Hearing shapes of drums - mathematical and physical aspects of isospectrality

1 MB·English

by  O. Giraud

Hearing shapes of drums — mathematical and physical aspects of
isospectrality
Olivier Giraud
∗
Univ. Paris-Sud, CNRS,LPTMS, UMR 8626, Orsay, F-91405, France
Laboratoire dePhysique Th´eorique (IRSAMC), CNRS and UPS,
Universit´e Paul Sabatier,F-31062 Toulouse, France
Koen Thas
†
Ghent University, Departmentof Pure Mathematics andComputer Algebra,
Krijgslaan 281, S25, B-9000 Ghent,Belgium
(Dated: August 6, 2010)
1 In a celebrated paper “Can one hear the shape of a drum?” M. Kac [Amer. Math.
1 Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric
0 billiards having the same spectrum of the Laplacian. This question was eventually
2 answeredpositivelyin1992bytheconstructionofnoncongruentplanarisospectralpairs.
Thisreviewhighlightsmathematical andphysicalaspects ofisospectrality.

n
a
J
6 CONTENTS J. Analyticdomains 26

] I. Introduction 2 VI. ExperimentalandNumericalInvestigations 27
h A. Numericalinvestigations 27
pII. APedestrianProofofIsospectrality 4 1. Mode-matchingmethod 27
- A. Paper-foldingproof 4
2. Expansionofeigenfunctions aroundthecorners
h B. Transplantationproof 6
withthedomain-decompositionmethod 27
t
a B. Experimentalrealizations 28
III. FurtherExamplesinHigherDimensions 8
m
A. Lattices andflattori 8 1. Electromagneticwaves inmetalliccavities 28
[ B. Constructionofexamples 9 2. Transversevibrationsinvacuumforliquidcrystal
C. Thefour-parameterfamilyofConwayandSloane 9 smecticfilms 28
1 D. Theeigenvaluespectrumasmoduliforflattori 10 3. Isospectralelectronicnanostructures 29
v
9IV. Transplantation 10 VII. SunadaTheory 29
3 A. Tiling 10 A. Permutations 29
2 1. Graphsandbilliardsbytiling 10
B. Commutator notions 29
1 2. TheexampleofGordonetal. 11
C. Finitesimplegroups 30
. 3. Theother knownexamples 11
1 4. EuclideanTI-domainsandtheirinvolution D. p-Groupsandextra-special groups 30
0 graphs 11 E. SunadaTheory 31
1 B. Someprojectivegeometry 12 F. ExamplesofSunadatriples 32
1 1. Finiteprojectivegeometry 12
: 2. Automorphismgroups 13 VIII. RelatedQuestions 33
v
3. Involutions infiniteprojectivespace 13 A. Boundaryconditions 33
i
X C. Projectiveisospectraldata 13 B. Homophonicpairs 34
1. Transplantationmatrices,projectivespacesand
r C. Spectral problemsforLiegeometries 34
isospectraldata 13
a D. Furtherquestions 35
2. Generalizedisospectraldata 15
3. Theoperator group 15
Acknowledgments 35
V. SemiclassicalInvestigation ofIsospectral Billiards 16
A. Meandensityofeigenvalues 16 A. Galleryofexamples 35
B. Periodicorbits 17 1. Somemodes 35
1. Greenfunction 17 2. The17familiesofisospectralpairsandtheir
2. SemiclassicalGreenfunction 18 mathematical construction 35
3. Semiclassicaldensityofeigenvalues 18
C. Diffractiveorbits 19 B. SpectralproblemsforLiegeometries 37
D. Greenfunction 20
1. Generalizedpolygons 37
E. ScatteringpolesoftheexteriorNeumannproblem 21
2. Dualityprinciple 37
F. Eigenfunctions 22
3. Automorphismsandisomorphisms 38
1. Triangularstates 22
2. Mode-matchingmethod 22 4. Pointspectraandorder 38
G. Eigenvaluestatistics 24 5. Concludingremarks 39
H. Nodaldomains 24
I. Isospectralityversusisolengthspectrality 24 C. Livsiccohomology 40
1. OkadaandShudo’sresultonisolengthspectrality24
2. Penrose–Lifshitsmushrooms 25 References 40
∗ [email protected]
† [email protected]
2
I. INTRODUCTION the particle be located at a position x. If the system
is described by the Hamiltonian H, the wave function
Elastic plates are probably some of the oldest supports satisfies the stationary Schr¨odinger equation HΨ =EΨ,
of sound production. They were used by most human where E is the energy of the particle. For a particle
cultures. Clay drums dated from the Chalcolithic have of mass m and momentum p evolving in a box defined
beenfoundingravesincentralEurope,andbronzedrums by its contour ∂B, the Hamiltonian describing the free
dated from the second millenary B.C. have been dis- motion inside the box reads H = p2/2m inside the box
covered in Sweden and Hungary. However, it is usu- enclosure ∂B and outside, and the time-independent
∞
ally acknowledged that the scientific study of the vi- Schr¨odinger equation takes the form (1).
