36]).
Sunada had interest in the coeﬃcients Ani (q) for n ̸= 0 since they might givenew invariants. But he found out, after tedious computations, that each Ani (q) canbe expressed as a polynomial of the A0i (q)’s.
It is interesting to point out that fn (q; τ ) is related to the discriminant ∆(λ),
which is deﬁned as the trace of the monodromy mapping ϕ(x) 7→ ϕ(x + 1) acting
on the solution space of the equation³´
d2− 2 + q(x) − λ ϕ = 0.
dx
Indeed, we have
Proposition 2.2. ([S7])
Z ∞

=

2

e−λτ (4πτ )−1/2 e−n /4τ fn (q; τ )dτ0p−∆′ (−λ) ³ ∆(−λ) − ∆(−λ)2 − 4 ´|n|p.
2∆(−λ)2 − 4

Using this proposition, one can prove that, if q(x) is a ﬁnite band potential;that is, if the equation ∆(λ)2 − 4 = 0 has only ﬁnitely many simple roots, then
(4πτ )−1/2 f0 (q; τ ) is a hypergeometric function of Pochhammer’s type.
There is an abstract form of trace formulae which is regarded as a generalizationof the class formula for a ﬁnite group ΓX|Γσ |−1 .
1=[σ]∈[Γ]A straightforward generalization is the following proposition, which turns out tobe useful in Sunada’s work on isospectral manifolds as will be seen in the next
subsection.

AN OVERVIEW OF SUNADA’S WORK

11

Proposition 2.3. [S17] Let V be a Hilbert space on which a ﬁnite group Γ acts
unitarily, and V Γ be the subspace of V consisting of Γ-invariant vectors. If a nonpositive operator A : V −→ V of trace class is Γ-equivariant, thenX|Γσ |−1 tr(σA).
(2.8)
tr(A|V Γ ) =[σ]∈[Γ]Even for a group Γ of inﬁnite order, there are cases that (2.8) may still makesense, at least in a formal sense. The trace formula for the covering set-up is
considered as such an example. To explain this, let π : X −→ M be a regularcovering map over a compact manifold M with covering transformation group Γ,
and let A0 be an integral operator on M with a lifting A on X. ThenX
trΓσ (σA).
(2.9)
tr A0 =
[σ]∈[Γ]Here, in general, the Γ-trace trΓ (T ) for Za Γ-equivariant integral operator T on
a manifold with a Γ-action is deﬁned byFt(x, x)dx, where F is a fundamental

domain for the Γ-action. If F is relatively compact, the Γ-trace is interpreted as a
von Neumann trace (see [1]). The trΓσ in (2.9) needs to be handled carefully sincea fundamental domain for Γσ -action is not relatively compact in general except for
σ = 1. In many P
practical cases, (2.9) is justiﬁed. For instance, the summation
formula 〈Θ, f 〉 = σ 〈Θ[σ] , f 〉 may be regarded as a disguised form of (2.9) for a
special A.
2.3. Density of states. While Sunada was studying trace formulae, he becameaware that a crude relation between an operator on a covering manifold and its basemanifold must be useful for a justiﬁcation of the notion of “(integrated) density ofstates”, which was ﬁrst introduced by physicists in the quantum theory of solids.
To simplify physicist’s explanation, we consider the Schrödinger operator H with aperiodic potential on the Euclidean space. Restricting H to a bounded domain andimposing a boundary condition, we count the number of eigenvalues not exceeding
λ. Then dividing this counting function by the volume of the domain, and blowing
up the domain to ﬁll the whole space, one gets the integrated density of states ϕ(λ).
The spectrum of the Schrödinger operator on the whole space is then characterized
completely by ϕ(λ). An interesting feature is that one has the same ϕ(λ) whatever
one chooses as a boundary condition (Dirichlet, Neumann or periodic boundary
condition). The fact behind this feature is that the volume growth of boundaries
is much less than that of the domains (see M. Shubin [46]).
A question here is whether the notion of the integrated density of states makes
sense for a more general set-up. What Sunada considered is the case of a regularcovering manifold X over a compact manifold M , which allows one to deﬁne theintegrated density of states associated with a periodic boundary condition as follows: Suppose that the covering transformation group Γ has a family of normal∞
subgroups {Γi }∞i=1 of ﬁnite index such that Γi+1 ⊂ Γi and ∩i=1 Γi = {1}. We thenhave a tower of ﬁnite-fold covering maps of closed manifolds · · · −→ Mi+1 −→Γi \X = Mi −→ · · · −→ M1 −→ M . We take a Γ-invariant function q. Let ϕMi (λ)
denote the number of eigenvalues of HMi = ∆Mi + q on the closed manifold Mi notexceeding λ.

12ATSUSHI KATSUDA AND POLLY WEE SY

Theorem 2.6. ([S33]) LetZ
HX =λdE(λ)

be the spectral resolution of HX , and putϕΓ (λ) = trΓ E(λ).
Then the measures (vol(Mi ))−1 dϕMi converge weakly to (vol(M ))−1 dϕΓ .
Thus the spectral distribution function trΓ E(λ) is essentially identiﬁed with the
integrated density of states associated with the periodic boundary condition.
A new question arises. What happens when we replace the periodic boundary
condition by the Dirichlet boundary condition ? A partial answer was given in his
joint work [S38] with Toshiaki Adachi. To explain this, let {Di }∞i=1 be a familyof bounded connected
open
sets
in
X
with
piecewisesmoothboundaries
satisfying
S∞Di ⊂ Di+1 , i=1 Di = X. Consider the Schrödinger operator HDi = ∆Di + q oneach Di with the Dirichlet boundary condition. We denote by ϕDi (λ) the numberof eigenvalues of HDi not exceeding λ.
Theorem 2.7. Deﬁne a “thick” boundary of a domain D by
∂h D = {x ∈ D; dist(x, ∂D) ≤ h}.
(1) The group Γ is amenable if and only if there exists a family {Di }∞i=1 ofbounded domains with piecewise smooth boundary satisfying the following property:
lim vol(∂h Di )/vol(Di ) = 0i→∞for every h > 0.
−1
dϕDi con(2) If a family {Di }∞i=1 satisﬁes the property in (1), then (vol(Di ))−1verges weakly to (vol(M )) dϕΓ .
For the deﬁnition of amenability, refer to R. J. Zimmer [55]. One may see J.
Dodziuk and V. Mathai [19] for a recent development.
The notion of Γ-trace which was eﬀectively used in the study of density of states
turns out to be also useful in the spectral study of covering manifolds. In his paper[S33], Sunada took up this notion to determine a criterion for a periodic Schrödinger
operator on a manifold to have band structure. Here band structure means thatthe spectrum is a union of mutually disjoint, possibly degenerate closed intervals,
such that any compact subset of R meets only ﬁnitely many. Subsequently he andJochen Brüning [S35] generalized the result to the case of periodic elliptic operators.
∗
In order to explain the criterion, we shall employ Cred
(Γ, K), the tensor product∗of the reduced group C -algebra of a discrete group Γ with the algebra K of compact operators on a separable Hilbert space of inﬁnite dimension. The C ∗ -algebra∗Cred(Γ, K) has a canonical von Neumann trace trΓ . We then deﬁne the Kadison
constant C(Γ) by∗
C(Γ) = inf{trΓ P ; P is a non-zero projection in Cred
(Γ, K)}.

By deﬁnition, Γ is said to have the Kadison property if C(Γ) > 0. Examples of suchΓ are abelian groups, free groups and surface groups (see [S32])3
We now let X be a Riemannian manifold of dimension n on which a discretegroup Γ acts isometrically, eﬀectively, and properly discontinuously. We assume
3It is a conjecture proposed by Kadison that, if Γ is torsion free, then C(Γ) = 1.

AN OVERVIEW OF SUNADA’S WORK

13

that the quotient space Γ\X (which may have singularities) is compact. Let E bea Γ-equivariant hermitian vector bundle over X, and D : C ∞ (E) −→ C ∞ (E) aformally self-adjoint elliptic operator which commutes with the Γ-action (in short,such a D is called a Γ-periodic operator).
Theorem 2.8. [S35] (1) If Γ has the Kadison property, then the spectrum of any
Γ-periodic elliptic operator has band structure.
(2) Suppose that D is a Γ-periodic elliptic operator of order p, and is boundedfrom below. Let N (λ) be the number of components of the spectrum of D whichintersect the interval (−∞, λ]. If Γ has the Kadison property, thenZA(x)dx,
lim sup N (λ)λ−n/p ≤ C(Γ)−1 Γ(1 + n/p)
λ→∞

Γ\X

where the function A(x) can be evaluated explicitly in terms of the principal symbol
σD(x, ξ) of D as given below.
Z
Z ∞−iγ¡¢−1 iτ
e dτ
A(x) = (2π)−n−1
tr σD(x, ξ) + iτ
dξ
Rn

−∞−iγ

(γ is an arbitrary real number).
In the deﬁnition of band structure, we do not exclude the possible existenceof eigenvalues. Indeed, one may construct a closed manifold with a free abelian
fundamental group such that the Laplacian on the universal covering manifold
possesses an eigenvalue ([S28]). On the other hand, the following theorem wasproven in [S30].
Theorem 2.9. Let X be the maximal abelian covering space over a closed Riemannian manifold M . Suppose that M admits a non-trivial S 1 -action whose generating
vector ﬁeld is parallel. Then the Schrödinger operator on X with a smooth periodic
potential has no eigenvalues.
The proof of this generalization of the classical result due to L. E. Thomas [49]was carried out by using the idea of twisted operators (see Section 2.5).
2.4. Isospectral manifolds. When Sunada was a guest researcher of the SFB(Sonderforchungsbereich) “Theoretische Mathematik” program in Bonn University(1979-80), he revisited the problem of random walks on a Riemannian manifold, andstudied the spherical mean operator which is considered as the transition operator of
the random walk. He observed that, if the radius is small enough, then the sphericalmean operator is a Fourier integral operator of negative order so that it is a compact
operator in the L2 space ([S9]). Since then, the problem of random walks, not only
on a Riemannian manifold but also on a graph has been his steadfast interest.
Among his papers, the most cited one is “Riemannian coverings and isospectral
manifolds” [S17] which appeared in Annals of Mathematics in 1985. The motivation
behind this work came up also during his stay in Bonn. At that time, he wanted tounderstand the class ﬁeld theory because it has been his dream to prove something
related to this theory ever since he read Takagi’s book on algebraic number theory.
For this sake, he tried to ﬁnd a geometric model of the class ﬁeld theory. Thus
the path he took is the reverse of what Hilbert had taken up to speculate a correct
formulation of his “absolute” class ﬁeld theory by looking at the theory of covering
surfaces. Sunada observed that closed geodesics under covering maps behave like

14

ATSUSHI KATSUDA AND POLLY WEE SY

prime ideals under ﬁeld extensions, and could soon formulate the class ﬁeld theory
in the Riemannian geometric setting4. He once said,”It is a simple toy model, but
I enjoy playing with it very much.”
Let us explain brieﬂy his idea. A prime cycle is a 1-cycle represented by a prime
closed geodesic. Given a ﬁnite-fold Riemannian covering map π : M −→ M0 , wesay that a prime cycle P in M lies above a prime cycle p in M0 if π(P) = pm (= mp)with a positive integer m, and write P|p. The integer m is called the degree of P,
and written as deg P. If π is n-fold, then there are at most n prime cycles lying
above p, andXn=deg P.
P|p

This identity already gives rise to a ﬂavor of algebraic number theory. This ﬂavor
becomes much stronger when we consider a regular covering map. Indeed, for agiven p, the covering transformation group G acts transitively on the set {P; P|p},and if deg P = f , then there exists a unique σ ∈ G such that c(t + f −1 ) = σc(t),where c is a representative of P. The element σ depends only upon P. We write(P|π) for σ and call it the Frobenius transformation associated with P. It is easily
checked that (P|π) is a generator of the stabilizer GP = {µ ∈ G; µP = P} so thatdeg P coincides with the order of GP . We also ﬁnd that (µP|π) = µ(P|π)µ−1 .
Thus if π is an abelian covering map, i.e. G is abelian, then (P|π) depends only
upon p. We write (p|π) instead of (P|π).
Taking account of the Dedekind theorem, we deﬁne the counterpart of ideal groupsimply as the free abelian group generated by prime cycles, which we denote by
IM . An element a in IM , therefore, takes the forma = p1 a1 · · · pk ak(ai ∈ Z),

which we call a geodesic cycle. A geodesic cycle a is said to be principal if the0homology class [a] is zero. We denote by IM
the subgroup of IM consisting ofprincipal geodesic cycles. An analogue of ideal class group is the quotient group0
CM = IM /IM. As a matter of fact, CM is nothing but the 1st integral homology
group H1 (M, Z).
Now for a covering map π : M −→ M0 and for a prime cycle P in M with P|p,deﬁne the norm Nπ (P) to be pdeg P , and extend it to the group homomorphism
IM −→ IM0 . In the geometric context, the fundamental theorem in class ﬁeld
theory is stated as follows.
Proposition 2.4. [S62] (1) For an n-fold covering map π : M −→ M0 , the index0[IM0 , IM· Nπ (IM )] is not greater than n. The equality holds if and only if π is0abelian.
(2) (Artin’s reciprocity law) If π is abelian, thenp 7→ (p|π)¡ 0the correspondence¢
yields an isomorphism of the quotient group IM0 / IM
·
N
(I
)
onto
G.
π
M
0
0
(3) For any subgroup H of ﬁnite index in IM0 containing IM, there exists an00abelian covering map π : M −→ M0 such that H = IM0 · Nπ (IM ).
In number theory, the quickest way to prove a similar inequality as in (1) is tomake use of some properties of the L-functions, and the proofs of (2) and (3) aresubstantially sophisticated. In the geometric case, the proof makes use of elementary features of the Hurewicz homomorphism: π1 (M0 ) −→ H1 (M0 , Z).
4This was discussed in the unpublished paper [S62].

AN OVERVIEW OF SUNADA’S WORK

15

During his stay in Germany, Sunada happened to attend a conference (Arbeitstagung) held in Bonn, thereby having a chance to listen to a talk by M. F. Vignéras, aFrench mathematician. In that occasion, she explained her construction of isospectral Riemann surfaces by means of the Selberg trace formula ([51]). Here we quotethe story disclosed by Sunada.
“Her method applies only to a very special class of manifolds; i.e., hyperbolic
surfaces and spaces. After some time, I came to think that there should be amore general method which could apply to a broader class of manifolds. Actually
I observed that Vignéras’ construction somehow ﬁts into my geometric model ofnumber theory. In my geometric context, what I would like to look for is a pairof Riemannian manifolds with the same spectral zeta function. Soon I started toseek a similar statement in number theory, and ﬁnally found a book which gave
me a satisfactory statement. It is the Proceedings edited by Cassel and Fröhlich[12] with exercise problems in the appendix. One of the problems is asking for apair of number ﬁelds with the same Dedekind zeta function. This is it ! The nicest
thing is that the answer to this problem is expressed in terms of Galois groups ofﬁeld extensions. Since I already had a geometric model of number theory, it wasalmost immediate to write down the desired statement in the geometric setting.
Of course, the proof in the geometric case should be diﬀerent from that in number
theory. But it is so convincing that I never have doubt about the validity of mystatement. Actually a week later, I could give a proof.”
The statement in number theory is the following
Proposition 2.5. Let K be a ﬁnite Galois extension of Q with Galois group G =G(K/Q), and let k1 and k2 be subﬁelds of K corresponding to subgroups H1 andH2 , respectively. Then the following two conditions are equivalent:(1) Each conjugacy class of elements in G meets H1 and H2 in the same numberof elements.
(2) The Dedekind zeta functions of k1 and k2 are the same.
It is known that many examples of the triplet (G, H1 , H2 ) arise from simple
algebraic groups. If G is a reductive algebraic group and H1 , H2 are nonconjugate
but associate parabolic subgroups, then (G, H1 , H2 ) satisﬁes condition (1).
In Proposition 2.5, Sunada replaced the Dedekind zeta function by the spectral
zeta function∞X
λk −s ,
ζM (s) =k=1

where 0 < λ1 ≤ λ2 ≤ · · · are non-zero eigenvalues of the Laplacian on a compact
Riemannian manifold M . He then proved the following by using Proposition 2.3.
Theorem 2.10. Let π : M −→ M0 be a regular Riemannian covering map withcovering transformation group G, and let π1 : M1 −→ M0 and π2 : M2 −→ M0 be
the covering maps corresponding to subgroups H1 and H2 , respectively. If the triplet(G, H1 , H2 ) satisﬁes condition (1) in the proposition above, then the zeta functions
ζM1 (s) and ζM2 (s) are identical.
Since ζM1 (s) = ζM2 (s) if and only if M1 and M2 are isospectral, this result
gives a chance to construct many non-isometric isospectral manifolds. Actually, bychoosing a suitable (G, H1 , H2 ) and M0 , we may construct many isospectral pairs,
especially isospectral surfaces of constant negative curvature whose genus is much

16

ATSUSHI KATSUDA AND POLLY WEE SY

smaller than Vignéras’ example (see R. Brooks and R. Tse [7] for an example ofgenus three).
We should note that Vignéras’ example also gave a negative answer to thequestion posed by I. M. Gel’fand [22], who asked if the induced representationof P SL2 (R) on L2 (Γ\P SL2 (R)) determines a discrete subgroup Γ up to conjugation. Sunada’s idea also allows a simple construction of a counterexample (see[S18]).
The excitement created by Sunada’s beautiful construction of isospectral manifolds is seen in several quotations given by mathematicians. As B. Cipra wrote, in
his review paper [14] on the famous problem “Can one hear the shape of a drum ?”proposed by M. Kac, ”Sunada’s method gives rise to a veritable cottage industry of
examples.” In the preface of his book [11], Peter Buser wrote “Sunada’s constructionof isospectral manifolds was fascinating, and I got hooked on constructing examples
for quite a while. So time went on and the book kept growing.... The editor, for instance, was interested, and so was my family.” Moreover, Robert Brooks, Sunada’slate friend, wrote, towards the end of a historical remark on isospectral manifolds
in his expository article [8], “The situation changed dramatically in [S17]. Here
Sunada showed how the phenomenon of isospectral manifolds could be understoodin a systematic way.” See also H. Pesce’s article [39] which contains an excellent
survey of Sunada’s construction.
It should be pointed out that there are many examples of isospectral manifolds
which are not obtained by Sunada’s method (see D. Schueth [44] for example). See
R. Brooks [9] for a graph version of the isospectral problem.
As for Kac’s problem, C. Gordon, D. Webb and S. Wolpert [23] gave a counterexample (thus one can not hear the shape of a drum in general) by using thetransplantation technique developed by Buser [10] and P. Bérard [4] which is closely
related to Sunada’s method.
Sunada was once (around 1988) asked by his colleague what a geometric analogueof the Riemann Hypothesis is. His answer was: λ1 (M ) ≥ λ0 (M̃ ) ([S33])5. Hereλ1 (M ) is the ﬁrst positive eigenvalue of the Laplacian on a compact manifold, andλ0 (M̃ ) is the bottom of the spectrum of the Laplacian on the universal covering
manifold M̃ . He admits that this was merely a joke. But it turns out that thisjoke has led him to a serious business to know more about the spectra of covering
manifolds.
2.5. Twisted Laplacians. It is quite natural for Sunada to proceed to the studyof a geometric analogue of the analytic number theory. He wanted to establish
an analogue of the Dirichlet theorem for arithmetic progression which asserts that,given positive integers a, d which are supposed to be coprime, there are inﬁnitely
many primes in the series a, a + d, a + 2d, . . .. More precisely, one hasx|{prime p ≤ x; p = a + kd for some k}| ∼ ϕ(d)−1(x ↑ ∞),
log x
where ϕ(d) is the Euler function. Remember that the proof relies on the propertiesof the L-functionY(1 − χ(p)p−s )−1 ,L(s, χ) =p
5Actually, for a closed surface, this condition is equivalent to that the Selberg zeta function
satisﬁes the Riemann Hypothesis.