bration of elastic plates goes back only to the end of
the 18th century, when the German researcher Ernst
Mathematically, solutions of the Helmholtz equation are
Chladni carried out the first systematic investigations
readily obtained in dimension d = 1. The problem of
on the production of sound by plates (Chladni, 1802;
vibrating strings had been solved in the 18th century by
Smilansky and St¨ockmann, 2007). When the plate was
Jean Le Rond d’Alembert. For a string of length L fixed
fixed in its middle and struck with a bow, it was set
at its two ends, solutions are simply given by f(x) =
into vibration. The mode that was being excited was
sin(nπx/L), where n is an integer. The sound produced
physically visualized by pouring sand on the plate: the
by the string has the possible frequencies nν , with the
0
sandaccumulatesatnodallines,thatislinesalongwhich
fundamental frequency given by ν =c/(2L).
0
the plate does not oscillate. Some insight was brought
Justas the one-dimensionalcase—whichcandescribe a
into the mathematical theory of vibrating plates by the
varietyofphysicalsituations—canbeseenasaproblem
French mathematician Sophie Germain, who published
of vibrating strings, the two-dimensional case is usually
Recherches sur la th´eorie des surfaces ´elastiques in 1821.
studiedfromtheperspectiveofitssimplestmathematical
In the course of the 19th century, Poisson, Kirchhoff,
equivalent, namely billiards. Billiards (in the mathemat-
Lam´e, Mathieu, and Clebsch, devised analytic expres-
ical sense) are two-dimensional compact domains of the
sions for the description of the oscillation for elementary
Euclidean plane R2. For instance, in quantum mechan-
shapes such as the rectangle,the triangle, the circle, and
ics, the billiard models the behavior of a particle moving
the ellipse.
freely in a box whose dimensions are such that it can
The motivation for studying this problem was mainly
be approximated by a two-dimensional enclosure. The
that the wave phenomenon at the heart of membrane
billiard problem is solved by looking for eigenfunctions
oscillations is in fact quite general. The stationary wave
ψ and eigenvalues E that are solutions of Eq. (1) inside
equationdescribingthe problemarisesinavarietyofsit-
thebilliard,imposingboundaryconditionsonthebound-
uations. In many fields of physics, such as acoustics,
ary∂B ofthe billiard. Physicalproblems impose specific
seismology, hydrodynamics, and heat propagation, the
boundary conditions. For instance hard wall domains in
mathematicalformulationoftheprobleminvolvespartial
quantum mechanics impose that the wave function van-
differentialequations,andgeneralsolutionsoftheseequa-
ishes on the boundary. In acoustics, clamping an elastic
tions can be found as superpositions of solutions of the
membraneimposes thatthe oscillationsandtheir deriva-
so-calledHelmholtzequation. Inad-dimensionalspacea
tive along the boundary vanish. The billiard problem
stationary solution to the wave equation is an unknown
usually considers the two following boundary conditions:
function of d variables describing the problem, and the
Dirichlet boundary conditions ψ = 0, for which the
Helmholtz equation reads ∂B
|
function vanishes on the boundary, or Neumann bound-
∆f +Ef =0, (1) ary conditions ∂nψ∂B =0, for which the normal deriva-
|
tive vanishes on the boundary. If such boundary condi-
tions areimposedthere is aninfinite but countablenum-
where ∆ is the d-dimensional Laplacian. Under suitable
ber of solutions to Eq. (1). We denote eigenfunctions of
approximations, numerous problems can be cast in that
the operator ∆ by ψ and eigenvalues by E , n N,
form. For instance, ina certainregimethe oscillationsof n n
− ∈
with 0 < E E E . Of course any combination
the height f = f(x,y) of a thin vibrating plate at point 1 2 3
≤ ≤ ···
of the above boundary conditions yields a different spec-
(x,y) can be described by (1).