AN OVERVIEW OF SUNADA’S WORK

17

where χ is a character of the multiplicative
group (Z/dZ)× , which is the Galois√
2π −1/d).
group of the cyclotomic ﬁeld Q(eA geometric analogue of the prime number theorem has been known as the asymptotic formula for counting function of prime closed geodesics in a closed surface
M of constant negative curvature. Let π(x) be the number of prime cycles p in M
with ℓ(p) ≤ x. Thenexπ(x) ∼(x ↑ ∞).
x
This is proven in a similar way as the usual proof of the prime number theorem byusing the Selberg zeta function.
Taking account of our convention to regard H1 (M, Z) as the (absolute) ideal classgroup, Sunada thinks that a geometric analogue of the Dirichlet theorem should be
an asymptotic formula for the counting function of prime cycles in a ﬁxed homology
class:¯¯
π(x, α) = ¯{p; [p] = α, ℓ(p) ≤ x}¯,
where α ∈ H1 (M, Z) and [p] stands for the homology class of p. There is, however,a big discrepancy between the number theory and its geometric model. That is, theideal class group H1 (M, Z) is of inﬁnite order, and so the conventional proof does
not work6. But one may still show the following theorem, which was independently
proven by Phillips and Sarnak [40].
Theorem 2.11. ([S24]) Let M be a closed surface of constant negative curvature
with genus g. Thenexπ(x, α) ∼ (g − 1)g g+1 (x ↑ ∞).
x
The key to its proof lies in the following properties of the geometric L-functionY
L(s, χ) =(1 − χ(p)e−sℓ(p) )−1 ,pwhere χ is a unitary character of H1 (M, R).
(1) L(s, χ) is analytically continued to the whole complex plane as a meromorphic
function.
(2) If χ is not a trivial character 1, then L(s, χ) is holomorphic in Re s ≥ 1.
(3) L(s, 1) has a simple unique pole at s = 1 in Re s ≥ 1.
(4) L(s, χ) has no zeros in Re s ≥ 1.
(5) The poles of L(s, χ) in Re s > 0 are
r
1
1
s= ±
− λi (χ) (i = 0, 1, 2, . . .),2
4
where λ0 (χ) ≤ λ1 (χ) ≤ · · · are eigenvalues of the Laplacian (twisted Laplacian) ∆χacting on sections of the ﬂat line bundle associated with χ.
These properties are also derived from those of the Selberg zeta function.
Crucial in the discussion is the fact that λ(1) = 0 is simple, and that λ(χ) = 0
if and only if χ = 1. Thus λ0 (χ) is simple and depends smoothly on χ wheneverχ is in a small neighborhood of 1. A perturbation technique allows us to computethe Hessian of λ0 (χ) at χ = 1 as
6A straightforward analogue of Tchebotarev’s density theorem, a generalization of the Dirichlettheorem to arbitrary ﬁnite Galois extensions, can be proven without any diﬃculty ([S62]).

18

ATSUSHI KATSUDA AND POLLY WEE SY(2.10)

8π 2(Hessλ0 )(ω, ω) =vol(M )

Z
M∥ω∥2 ,

where ω is a harmonic 1-form on M . Put J(M ) = H 1 (M, R)/H 1 (M, Z) which is
identiﬁed with the group of unitary characters and equipped with the ﬂat metricinduced from L2 -norm on the space of harmonic 1-forms. ConsiderZ³d ´g L′ (s, χ)χ(−α) −Fα (s) = −dχ,ds L(s, χ)J(M )where dχ is the normalized Haar measure. On the one hand, we have, in view oforthogonal relations of characters,
X X
k g ℓ(p)g+1 e−kℓ(p) .
Fα (s) =k=1
k[p]=αp

On the other hand, if we put
f0 (χ) =then we ﬁnd(2.11)p¢1¡
1 + 1 − 4λ0 (χ) ,2ZFα (s) = −³´d ´g ³1
χ(−α) −−dχ + h(s),dss − f0 (χ)
Vwhere V is a small neighborhood of 1, and h(s) is holomorphic in Re s ≥ 1.
Applying the Morse Lemma to the function f0 , and estimating the integral in(2.11), we conclude that there is a locally integrable function h(t) with¯√(g − 1)g ¯¯¯¯Fα (s) −¯ ≤ h(t) (s = 1 + ϵ + −1t)s−1
for a small positive ϵ. Finally applying a version of Ikehara’s Tauberian theorem,
we obtain Theorem 2.11.
Sunada says that the key idea of the proof came up when he was discussing the
problem with Atsushi Katsuda inside a bullet train (Shinkansen) on their way to
the Taniguchi Symposium held at Katata in 1985. Just as Sunada explained what
he had wanted for the function Fα , Katsuda said ”It seems OK if the Hessian of
λ0 (χ) at χ = 1 is positive deﬁnite.” This is actually the end of the proof !
After completing the paper [S24], Sunada and Katsuda extended the above result
to the case of a manifold with variable negative curvature. They knew that, for thissake, it was indispensable to go over to the theory of dynamical systems. In fact,a prime closed geodesic is identiﬁed with a closed (periodic) orbit of the geodesic
ﬂow. Therefore it is natural to consider the counting problem of closed orbits of ageneral dynamical system. The most appropriate set-up is the dynamical systemsof Anosov type. Indeed, a dynamical analogue of the prime number theorem has
been known in such a set-up (W. Parry and M. Pollicott [38]).
The precise set-up is described as follows. Let {ϕt } be a smooth, transitive,
weakly mixing Anosov ﬂow on a compact manifold X. Given a homology classα ∈ H1 (X, Z), we consider the counting function π(x, α) deﬁned in the same way
as before where, in this turn, ℓ(p) is the period of a closed orbit p. We denoteAN OVERVIEW OF SUNADA’S WORK19

by h the topological entropy and by m the (unique) invariant measure on X withmaximal entropy. Deﬁne the winding cycle Φ, a linear functional on H 1 (X, R), byZ
Φ([ω]) =〈Z, ω〉dm,
X

where ω is a closed 1-form, and Z is the vector ﬁeld generating {ϕt }. We also introduce the covariance form δ, a positive semi-deﬁnite quadratic form on H 1 (X, R),
by setting
Z
³Z t
´2
δ([ω], [ω]) = lim
dm(x)〈Z, ω〉ϕτ (x)dτ − tΦ([ω]) .
t→∞

X

0

It is checked that, if Φ vanishes (this is the case for geodesic ﬂows), then δ is positive
1
1
deﬁnite¡ so that¢ it induces a ﬂat metric on J(X) = H (X, R)/H (X, Z). We denoteby vol J(X) the volume with respect to this metric.
The following is the main theorem that Sunada and Katsuda proved after an
exchange of a bunch of letters between IHES and Okayama University (e-mail was
not yet available at that time).
Theorem 2.12. [S27] If Φ vanishes, then¡¢−1 −1 ehxhπ(x, α) ∼ (2π)−b/2 vol J(X)(b = rank H 1 (X, Z)).
x(b/2)+1
The idea of the proof is almost the same as the case of surfaces of constantnegative curvature except for the use of twisted Laplacians. They made use of atwisted version of what is called the Ruelle operator instead of twisted Laplacians,
which is deﬁned in terms of symbolic dynamics. Reﬁnements and generalizations
have been developed by M. Pollicott, R. Sharp, N. Anantharaman and Motoko
Kotani. (see [45] and [30] for instance).
In connection with the existence problem of closed geodesics in a homology class,
it is worthwhile to point out that, if there is no closed geodesics homologous tozero in an n-dimensional compact Riemannian manifold M , then M has the same
homotopy type as the torus T n .
Another eﬀective use of the twisted Laplacians ∆χ was made in the work [S47],
in which the long time behavior of the heat kernel was studied. It is known thatthe heat kernel k(t, x, y) on a general Riemannian manifold X has the following
short time asymptotics¡ d(x, y)2 ¢k(t, x, y) ∼ (4πt)−m/2 exp −4t
× (a0 (x, y) + a1 (x, y)t + a2 (x, y)t2 + . . . ) (t ↓ 0),
where m = dim X and d(x, y) denotes the Riemannian distance between x and y(here d(x, y) is supposed to be small enough). On the one hand, the coeﬃcientsai (x, y) have “local” nature in the sense that they are described by quantities deﬁned
only on a neighborhood of the shortest geodesic joining x and y. On the other hand,the behavior of k(t, x, y) as t goes to inﬁnity have to be controlled by the global
properties of the manifold. Sunada observed that this is the case for abelian covering
manifolds over compact manifolds, and gave an explicit asymptotic formula in the
joint work [S42] with Motoko Kotani.
Let π : X −→ M be a regular covering map whose covering transformationgroup Γ is free abelian, associated with which we have a surjective homomorphism

20

ATSUSHI KATSUDA AND POLLY WEE SY

of H1 (M, Z) onto Γ and its extension to a surjective linear map of H1 (M, R) =H1 (M, Z) ⊗R onto Γ ⊗ R. We introduce an inner product on H 1 (M, R) by identifying it with the space of harmonic 1-forms on M . The dual inner product isequipped on H1 (M, R) = Hom(H 1 (M, R), R). As the inner product on Γ ⊗ R, wetake up the quotient inner product derived from the inner product on H1 (M, R).
To describe long time asymptotic behavior of the heat kernel on X, we need adistance function diﬀerent from the Riemannian one. For this sake, deﬁne the mapΦ̃Γ : X → Γ ⊗ R by using the paring of Γ ⊗ R and Hom(Γ, R) as
Z x〈Φ̃Γ (x), ω〉 =
ω̃,
x0

1

where ω ∈ Hom(Γ, R) ⊂ H (M, R), and ω̃ denotes its lift to X. It should bepointed out that the integral in the right-hand side does not depend on the choiceof a path joining x0 and x. We then put dΓ (x, y) = ∥Φ̃Γ (y) − Φ̃Γ (x)∥, where ∥ · ∥denotes the Euclidean norm on Γ ⊗ R associated with the inner product deﬁned
above.
Consider the ﬂat torus Γ ⊗ R/Γ ⊗ Z with the ﬂat metric induced from theinner product above. Since Φ̃Γ (σx) = Φ̃Γ (x) + σ ⊗ 1 for σ ∈ Γ, we obtain a mapΦΓ : M → Γ ⊗ R/Γ ⊗ Z whose lift to X is Φ̃Γ . We call the ﬂat torus Γ ⊗ R/Γ ⊗ Z
the Γ-Albanese torus and denote it by AlbΓ . We also call ΦΓ the Γ-Albanese map.
The map ΦΓ is harmonic.
Let r = rank(Γ) (> 0) andC(X) = vol(M )r/2−1 vol(AlbΓ ).
The following theorem tells that a homogenization takes place not only for theheat kernel but also for the space itself as time t goes to inﬁnity.
Theorem 2.13. [S47](Local Central Limit Theorem)³¡ vol(M )¢´lim (4πt)r/2 k(t, x, y) − C(X) exp −dΓ (x, y)2 = 0,t↑∞
4t
uniformly for all x, y ∈ X.
This
¡ also tells that,¢ if B(t) is the Brownian motion on X, then the processδ Φ̃Γ B(δ −1 vol(M )t) goes, in distribution, to the Brownian motion on the Euclidean space Γ ⊗ R as δ goes to zero (see [33] for a probabilistic¡¢ interpretation).
It is worthwhile to mention that the stochastic process Φ̃Γ B(t) on Γ ⊗ R is amartingale since Φ̃Γ is a harmonic map.
One may establish an asymptotic expansion of k(t, x, y) as a byproduct.
Theorem 2.14. [S47]k(t, x, y) ∼ (4πt)−r/2 C(X)(1 + c1 (x, y)t−1 + c2 (x, y)t−2 + . . . )cl (x, y) ∼1 ¡ vol(M ) ¢ldΓ (x, y)2l−l!
4(t ↑ ∞),(d(x, y) ↑ ∞).