tral problem. In this review however, we will be mainly
At the end of the 19th century, James C. Maxwell
concerned with Dirichlet boundary conditions.
showed that the electric and the magnetic field behave
like waves and established equations governing the time In the second half of the 20th century, quantum billiards
evolution of the electromagnetic field. From Maxwell’s were studied in the framework of quantum chaos.
equations it is easy to prove that the electric and the Quantum properties of classical systems were inves-
magnetic field components also obey the same wave tigated, and different behaviors were found according
equation(1). Further interest developed in this equation to the properties of integrability or chaoticity of the
when the wave-like behavior of matter was discovered in underlying classical dynamics. This quantum-classical
the early years of quantum mechanics. The Schr¨odinger correspondence led to various conjectures for integrable
equation was established in 1926 by Erwin Schr¨odinger systems (Berry and Tabor, 1977) and chaotic systems
to describe the spacetime evolution of a quantum sys- (Bohigas et al., 1984). These conjectures rest on pow-
tem. The behavior of a particle can be described, in the erful mathematical tools that allow insight into the
frameworkof quantummechanics, by a wavefunction ψ, properties of solutions of the Helmholtz equation (1).
which is a function of the position of the particle, and For instance, the Weyl formula (see section V.A), or
which characterizes the probability amplitude ψ(x) that semiclassical trace formulas (see section V.B.2), provide
3
a connection between the density of energy levels and (These examples are also described by (Protter, 1987).)
classical features of the domains such as area, perimeter More specifically, it is provedthat there exist domainsC
or properties of classical trajectories in the domain. and C in the unit sphere Sn 1 in Rn, n 4, which are
′ −
≥
The existence of such formulas and the conjectures on Dirichlet and Neumann isospectral but not congruent in
the quantum-classical correspondence indicate that the Sn 1. This existence follows from the observation that
−
spectrum of a billiard contains a certain amount of there are finite reflection groups W and W that act on
′
information about the shape of the billiard. Therefore the same Euclidean space Rn, n 4, for which the sets
≥
it is natural to ask how much information about the of exponents coincide, and the intersections (C and C )
′
billiardcanberetrievedfromknowledgeoftheeigenvalue of their chambers with Sn 1 are not congruent in Sn 1.
− −
spectrum. For rectangular or triangular billiards, it Then work of B´erardand Besson (1980) is applied.
is known that a finite number of eigenvalues suffices In the late 1980s, various other papers appeared, giving
to entirely specify the shape of the billiard (see e.g. necessary conditions that any family of billiards sharing
(Chang and Deturck, 1989)), but is this true for more the same spectrum should satisfy ((Melrose, 1983),
complicated shapes? (Osgood et al.,1988a), (Osgood et al., 1988b)), andnec-
essary conditions givenas inequalities on the eigenvalues
were reviewed in (Protter, 1987).
In1966,inacelebratedpaper(Kac,1966),MarkKacfor-
mulatedthe famousquestion“Canone hearthe shape of
But it was almost 30 years after Kac’s paper that the
adrum?”. Thisprovocativequestionisofcoursetobeun-
first example of two-dimensional billiards having exactly
derstoodmathematically as follows: Is it possible to find
the same spectrum was finally exhibited in 1992. The
two (or more) non-isometric Euclidean simply connected
pair was found by C. Gordon, D. Webb and S. Wolpert
domains for which the sets E n N of solutions of
n
{ k ∈ } in their paper “Isospectral plane domains and surfaces
(1) with Ψ = 0 are identical? More broadly,
Boundary
via Riemannian orbifolds” (Gordon et al., 1992a). They
the questio|n raises the issue of the inverse problem of
gave a no as a final answer to Kac’s question, and as a
retrieving information about a drum from knowledge of
reply to Kac’s paper,they published a paper titled “One
its spectralproperties. As the spectroscopistA.Schuster
cannotheartheshapeofadrum”(Gordon et al.,1992b).
put it in an 1882 report to the British Association for
ThemostpopularizedexampleisshowninFig.1. Crucial
the Advancement of Science: ”To find out the different
tunes sent out by a vibrating system is a problem which
may or may not be solvable in certain special cases, but
it would baffle the most skillful mathematicians to solve
theinverseproblemandtofindouttheshapeofabellby
means of the sounds which it is capable of sending out.
And this is the problem which ultimately spectroscopy
hopes to solve in the case of light. In the meantime
we must welcome with delight even the smallest step in
the desired direction.” (Mehra and Rechenberg, 2000).
Actually, it was known very early, from Weyl’s formula,
that one can “hear” the area of a drum and the length
of its perimeter (see section V.A, and (Vaa et al., 2005)
for a historical account of the problem). But could the
shape itself be retrieved from the spectrum? That is,
whatkindofinformationonthegeometryisitpossibleto
gatherfromthe knowledgeofthe spectrum, for instance,
FIG. 1 Paradigmatic pair of isospectral billiards with seven
using semiclassical methods that allow investigation of
half-squareshapedbasetiles. Thedottedlinesarejustforthe
the quantum-classical correspondence? And what kind
eye.
of sufficient conditions allow the geometry to be entirely
specified from the spectrum?
for finding the example was a theorem by Sunada (see
Formally, an answer “no” to Kac’s question amounts to section VII.3) asserting that when two subgroups are
findingisospectral billiards,thatisnon-isometricbilliards “almost conjugate” in a group that acts by isometries
having exactly the same eigenvalue spectrum. Since the on a Riemannian manifold, the quotient manifolds are
appearance of Kac’s paper (Kac, 1966), far more than isospectral. In fact, the other examples which were con-
500 papers have been written on the subject, and innu- structed after 1992all usedSunada’s method. Later,the
merable variations on “hearing the shape of something” so-called transplantation technique was used, giving an
can be found in the literature. Early examples of flat easier way for detecting isospectrality of planar billiards.
tori sharing the same eigenvalue spectrum were found Still, essentially only 17 families of examples that say
in 1964 by Milnor in R16 from nonisometric lattices notoKac’squestionwereconstructedina40yearperiod.
of rank 16 in R16 (see section III). Other examples
of isospectral Riemannian manifolds were constructed Since the literature on isospectrality is large, and covers
later, for example on lens spaces (Ikeda, 1980) or on a broad spectrum of mathematical topics, we have cho-
surfaces with constant negative curvature (Vign´eras, sen here to put the focus on isospectral billiards, that
1980). In 1982, H. Urakawa produced the first examples is, two-dimensional isospectral domains of the Euclidean
of isospectral domains in Rn, n 4 (Urakawa, 1982). plane, with Dirichlet boundary conditions. It is worth
≥
4
noting that simple examples of isospectral domains can spectral theory associated with quantum graphs. Many
be constructed in the case of mixed Dirichlet-Neumann results exist, and we just mention a few striking ones.
boundary conditions. Such constructions were proposed One of the main results in that spectral theory can be
by(Levitin et al.,2006)(seesectionVIII.A). Wenowre- found in (Gutkin and Smilansky, 2001), where the trace
view some results on related topics, to which we will not formula is used to show that (under certain conditions)
return in this paper. a quantum graph can be recoveredfrom the spectrum of
First we mention severalfundamental results on isospec- its Laplacian. (Necessary conditions include the graph
trality that will be omitted. Zelditch (1998) proved that beingsimpleandtheedgeshavingrationallyindependent
isospectral simple analytic surfaces of revolution are iso- lengths.) Using a spectral trace formula, Roth (1984)
metric. That is, he considered the moduli space of in an early paper constructed isospectral quantum
metrics of revolution (S2,g) with the following proRper- graphs. von Below (2001), on the other hand, used
ties. Suppose that there is an effective action of S1 by the connection between spectra of discrete graphs and
isometries of (S2,g). The two fixed points are N and S. spectra of (equilateral) quantum graphs to transform
Denote by (r,θ) geodesic polar coordinates centered at isospectral discrete graphs into isospectral quantum
N, with θ = 0 being some fixed meridian γ from N to graphs. Finally, we note that Parzanchevskiand Band
M
S. Themetricgcanthenbewrittenasg =dr2+a(r)dθ2, (2010) presented a method for constructing isospectral
where a:[0,L] R+ is defined by a(r)= S (N)/(2π), quantum graphs, based on linear representations of
r
with S (N) th7→e length of the distance ci|rcle of|radius finite groups. Note that a different notion of graph
r
r cent|ered a|t N. The properties now are as follows: (i) isospectrality was considered by Thas (2007b) based on
g is real analytic, (ii) a has precisely one critical point the spectrum of the adjacency matrix of the graph. We
r ]0,L[, with a (r ) < 0, corresponding to an equa- return to this point in section IV.A.4.