The explicit forms of the coeﬃcients ci (x, y) in Theorem 2.14 are complicatedin general. But one may still give the exact shape of c1 (x, y) as follows: Let
ω1 , . . . , ωr be an orthonormal basis of the space Hom(Γ, R)(⊂ H 1 (M.R)) and let
G : C ∞ (M ) → C ∞ (M ) be the Green operator.

AN OVERVIEW OF SUNADA’S WORK

21

Theorem 2.15. [S47]rrX
X
vol(M )
vol(M )
|ωi |2 )(π(x)) + G(
|ωi |2 )(π(y)))
dΓ (x, y)2 −
(G(
4
2
i=1
i=1
Z
r Z
r
r
´
³
X
X
X
vol(M )
G(〈ωi , ωj 〉)〈ωi , ωj 〉 .
|ωi |2 + 2+|ωi |2 )G(
4M
i,j=1 Mi=1
i=1

c1 (x, y) = −This theorem is seemingly technical. However its discrete analogue turns out tobe useful in Sunada’s study of geometric crystallography as seen in Subsection 2.8.
For the proof of these theorems, the perturbation theory is again employed. LetLχ be the ﬂat line bundle associated with χ. One may take an orthonormal basis
2{sχ,k }∞
k=0 of L (Lχ ) such that(1) ∆χ sχ,k = λk (χ)sχ,k ,(2) sχ,k is bounded and integrable with respect to χ,(3) sχ,0 is smooth in χ around χ = 1 and s1,0 ≡ vol(M )−1/2 .
The core of the proof is in the fact that the heat kernel k(t, x, y) on X is expressedas∞ ZXk(t, x, y) =exp(−λk (χ)t)s̃χ,k (x)s̃χ,k (y)dχ,k=0

Γ̂

b A careful
where s̃χ,k is the lift of sχ,k , and dχ is the normalized Haar measure of Γ.
analysis of this integral expression yields the theorems above (for Theorem 2.15,
one needs to know the coeﬃcients in the Taylor expansion of λ0 (χ) at χ = 1 up to
the 4th order).
The idea of twisted Laplacians has been developed further by Sunada for the
spectral study of general covering manifolds. The starting point of his study was
to understand a “mechanism” of the following result by R. Brooks [6]: Let M̃ be
the universal covering manifold of a compact Riemannian manifold M . The bottomof the spectrum λ0 (M̃ ) of the Laplacian ∆M̃ on M̃ is zero if and only if π1 (M ) isamenable. Brooks’ idea is to employ the theory of integral currents and Fφlner’s
theorem on amenability.
The observation Sunada made is that ∆M̃ is unitarily equivalent to the twisted
Laplacian associated with the regular representation of π1 (M ). To explain this ina more general set-up, we consider a regular covering map π : X −→ M over acompact Riemannian manifold M with covering transformation group G. Givena unitary representation ρ : G −→ RU (H) on a Hilbert space H, we set ℓ2 (ρ) ={f ; X −→ H; f (σx) = ρ(σ)f (x), M ∥f ∥2 < ∞}, which has a natural Hilbert
space structure. The twisted Laplacian ∆ρ is deﬁned to be the restriction to ℓ2 (ρ)
of the Laplacian acting on H-valued functions on X. Actually ∆ρ is the Laplacian
acting on sections of the ﬂat vector bundle (possibly of inﬁnite rank) associatedwith ρ. One can establish the following estimates for the bottom of the spectrum
λ0 (ρ) = inf σ(∆ρ ) from below and above.
Theorem 2.16. ([S25])(2.12)c1 δ(ρ, 1)2 ≤ λ0 (ρ) ≤ c2 δ(ρ, 1)2 .

Here c1 , c2 are positive constants independent of ρ

22

ATSUSHI KATSUDA AND POLLY WEE SY

The quantity δ(ρ, 1) is the Kazhdan distance between ρ and the trivial representation 1 deﬁned by δ(ρ, 1) = inf{supg∈A ∥ρ(g)v − v∥; v ∈ H, ∥v∥ = 1}, where
A is a ﬁnite set of generators of G. The inequalities in (2.12), in particular, saythat λ0 (ρ) = 0 if and¡ only if δ(ρ,¢ 1) = 0. Applying this to the regular representation7 ρH : G −→ U ℓ2 (H\G) associated with a subgroup H of G, we conclude
that λ0 (∆H\X ) = 0 if and only if δ(ρH , 1) = 0 since ∆ρH is unitarily equivalent to
the discrete Laplacian ∆H\X on the quotient manifold H\X 8. In particular, sinceδ(ρG , 1) = 0 if and only if G is amenable9, we ﬁnd that λ0 (∆X ) = 0 if and only if
G is amenable.
Sunada says that he came up Zwith the estimates (2.12) as a natural generalizationof (2.10). Indeed, the integralM∥ω∥2 on the right-hand side of (2.10) gives riseto the distance between χ and the trivial character.
In connection with Theorem 2.16, Sunada also established a relationship between
spectra and the notion of weak containment. As a corollary, he proved that, if
π : X −→ M is an amenable covering map, then σ(∆M ) ⊂ σ(∆X ).
A discrete version of Theorem 2.16 was given in [S31]. The estimate from below
in this case is related to a construction of expanders, a model of eﬃcient communication networks (see [S57]).
Sunada was greatly impressed by the beautiful relation between spectra andgroup structure obtained by Robert Brooks in the paper mentioned above. He
remembers that Brooks was the ﬁrst to read through his preprint on isospectral
manifolds. Sunada and Brooks had mutually inﬂuenced one another in many occasions thereafter, until Brooks passed away in 2002.
As for the development of the spectral study of covering spaces towards a diﬀerent
direction, one may refer to the paper by P. Kuchment and and Y. Pinchover [32].
The idea of “twisting” was used by M. A. Shubin and Sunada [S55] in theirrigorous derivation of the classical T 3 -law for speciﬁc heat of crystals, a typicalthermodynamical quantity in solid state physics, which dates back to Einstein’s
pioneering work in 1907 and its subsequent reﬁnement by Debye in 1912.
2.6. Ihara Zeta functions. In the mid-80s, Sunada started to study the zetafunction associated with a cocompact torsion-free discrete subgroup of P SL2 (Qp ),which was originally introduced by Y. Ihara [28] in 1966 as an analogue of theSelberg zeta function. In the background of Ihara zeta functions, there is a consensus that the p-adic version of the real hyperbolic plane GL2 (R)/(R× × O(2))
10
is GL2 (Qp )/(Q×p × GL2 (Zp )) . The latter space is, as a special case of BruhatTits buildings, regarded as the set of vertices in the regular tree of degree p + 1,
and hence its compact quotient yields a ﬁnite regular graph. Thus it is natural tointerpret the Ihara zeta functions in terms of ﬁnite regular graphs.
According to Sunada, his idea in this study is simple. He says that he justmimicked the geometric interpretation of Selberg zeta functions in terms of closed7ρ

G
H is the induced representation IndH (1) of the trivial representation.

8This implies that, if ρ

H1 and ρH2 are equivalent for two subgroups H1 , H2 , then ∆H1 \X and∆H2 \X are unitary equivalent. This gives another proof of Theorem 2.10.
9This is equivalent to that G has an invariant mean.
10GL (Z ) and O(2) are maximal compact subgroups of GL (Q ) and GL (R), respectively.
p
p
2
2
2

AN OVERVIEW OF SUNADA’S WORK

23

geodesics. Since he already had a geometric model of number theory, it was a sortof exercise for him to carry out the idea.
Let P be the set of all prime cycles in a ﬁnite regular graph X of degree q + 1.
Here a prime cycle is an equivalence class of a closed path without backtracking(we call closed geodesic for simplicity) which is not a power of another one. Twoclosed paths are said to be equivalent if one is obtained by a cyclic permutationof edges in another. In 1985, Sunada [S22], following the suggestion stated in the
preface of J.-P. Serre [47], expressed the Ihara zeta function as the Euler productY
Z(u) =(1 − u|p| )−1 ,p∈P

and gave a graph theoretic proof for the following determinant expression in termsof the adjacency operator A¡¢−1
Z(u) = (1 − u2 )(1−q)N/2 det I − uA + qu2 I,
where N is the number of vertices. From this identity, it follows that the Ihara zetafunction of a regular graph X satisﬁes the Riemann Hypothesis if and only if every√
eigenvalue µ of A satisﬁes |µ| ≤ 2 q except for µ = ±(q + 1). Graphs satisfying
the Riemann Hypothesis was later called Ramanujan graphs by A. Lubotzky, R.
Phillips and P. Sarnak [34].
Sunada’s observation was immediately generalized by K. Hashimoto and A. Horito the case of semi-regular graphs which correspond to p-adic semi-simple groupsof rank one. H. Bass [3] noticed that “regularity” of graphs is not necessary for adeterminant expression. Sunada in the collaborative work with Kotani [S48] used
the idea of discrete geodesic ﬂows to give another proof of Bass’ result. Actually,
they interpreted the Ihara zeta function as the dynamical zeta function associatedwith a symbolic dynamical system. An instructive proof together with a generalization to weighted graphs can be found in M. D. Horton, H. M. Stark and A. A.
Terras [27]. See also [37].
There have been various attempts to introduce Ihara zeta functions for inﬁnite
graphs. See for instance D. Guido, T. Isola, and M. L. Lapidus [26], R. I. Grigorchuk
and A. Zuk [24].
2.7. Quantum ergodicity. Mathematics is developed through interactions andcommunications among mathematicians. Sunada’s work on quantum ergodicity is
one of such examples. Around 1990, he read an article [53] of Steven Zelditch who
has been an appreciative reader of Sunada’s work. The subject of that paper is on
asymptotic behavior of eigenfunctions of Laplacians.
Before explaining Zelditch’s result and its generalization, let us recall an elementary fact ([S39]). Let 0 = λ0 < λ1 ≤ λ2 ≤ · · · be eigenvalues of ∆ on a compact
Riemannian manifold M , and let {ϕk }∞k=0 be an orthonormal basis of eigenfunctions
with ∆ϕk = λk ϕ. We then have the asymptotic formulaZZ∞X
f (x)|ϕk (x)|2 dx =f (x)k(t, x, x)dxe−λk tk=0∼ (4πt)−n/2

M

M

Z

M

f (x)dx(n = dim M )24

ATSUSHI KATSUDA AND POLLY WEE SY

as t ↓ 0. Applying the Hardy-Littlewood Tauberian theorem, we obtainZ
X Z
f (x)|ϕk (x)|2 dx = vol(M )−1
f (x)dx,
lim φ(λ)−1
λ→∞

λk ≤λ

M

M

where φ(λ) = |{k; λk ≤ λ}|. This identity holds without any condition on M . The
following tells that a dynamical property of the geodesic ﬂow allows us to obtain areﬁned limit formula.
Proposition 2.6. (Quantum ergodicity [53], [15]) If the geodesic ﬂow is ergodic,
then there exists a subsequence {λkj } of {λk } of full density such that for everyf ∈ C ∞ (M ),ZZlimf (x)|ϕkj (x)|2 dx = vol(M )−1
f (x)dx.
j→∞

M

M

The meaning of “full density” is that|{kj ; λkj ≤ λ}|= 1.
λ→∞ |{k; λk ≤ λ}|lim

Ergodicity in classical mechanical system means that the time average of anobservable is equal to the space average. In his attempt to give a more natural
ﬂavor of ergodicity to the limit theorem above, Sunada introduces the concept ofquantum ergodicity at inﬁnite energy level.
To explain this in full generality, Sunada considers a positive self-adjoint elliptic
b of order one with eigenvalues 0 ≤ e1 ≤ e2 ≤ · · · .
pseudo-diﬀerential operator Hb the principal symbol of H,b and denote by ϕt the Hamiltonian ﬂow
Put H = σ(H),on T ∗ M \0 generated by the Hamiltonian H. A quantum observable is deﬁned to
be a pseudo-diﬀerential operator√ A of order zero. The time evolution of A is givenbby Ut ∗ AUt where Ut = exp t −1H
To follow the deﬁnition of classical ergodicity, we deﬁne the time average up to
time t byZ
1 t ∗Us AUs ds,t 0Xbwhich converges weakly to A =Pe APe . Here e runs over the spectrum of H,eb = eϕ}. We
and Pe is the orthogonal projection onto the eigenspace Ve = {ϕ; Hϕalso deﬁne the space average by
X〈Aϕk , ϕk 〉〈A〉 = lim N (e)−1
e↑∞

k
ek ≤e

where N (e) = |{k; ek ≤ e}| and {ϕk } is a complete orthonormal basis of eigenfuncb with Hϕb k = ek ϕk . It is worthwhile to note that 〈A〉 is also expressedtions of H
asZ〈A〉 = lime→∞

S(V≤e )〈Aϕ, ϕ〉dS(ϕ),

where dS
P is the normalized uniform measure on the unit sphere S(V≤e ) of the spaceV≤e = λ≤e Vλ .

AN OVERVIEW OF SUNADA’S WORK25

Now the quantum dynamical system {Ut } is said to be quantum ergodic (at∗inﬁnite energy level) if, for every observable A, the space average 〈A A〉 exists, and∗〈A A〉 = |〈A〉|2 .
This is an analogue of the criterion for the classical ergodicity which can be stated
as “〈|f |2 〉 = |〈f 〉|2 for every classical observable f ”, where f is the time average,
and 〈f 〉 is the space average.
Sunada [S43] proved the followings.
Theorem 2.17. The ﬂow ϕt restricted to the hypersurface Σ1 = H −1 (1) is ergodic
if and only if {Ut } is quantum ergodic, and∗lim 〈At ∗ At 〉 = 〈A A〉.

t→∞

Theorem 2.18. (1) Quantum ergodicity is equivalent to the following near-diagonal
asymptotic.
Z
¯2
X ¯
¯2 ¯¯
¯
−1
¯
¯
lim N (e)〈Aϕi , ϕj 〉 = ¯
σ(A)dω1 ¯ .
e→∞i,jei =ej ≤eΣ1where ω1 is the normalized Liouville measure on Σ1 .
∗(2) The condition limt→∞ 〈At ∗ At 〉 = 〈A A〉 is equivalent to the following oﬀdiagonal asymptotic.
XX¯¯¯〈Aϕi , ϕj 〉¯2 = 0.
lim lim sup
δ↓0

e↑∞

i
j
ei ≤e 0<|ei −ej |≤δOne can prove that if {Ut } is quantum ergodic (hence if ϕt is ergodic), then there
exists a subsequence {kj } of full density such that, forR every pseudo-diﬀerential
the sequence 〈Aϕkj , ϕkj 〉 = M Aϕkj ϕkj ∗ converges toRoperator A of order zero,∗σ(A)dω1 , where ϕ is the complex conjugate of ϕ.
Σ1b = ∆1/2 , we have H(x, ξ) = ∥ξ∥ so that, in this case, Σ1 is theNote that, for H
tangent unit sphere bundle, and ϕt is the geodesic ﬂow.
The notion of quantum ergodicity at ﬁnite energy level was formulated by Tatsuya Tate [48]. See also S. Zelditch [54] for an abstract set-up.
Sunada’s work of quantum ergodicity looks isolated from others. One can see,however, that he always demonstrates high capabilities common to all his work topush to the completion the original concepts.
2.8. Discrete geometric analysis. Sunada says that there are several sources
available for him to start with the study of discrete geometric analysis, a ﬁeld
dealing with analysis on graphs by using geometric ideas. One is, as a matter ofcourse, Ihara zeta functions, where adjacency operators play a role of Laplacians.
Another is the symbolic dynamics used in the study of unstable dynamical systems
in which Ruelle operator on oriented graphs is an indispensable tool. There is one
more source, a bit quirky one. He wrote a book (in Japanese) on the elementary
aspects of area and volume including Dehn’s theorem on scissors congruence oftetrahedra ([S53]). While he was writing the part on the area theory, this question
came to his mind: when a rectangle K is divided into ﬁnitely many squares, is theratio of two sides rational ? Sunada asked Prof. Koji Shiga, his supervisor when
he was in TIT, whether the answer is known. Shiga immediately told him that

26

ATSUSHI KATSUDA AND POLLY WEE SY

the problem was already solved by M. Dehn in 1902 aﬃrmatively ([18]). Sunada
tried in vain to understand the proof, and so he tried to ﬁnd his own proof. Aftersome trials, he found out that he could reduce the problem to a discrete version ofPoisson’s equation ∆f = g ([SB6]). This was not surprising to him since, as he wasalso told by Shiga, Dehn’s theorem can be shown by an idea of electric circuits ([5]).
Note that the fundamental laws of electric circuits are also described in terms ofa discrete analogue of “the method of orthogonal projections” as H. Weyl alreadyobserved in 1920s (see [S57]).
A much serious reason came up when he learned of a strange phenomenon forthe spectra of discrete magnetic Schrödinger operators deﬁned on Z2 :(Hα1 ,α2 ϕ)(m, n)=√
1 √−1α1 nϕ(m + 1, n) + e− −1α1 n ϕ(m − 1, n)[e
4 √√+e −1α2 m ϕ(m, n + 1) + e− −1α2 m ϕ(m, n − 1)],

(α1 , α2 ∈ R). The operator Hα1 ,α2 : ℓ2 (Z2 ) −→ ℓ2 (Z2 ) is bounded and self-adjoint.
The operator I − Hα1 ,α2 is regarded as a discretization of the following Schrödinger
operator with the uniform magnetic ﬁeld B = (α2 − α1 )dx ∧ dy√√
¡ ∂¢2 ¡ ∂¢2−+ −1α1 y −+ −1α2 x ,∂x∂ywhose spectrum consists only of eigenvalues of inﬁnite multiplicity provided thatθ = α2 − α1 is not zero; say¡
1¢{2|θ| k + ; k = 0, 1, 2, . . .}.
2
On the other hand, the spectrum σ(Hα1 ,α2 ), a closed subset of [−1, 1], is quitesensitive to the parameter θ = α2 − α1 , which physicists call the magnetic ﬂux. If
θ/2π is rational, then σ(Hα1 ,α2 ) has band structure, while if θ/2π is a Liouville
number, then σ(Hα1 ,α2 ) is a Cantor set.
Having this example in mind11, Sunada formulates a discrete analogue of general
Schrödinger operators with periodic magnetic ﬁelds ([S40]) and deﬁnes the notionof magnetic ﬂux. For this end, he starts with a realization Φ of a graph X ina Riemannian manifold M (thus Φ is a piecewise smooth map of X into M ). We
denote by V the set of vertices, and by E the set of all oriented edges of X. Both X
and M are supposed to have a free Γ-action, and Φ is supposed to be Γ-equivariant.
A magnetic ﬁeld on M is a closed 2-form B. We assume that B is Γ-invariant,and that B has a global vector potential (1-form) A so that B = dA (A is notnecessarily Γ-invariant). We also assume H 1 (M, R) = {0}. Deﬁne function fA on
E by
´
³√ Z
fA (e) = exp −1 Φ∗ A ,ewhich is a U (1)-valued 1-cochain on X. Using the assumptions on B and M , we
ﬁnd a function ϕσ on M for each √σ ∈ Γ such that dϕσ = σ ∗ A − A. If we deﬁnesσ : X −→ U (1) by sσ (x) = exp(− −1ϕσ (Φ(x))), thenfA (σe) = fA (e)sσ (o(e)),
sσ (t(e))

11Sunada’s interest in geometry and analysis of magnetic ﬁelds is already seen in [S37] and[S61].

AN OVERVIEW OF SUNADA’S WORK

27

where o(e) (resp. t(e)) denotes the origin (resp. terminus) of e. This implies
that the cohomology class [fA ] ∈ H 1 (X, U (1)) is Γ-invariant. We then deﬁne theoperator HfA : ℓ2 (X) −→ ℓ2 (X) by
X
1(HfA ϕ)(x) =fA (e)ϕ(t(e)),
deg x
e∈Ex

where Ex = {e ∈ E; o(e) = x} and deg x = |Ex |. PThe space ℓ2 (X) is theHilbert space consisting of functions ϕ on V such that x∈V |ϕ(x)|2 deg x < ∞.
The operator I − HfA is considered√a discretization of the magnetic Schrödinger
operator ∇∗A ∇A , where ∇A = d − −1A. (see [31]). One may easily check that
HfA coincides with Hα1 ,α2 for the square lattice X = Z2 realized in M = R2 , andfor the vector potential A = α1 ydx + α2 xdy (B = dA = (α2 − α1 )dx ∧ dy).
Taking account of the discussion above, Sunada introduces an abstract set-up.
On the one hand, he begins with a regular covering graph X = (V, E) over a ﬁnite
graph X0 with covering transformation group Γ, and a 1-cochain f ∈ C 1 (X, U (1))
on X whose cohomology class [f ] ∈ H 1 (X, U (1)) is Γ-invariant. On the other hand,
he considers a random walk on X withP a Γ-invariant transition probability p : E −→R; namely p(σe) = p(e), p(e) ≥ 0, e∈Ex p(e) = 1. Then he deﬁnes the operatorP
Hf : ℓ2 (X) −→ ℓ2 (X) by (Hf ϕ)(x) = e∈Ex p(e)f (e)ϕ(t(e)). If the random walkis symmetric (reversible) in the sense that there is a Γ-invariant positive-valued
function m on V with p(e)m(o(e)) = p(e)m(t(e)) (e being the inverse edge of e),then Hf isPbounded self-adjoint operator acting in the Hilbert space ℓ2 (X) = {ϕ :V −→ C; x∈V |ϕ(x)|2 m(x) < ∞}. He calls Hf a magnetic transition operator.
The notion of magnetic ﬂux is introduced in this set-up as follows. From theassumption that [f ] is Γ-invariant, it follows that there exists a U (1)-valued functionsσ on V such that f (σe) = f (e)sσ (t(e))sσ (o(e))−1 . PutΘf (σ, γ) =sσ (γx)sγ (x).
sσγ (x)

It is observed that the right-hand side does not depend on x and Θf (σ, γ) is a group2-cocycle of Γ with coeﬃcients in U (1); that is,Θf (σ1 , σ2 σ3 )Θf (σ2 , σ3 ) = Θf (σ1 , σ2 )Θf (σ1 σ2 , σ3 ).
Thus we obtain [Θf ] ∈ H 2 (Γ, U (1)). This is what Sunada calls the magnetic ﬂux
class.
The following is a partial generalization of what we have mentioned for Hα1 ,α2 .
Theorem 2.19. [S40] If Γ is abelian and [Θf ] ∈ H 2 (Γ, Q/Z), then the spectrumof Hf has band structure.
The magnetic ﬂux class [Θf ] depends only on [f ] so that we have a homomorphism Θ : H 1 (X, U (1))Γ −→ H 2 (Γ, U (1)). In connection with this, we have thefollowing exact sequenceπ∗Θ

1 → H 1 (Γ, U (1))→H 1 (X0 , U (1)) → H 1 (X, U (1))Γ → H 2 (Γ, U (1)) → 1,
where H 1 (Γ, U (1))→H 1 (X0 , U (1)) is the homomorphism induced from the surjective homomorphism µ : π1 (X0 ) −→ Γ associated with the covering map π :X −→ X0 . If X is the maximal abelian covering graph of X0 , we observe thatΘ
H 1 (Γ, U (1))→H1 (X0 , U (1)) is an isomorphism so that H 1 (X, U (1))Γ → H 2 (Γ, U (1))

28

ATSUSHI KATSUDA AND POLLY WEE SY

is also an isomorphism. Therefore the magnetic ﬂux class [Θf ] determines the unitary equivalence class of Hf . This is not the case in general (an analogue of the
Aharonov-Bohm eﬀect).
The study of magnetic transition operators stimulated Sunada’s interest in thetheory of random walks itself. His interest was multiplied when he was asked by aJapanese probabilist about a geometric meaning of the constant which appears in
the local limit formula for classical lattices in plane, based on a few explicit values
computed in an ad hoc manner (see the table below). The local limit formula forthe simple random walk on a lattice X claims that there exists a positive constantC(X) such that12
lim (4πn)p(n, x, y)(deg y)−1 = C(X),n→∞where p(n, x, y) is the n-step transition probability13. The constant C(X), which,by deﬁnition, depends only on the graph structure of X, is surely an interesting
quantity in its own right.
X
C(X)hexagonal triangular quadrilateral kagome
√
√
√
3
2 3
2 3
2
3
3

The paper [S44] written jointly with Kotani and Tomoyuki Shirai gave an answer
to the question. They treated the local limit formula for a symmetric random walkon abelian covering graphs X over ﬁnite graphs (which they call crystal lattices),and gave an expression of C(X) in terms of a discrete analogue of Albanese tori(see Theorem 2.22 below for another expression and [S45], [S60]).
In the article [S47] which we have mentioned in Subsection 2.5, Sunada andKotani established more precise asymptotics for p(n, x, y); namely, a local central
limit theorem and an asymptotic expansion. Actually the idea for the heat kernel
can be applied without almost any change to the discrete case. To state the asymptotics, what we need is a discrete version of harmonic theory described below14.
For simplicity, let us restrict ourselves to the case of simple random walks on
crystal lattices; that is, the case that p(e) = (deg o(e))−1 . Let X be an abelian
covering graph over a ﬁnite graph X0 whose covering transformation group is Γ.
We assume that Γ is free abelian. As in the case of manifolds, we have a surjective
linear map of H1 (X0 , R) onto Γ ⊗ R and its transpose Hom(Γ, R) −→ H 1 (X0 , R).
Recall that the ﬁrst cohomology group H 1 (X0 , R) is deﬁned as C 1 (X0 , R)/Image d,where d : C 0 (X0 , R) −→ C 1 (X0 , R) is the coboundary map of the cochain groupsof X0 as a CW-complex.
12For simplicity, X is assumed to be non-bipartite from now on. In bipartite case, a minor

modiﬁcation of statement is required.
13If we consider a d-dimensional lattice, we should replace 4πn by (4πn)d/2 in the limit formula.
14A partial generalization to “nilpotent crystal lattice” was established by Satoshi Ishiwata.

AN OVERVIEW OF SUNADA’S WORK29

Deﬁne the inner products on C 0 (X0 , R) and C 1 (X0 , R) respectively byX(2.13)〈f1 , f2 〉 =f1 (x)f2 (x)deg x,x∈X0(2.14)〈ω1 , ω2 〉 =1 X
ω1 (e)ω2 (e),
2
e∈E0

∗

1

0

and let d : C (X0 , R) −→ C (X0 , R) be the adjoint operator of the coboundary
operator d with respect to these inner products. PutXω(e) = 0}.
H1 (X0 ) = {ω ∈ C 1 (X0 , R); d∗ ω = −(deg x)−1e∈(E0 )xThis is a discrete analogue of the space of “harmonic 1-forms” on X0 . We now identify H 1 (X0 , R) with H1 (X0 ) via the canonical isomorphism H1 (X0 ) onto H 1 (X0 , R)(the discrete Hodge-Kodaira theorem). Through this identiﬁcation, we obtain an
inner product on H 1 (X0 , R) and an inner product g0 on Γ ⊗ R.
Next deﬁne Φ : X −→ Γ ⊗ R bynX
ω̃(ei ), (ω ∈ Hom(Γ, R) ⊂ H 1 (X0 , R)),〈Φ(x), ω〉 =i=1

where c = (e1 , . . . , en ) is a path in X joining a reference point x0 and x, and ω̃ is thepull-back of ω to X. We readily check that the piecewise linear map Φ interpolated
by line segments is a periodic realization of X in the sense that Φ(σx) = Φ(x) + σ.
We call Φ : X −→ (Γ ⊗ R, g0 ) the standard realization of X.
Now the materials are ready to state the asymptotics for p(n, x, y). Indeed,
instead of the Laplacian, we only have to treat the discrete Laplacian deﬁned by
I − H1 (remember that H1 is just the ordinary transition operator for the randomwalk). But we are not going to repeat almost the same statements here. Instead westate a consequence of the results and a peculiar feature of the standard realization
Φ.
Theorem 2.20. [S50] Let ∥ · ∥ be the norm associated with the inner product g0 .
Thenn p(n, x, x)op(n, y, y)∥Φ(x) − Φ(y)∥2 = lim 2n
+
−2 .
n→∞
p(n, y, x) p(n, x, y)This theorem, shown by means of the graph versions of Theorem 2.14 and 2.15,
says that the left-hand side does not depend on the choice of the “periodic lattice”
Γ since the right-hand side depends only on the graph structure. Furthermore,
we may conclude that the standard realization Φ has maximal symmetry in thefollowing sense.
Theorem 2.21. [S50] Let Aut(X) be the automorphism group of the crystal lattice
X as an abstract graph, and let Mg0 be the group of congruent transformations of(Γ ⊗ R, g0 ). Then there exists a homomorphism ρ : Aut(X) −→ Mg0 such that
Φ(gx) = ρ(g)Φ(x) for g ∈ Aut(X).
In [S56], Sunada observed that there is a remarkable relation between the constant C(X) appearing in the local limit formula and a certain energy of crystals
which characterizes the standard realization by a minimal principle.
We think of a crystal (a periodically realized crystal lattice) in Rd as a systemof harmonic oscillators, that is, each edge represents a harmonic oscillator whose

30

ATSUSHI KATSUDA AND POLLY WEE SY

energy is the square of its length. We shall deﬁne the energy of a crystal “per a unitcell” in the following way. Given a bounded domain D in Rd , denote by E(D) thesum of the energy of harmonic oscillators whose endpoints are in D, and normalize
it in such a way as
E0 (D) =E(D)deg(D)1−2/d vol(D)2/d,

where deg(D) is the sum of degree (valency) of vertices in D. Take an increasing∞dsequence of bounded domains {Di }∞i=1 with ∪i=1 Di = R (for example, a family ofconcentric balls). The energy of the crystal (per a unit cell) is deﬁned as the limit
Ener = lim E0 (Di ).
i→∞

Indeed the limit exists under a mild condition on {Di }∞i=1 , and Ener does notdepend on choices of {Di }∞
.
We
write
E
(Φ)
for
the
energywhen the crystal isner
i=1
given by a periodic realization Φ. It is easy to observe that Ener (T ◦ Φ) = Ener (Φ)
for every homothetic transformation T .
Theorem 2.22. [S60] For every periodic realization Φ of X in Rd , we have15
Ener (Φ) ≥ dC(X)−2/d .
The equality holds if and only if Φ is standard.
The proof of this remarkable inequality, available at present, is not carried out
by ﬁnding a direct link between the two quantities, but is based upon a canonicalexpression of the standard realization, an analogue of Albanese maps in algebraic
geometry ([S50], [S60]).
Example 2.2. Figure 1 exhibits three kinds of periodic realizations of the hexagonallattice (the maximal abelian covering graph over the graph with two vertices joinedby three parallel edges). The ﬁgure (honeycomb) on the right side is its standard
realization.

Figure 1. Various periodic realizations of the hexagonal lattice
We also observe that the standard realizations of the triangular lattice and thequadrilateral lattice are the regular triangular lattice and the regular square lattice,respectively.
15This inequality is for non-bipartite crystal lattices. In bipartite case, the right-hand sideshould be replaced by d(C(X)/2)−2/d .

AN OVERVIEW OF SUNADA’S WORK

31

These examples tell that “a random walker on a crystal lattice X can detect themost natural way for X to sit in space”.
The notions of Γ-Albanese tori and Γ-Albanese maps are deﬁned in the samemanner as the continuous case. For the maximal abelian covering graphs, the ΓAlbanese maps are closely related to the Abel-Jacobi maps introduced by R. Bacher,
P. De La Harpe, and T. Nagnibeda [2] (see [S60]).
The paper [S54] jointly written with Kotani handled large deviations of random
walks on crystal lattices which exhibit another aspect of asymptotics. The theoryof large deviations, in general, concerns the asymptotic behavior of remote tails of
sequences of probability distributions. For the simple random walk16 the problem
is to ﬁnd an asymptotic behavior of the transition probability p(n, x, y) while y is
kept near the boundary ∂Bn (x) = {y ∈ V ; d(x, y) = n}, where d(x, y) is the graphdistance between x and y. Note that Bn (x) = {y ∈ V ; d(x, y) ≤ n} is the reachable
region of the random walker starting from x.
What they did ﬁrst was to look at the shape of Bn (x) in space when n goesto inﬁnity. More precisely, by ﬁxing a periodic realization Φ : X −→ Γ ⊗ R with
Φ(x) = 0, and considering the scaling-limit ﬁgure¢
1 ¡Φ Bn (x) ,n→∞ n

D = limthey observed that this coincides with the unit ball for the norm on Γ ⊗ R deﬁnedby∥ξ∥1 = inf{∥x∥1 | x ∈ H1 (X0 , R), µR (x) = ξ},where ∥x∥1 denotes the ℓ1 -norm of x as a 1-chain of° the ﬁnite graph
X0 . To be°P°P°exact, the norm ∥ · ∥1 on C1 (X0 , R) is deﬁned by ° e∈E o ae e° =e∈E o |ae |1(E o ⊂ E is an orientation, i.e., E o ∩ E o = ∅ and E o ∪ E o = E). Thus it isconcluded that D is a convex polyhedron. The detailed combinatorial structure of
the convex polyhedron D in the case of the maximal abelian covering graph was
discussed in their paper.
The metric space Γ ⊗ R with the distance d∞ (ξ1 , ξ2 ) = ∥ξ1 − ξ2 ∥1 turns out tocoincide with the Gromov-Hausdorﬀ limit of (X, n−1 d) as n goes to inﬁnity (M.
Gromov [25]).
Now what about the large deviation asymptotic of p(n, x, y) ? Their answer isthe following. There exists a convex function H on Γ ⊗ R (possibly assuming +∞)such that, for ξ in the interior of D, and {yn }∞n=1 in V such that {Φ(yn ) − nξ} isbounded, we have1lim
log p(n, x, yn ) = −H(ξ),
n→∞ nand D = {ξ; H(ξ) < ∞}. The convex function H is explicitly described in termsof the maximal positiveµ0 (ω)
transition operators17 Lω
¡ of ¢the twisted
P eigenvaluesω(e)deﬁned by Lω f (x) = e∈E0x p(e)ef t(e) (ω ∈ C 1 (X0 , R)). More precisely, if
1
we think of µ0 as a function on H (X0 , R) using the fact that Lω+df = e−f Lω ef
16With a suitable modiﬁcation, one may handle general symmetric and non-symmetric random
walks.
17Actually, L is linearly equivalent to the twisted operator associated with a real characterωof H1 (X0 , Z).

32

ATSUSHI KATSUDA AND POLLY WEE SY(f ∈ C 0 (X0 , R)), thenH(ξ) =

supx∈Hom(Γ,R)£¤(ξ, x) − log µ0 (x)(recall that Hom(Γ, R), the dual of Γ ⊗ R, is regarded as a subspace of H 1 (X0 , R)).
See [S57] for an overview of discrete geometric analysis including Sunada’s own
work, which is based on his lectures given at Gregynog Hall, University of Walesin 2007, as an activity of the Project “Analysis on graphs and its applications” in
the Isaac Newton Institute.
2.9. Strongly isotropic crystals (a diamond twin). A mathematician who
declares himself to be a geometer surely would like to follow the tradition of ancient
Greek mathematics; namely, the wish to classify beautiful ﬁgures as the Greek
mathematicians did for regular polyhedra. This has been Sunada’s desire since hewas studying the standard realizations of crystal lattices.
The crystal he ﬁrst looked at is the diamond crystal since the diamond as a real
crystal is not only beautiful, but also has many interesting physical properties. He
had thought that there should be some peculiar feature in its microscopic structure,and eventually found out that it possesses the following three properties:(1) The diamond crystal has maximal symmetry in the sense that every automorphism of the diamond lattice as an abstract graph extends to a congruent
transformation.
(2) It is vertex-transitive in the sense that the automorphism group acts transitively on the set of vertices.
(3) It has the strong isotropic property in the sense that every permutation of
the edges with the same endpoint extends to an automorphism (thus extends to acongruent transformation which leaves the crystal invariant).
Sunada’s desire was to list all crystals having these properties. His answer isstated in the following theorem.
Theorem 2.23. [S56] A crystal having the properties (1), (2), (3) must be either the
diamond crystal or the standard realization of the maximal abelian covering graphover the complete graph K4 .
It is worthy to note that the diamond crystal is the standard realization of the
maximal abelian covering graph over the graph with two vertices joined by 4 paralleledges, and thus (1) is a consequence of Theorem 2.21. The crystal mentioned in
the above theorem, entitled to be called a diamond twin as a hypothetical crystal,is what Sunada called the K4 crystal18. At the time when he was writing the paper[S56], he did not know that it has been known in crystallography. Actually, there
is an interesting history about (re)discovery of this crystal.
It is believed that the crystallographer who discovered this crystal structure
for the ﬁrst time is Laves (1923). In 1954, A. F. Wells mentioned the structure
and called it “Net 1f” in his article “The geometrical basis of crystal chemistry”
appeared in Acta Crystallography. It is not deﬁnite whether he knew of Laves’
work. In his book entitled “Three Dimensional Nets and Polyhedra” published in
1977, Wells renamed it “(10,3)-a”. H. S. M. Coxeter [17] called it “Laves’ graph ofgirth ten”. M. O’Keeﬀe and his colleagues discussed this structure in some details
18The K crystal has chirality.
4AN OVERVIEW OF SUNADA’S WORK

33

and renamed it “srs” due to its chemical relevance in Acta Cryst. A59 (2003), 22.
Ludwig Danzer discovered the structure once again around 1994. John H. Conway
calls this crystal the “triamond” (see the book [16] published recently).

Figure 2. K4 crystal (the image created by Hisashi Naito)
The remarkable feature of the K4 crystal19 has recently attracted the attentionof researchers in the material science who are now evaluating its various physicalproperties by ﬁrst principles calculations ([S58], [S63]). For instance, their calculations show that, as an sp2 carbon crystal20, the K4 crystal has a meta-stable state,
and therefore suggest that there is possible pressure which may induce structuralphase transition from graphite to K4 . It is also observed that valence electrons aremainly localized along the bonding, which gives rise to the metallic property of thiscarbon crystal.
> > > > > > > > > > > > > > > > > > >>From our description of Sunada’s work, one can get some picture of how newconcepts and ideas were formulated and how elegant techniques of proofs weregenerated, which therefore allows one to have a sense of the depth and beauty of
his ﬁndings. More importantly, we notice that, though the topics Sunada has been
concerned with are diverse, one can still ﬁnd a consistent story in his study. His
new awareness of mathematical issues has always been emerging from his previous
work. This is one main reason why Sunada’s work is considered highly original.
To this day, it is indeed more than a dream comes true for Sunada. For, his
numerous contributions have had and will continue to make a signiﬁcant impact on
discrete geometric analysis, spectral geometry, dynamical systems, probability, andothers in mathematics.
19Sunada’s observation was reviewed by three magazines; namely, Nature Materials, Science,and Scientiﬁc American.
20The diamond is an sp3 carbon crystal.

34

ATSUSHI KATSUDA AND POLLY WEE SY

Once again, we congratulate Professor Sunada on his 60th birthday. May hehave many more good years to come.
Acknowledgement The authors are indebted to Professor Toshikazu Sunada forproviding them with a memorandum on his own work, and for answering manyquestions they raised during the preparation of this article.
References[S1] T. Sunada, Non-linear eliptic operators on a compact manifold and an implicit function
theorem, Nagoya Math.J. 57 (1974), 175-200.
[S2] T. Sunada, Holomorphic mappings into a compact quotient of symmetric bounded domain,
Nagoya Math.J. 64 (1976), 159-175.
[S3] T. Sunada, Closed geodesics in a locally symmetric space, Tohoku Math.J. 30 (1978), 59-68.
[S4] T. Sunada, Holomorphic equivalence problem for bounded Reinhaldt domains, Math. Ann.
235 (1978), 111-128.
[S5] T. Sunada, Spectrum of a compact ﬂat manifold, Comment. Math. Helv. 53 (1978), 613-621.
[S6] T. Sunada, Rigidity of certain harmonic mappings, Invent. Math. 51 (1979), 297-307.
[S7] T. Sunada, Trace formula for Hill’s operators, Duke Math. J. 47 (1980), 529-546.
[S8] T. Sunada, Geodesic chains and the spherical mean operator, Proc. ”Global Diﬀerential
Geometry/Global Analysis”, Lect. Note in Math. 837 (1981), 229-232.
[S9] T. Sunada, Spherical means and geodesic chains in a Riemannian manifold, Trans. A.M.S.
267 (1981), 483-501.
[S10] T. Sunada, Trace formula and heat equation asymptotics for a non-positively curved manifold, Amer. J. Math. 104 (1982), 795-812.
[S11] J. Noguchi and T. Sunada, Finiteness of the family of rational and meromorphic mappings
into algebraic varieties, Amer. J. Math. 104 (1982), 887-900.
[S12] T. Sunada, Trace formula, Wiener integrals and asymptotics, Proc. Japan-France Seminer”Spectra of Riemannian Manifolds”, Kaigai Publ. Tokyo 1983, 159-169.
[S13] T. Sunada, Mean-value theorems and ergodicity of certain geodesic random walks, Compositio Math. 48 (1983), 129-137.
[S14] T. Sunada, Geodesic ﬂows and geodesic random walks, Advanced Studies in Pure Math.
”Geometry of Geodesics and Related Topics”, Vol.3 (1984), 47-85.
[S15] T. Adachi and T. Sunada, Energy spectrum of certain harmonic mappings, Compositio
Math. 56 (1985), 153-170.
[S16] T. Adachi and T. Sunada, Geometric bounds for the number of certain harmonic mappings,
Proc. ”Diﬀerential Geometry of Submanifolds”, 1984, Springer Lect. Note in Math. 1090
(1985), 24-36.
[S17] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985),
169-186.
[S18] T. Sunada, Gel’fand’s problem on unitary representations associated to discrete subgroupsof P SL(2, R), Bull. A.M.S. 12 (1985), 237-238.
[S19] T. Adachi and T. Sunada, Homology of closed geodesics in a negatively curved manifold, J.
Diﬀ. Geom. 26 (1987), 81-99.
[S20] A. Katsuda and T. Sunada, Homology of closed geodesics in certain Riemannian manifolds,
Proc. A.M.S. 96 (1986), 657-660.
[S21] T. Adachi and T. Sunada, Twisted Perron-Frobenius theorem and L-functions, J. Funct.
Anal. 71 (1987), 1-46.
[S22] T. Sunada, L-functions in geometry and some applications, Proc. Taniguchi Symp. 1985,”Curvature and Topology of Riemannian Manifolds”, Springer Lect. Note in Math. 1201
(1986), 266-284.
[S23] T. Sunada, Number theoretic analogues in spectral geometry, Proc. The 6th Symp. ”Diﬀerential Geometry and Diﬀerential Equations” , held in Shanghai, 1985, Springer Lect. Note in
Math. 1255 (1987), 96-108.
[S24] A. Katsuda and T. Sunada, Homology and closed geodesics in a compact Riemann surface,
Amer. J. Math. 110 (1987), 145-156.
[S25] T. Sunada, Unitary representations of fundamenatl groups and the spectrum of twisted
Laplacians, Topology 28 (1989), 125-132.

AN OVERVIEW OF SUNADA’S WORK

35

[S26] T. Sunada, Fundamental groups and Laplacians, Proc. Taniguchi Symp. ”Geometry andAnalysis on Manifolds”, 1987, Springer Lect. Note in Math. 1339 (1988), 248-277.
[S27] A. Katsuda and T. Sunada, Closed orbits in homology clasees, Publ. Math. IHES. 71 (1990),
5-32.
[S28] T. Kobayashi, K. Ono and T. Sunada, Periodic Schrödinger operator on a manifold, Forum
Math. 1 (1989), 69-79.
[S29] T. Sunada, Dynamical L-functions and homology of closed orbits, Bull. A.M.S. 20 (1989),
73-77.
[S30] T. Sunada, A periodic Schrödinger operator on an abel cover, J. Fac. Sci. Univ. Tokyo 37(1990), 575-583.
[S31] T. Sunada and P. W. Sy, Discrete Schrödinger operators on a graph, Nagoya Math. J. 125(1992), 141-150.
[S32] T. Sunada, Group C ∗ -algebras and the spectrum of a periodic Schrödinger operator on amanifold, Canad. J. Math. 44 (1992), 180-193.
[S33] T. Sunada and M. Nishio, Trace formulae in spectral geometry, Proc. ICM-90 Kyoto,
Springer-Verlag, Tokyo, (1991), 577-585.
[S34] T. Sunada, Homology, Wiener integrals and density of states, J. Funct. Anal. 106 (1992),
50-58.
[S35] J. Brüning and T. Sunada, On the spectrum of periodic elliptic operators, Nagoya Math. J.
126 (1992), 159-171.
[S36] T. Adachi, T. Sunada and P. W. Sy, On the regular representation of a group applied tothe spectrum of a tower, Proc. ”Analyse algebrique des perturbations singulieres”, (1993),
125-133.
[S37] J. Brüning and T. Sunada, On the spectrum of gauge-periodic elliptic operaors, Asterisque210 (1992).
[S38] T. Adachi and T. Sunada, Density of states in spectral geometry, Comment. Math. Helv.
68 (1993), 480-493.
[S39] T. Sunada, Euclidean versus non-Euclidean aspects in spectral geometry, Prog. Theor. Phys.
Supplement 116 (1994), 235-250.
[S40] T. Sunada, A discrete analogue of periodic magnetic Schrödinger operators, Contemporary
Math. 173 (1994), 283-299.
[S41] T. Sunada, On the number theoretic method in geometry, Amer. Math. Soc. Transl. 160(1994), 1-17.
[S42] T. Sunada, Fundamental groups and Laplacians, Amer. Math. Soc. Transl. 160 (1994),
19-32.
[S43] T. Sunada, Quantum ergodicity, Trend in Mathematics, Birkhauser Verlag, Basel, 1997, 175- 196.
[S44] M. Kotani, T. Shirai and T. Sunada, Asymptotic behavior of the transition probability of arandom walk on an inﬁnite graphs, J. Funct. Analy. 159 (1998), 664-689.
[S45] M. Kotani and T. Sunada, Jacobian tori associated with a ﬁnite graph and its abelian
covering graphs, Adv. in Appl. Math. 24 (2000), 89-110.
[S46] M. Kotani and T. Sunada, Standard realizations of crystal lattices via harmonic maps,
Trans. A.M.S. 353 (2000), 1-20.
[S47] M. Kotani and T. Sunada, Albanese maps and an oﬀ diagonal long time asymptotic for theheat kernel, Comm. Math. Phys. 209 (2000), 633-670.
[S48] M. Kotani and T. Sunada, Zeta functions of ﬁnite graphs, J. Math. Sci. Univ. Tokyo 7(2000),7-25.
[S49] M. Kotani and T. Sunada, The pressure and higher correlation for an Anosov diﬀeomorphisms, Ergod. Th. & Dynam. Sys. 21 (2001), 807-821.
[S50] M. Kotani and T. Sunada, Spectral geometry of crystal lattices, Contemporary. Math. 338(2003), 271-305.
[S51] M. Kotani and T. Sunada, Geometric aspects of large deviations for random walks ona crystal lettice, Proc. of the “Microlocal Analysis and Complex Fourier Analysis”, World
Scientiﬁc, (2003), pp215-223.
[S52] T. Sunada, Book Review in Bull. Amer. Math. Soc. “Random walks on inﬁnite graphs andgroups” by W. Woess.
[S53] K. Shiga and T. Sunada, A Mathematical Gift, III, originally published in Japanese in 1996
by Iwanami Shoten, and translated by E. Tyler, Amer. Math. Soc., 2005.

36

ATSUSHI KATSUDA AND POLLY WEE SY[S54] M. Kotani and T. Sunada, Large deviation and the tangent cone at inﬁnity of a crystal
lattice, Math. Z. 254 (2006), 837-870.
[S55] M. Shubin and T. Sunada, Mathematical theory of lattice vibrations, Pure and Appl. Math.
Quaterly 2 (2006), 745-777.
[S56] T. Sunada, Crystals that nature might miss creating, Notices of the AMS, 55 (2008), 208215.
[S57] T. Sunada, Discrete geometric analysis, Proceedings of Symposia in Pure Mathematics 77(2008), 51-86.
[S58] M. Itoh, M. Kotani, H. Naito, Y. Kawazoe, T. Sunada, and T. Adschiri, A diamond twin–Its possible physical properties–, preprint[S59] T. Sunada and H. Urakawa, Ray-Singer zeta functions for compact ﬂat manifolds, preprint.
[S60] M. Kotani and T. Sunada, Graph-theoretic Albanese maps revisited, preprint.
[S61] T. Sunada, Magnetic ﬂows on a Riemann surface, Proc. KAIST Math. Workshop, vol. 8(1993), 93-108.
[S62] T. Sunada, Tchebotarev’s density theorem for closed geodesics in a compact locally symmetric space of negative curvature, unpublished (1980).
[S63] M. Itoh, M. Kotani, H. Naito, T. Sunada, Y. Kawazoe, and T. Adschiri, New metallic carbon
crystal, preprint 2008.
[SB1] T. Sunada, Fundamental Groups and Laplacians, in Japanese, Kinokuniya Shoten, 1988.
[SB2] T. Sunada, Matrices and Determinants, in Japanese, Iwanami Shoten, 1996.
[SB3] T. Sunada, Introduction to Geometry, in Japanese, Iwanami Shoten, 1996.
[SB4] T. Sunada, Geometry of Surfaces, in Japanese, Iwanami Shoten, 1997.
[SB5] T. Sunada, Banach-Tarsky Paradox, in Japanese, Iwanami Shoten, 1997.
[SB6] T. Sunada, Geometry of Partitions –Two Theorems due to M. Dehn–, in Japanese, Nihon
Hyoronsha, 2000.
[SB7] T. Sunada, Physics from the Mathematical Viewpoint, 5 volumes including Newtonianmechanics, classical mechanics, mechanics of continuums, theory of relativity, statistical mechanics, thermodynamics, quantum mechanics, in Japanese, Iwanami Shoten, 2003, 04, 05.
[SB8] T. Sunada, Introduction to Modern Geometry, in Japanese, Air University Press, 2004.
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Department of Mathematics, Okayama University, Okayama, Japan
Institute of Mathematics, University of the Philippines, Diliman, Philippines