0 ′′ 0
∈
torial geodesic γ , and (iii) the nonlinear Poincar´e map
E
for γ is of twist type. To end this section, we give a short description of the
PγE E contents of the paper.
Denote by the subset of metrics with simple
∗
R ⊂ R To familiarize the reader with the notions involved, we
length spectra in the sense of (Zelditch, 1998). Then
Zelditch proved that Spec : RN is 1-1. Further- startbypresentingasimpleproofofisospectralityforthe
R∗ 7→ + seminal example of Gordon et al. (1992a) in section II.
more, in (Zelditch, 1999) — see also (Zelditch, 2000),
Then the first historical examples of higher-dimensional
Zelditch showed that real plane domains Ω that (1) are
isospectral pairs of flat tori are constructed (section III).
simply connected and real analytic, (2) are Z Z -
2 2
× (Much more work has been done on isospectrality for
symmetric(i.e.,havethesymmetryofanellipse),and(3)
the Laplace-Beltrami operator on flat tori in higher di-
have at least one axis that is a nondegenerate bouncing
mensions than just the material we cover in section II.
ball orbit, the length of which has multiplicity 1 in the
We refer to that section for more commentaries on that
length spectrum Lsp(Ω), are indeed determined by their
matter.) Section IV is devoted to the mathematical as-
spectrum. In recent work Zelditch (2004a) pursued his
pects lying behind the construction of the known exam-
goal of eventually solving the inverse spectral problem
ples of isospectral pairs. Then we review various aspects
for general real analytic plane domains. We will return
of the properties of isospectral pairs (section V), as well
to this issue in more detail in section V.J.
as experimental implementations and numerical checks
of isospectrality (section VI). As the first examples of
Concerning the known counterexamples in the plane, it
isospectral billiards were produced by applying Sunada
should be remarked that the constructed domains are
theory, a review of this theory is given in section VII.
not convex (see e.g. Appendix A). The objective of
In the last sectionwe examine questions relatedto Kac’s
Gordon and Webb (1994) is to exhibit pairs of convex
problem.
domainsinthehyperbolicplaneH2 thatarebothDirich-
let and Neumann isospectral. They are obtained from
nonconvex examples in the real plane by modifying the
II. A PEDESTRIAN PROOF OF ISOSPECTRALITY
shape of a fundamental tile. Other interesting variations
on the problem include the construction of a pair of
The first examples of isospectral billiards in the Eu-
isospectral (nonisometric) compact three-manifolds,
clideanplanewereconstructedusingpowerfulmathemat-
called “Tetra” and “Didi”, which have different closed
ical tools. We postpone these historical constructions to
geodesics (Doyle and Rossetti, 2004).
sectionVII.E. Thepresentsectionaimsatillustratingthe
main ideas involved in isospectrality, so that the reader
The related question of graph isospectrality has also canacquiresomeintuitionaboutit. Morerigorousmath-
attracted much interest. We mention here a few results. ematical grounds will be provided in the next sections.
A quantum graph is a metric graph equipped with a
differential operator (typically the negative Laplacian)
and homogeneous differential boundary conditions at A. Paper-folding proof
the vertices. (Recall that a metric graph is a graph
such that to each edge e is assigned a finite (strictly We start with a simple construction method that was
positive) length ℓ R, so that it can be identified with proposed by Chapman (1995). It is based on the so-
e
∈
the closed interval [0,ℓ ] R. Without the boundary called ”paper-folding” method. To illustrate it we follow
e
⊂
conditions, the graph “consists of” edges with functions (Thain,2004),wherethemethodisillustratedonasimple
defined separately on each edge.) So there is a natural example.
5
ConsiderthetwobilliardsinFig.2. Eachbilliardismade φ = φ since tiles 1 and 2 are glued together.
tile1 tile2
| |
Thus, ψ givenby Eq.(2) indeedvanishes onthe leftmost
1 2 3 4 1 2 3 4 vertical boundary of the right billiard. After we have
5 6 7 5 6 7 checked by inspection all (inner and outer) boundaries,
we have proved that the two billiards of Fig. 2 are
isospectral.
FIG. 2 The pair 73 (see Appendix A) of isospectral billiards
with a rectangular base shape.
Withthepaper-foldingmethod,itisclearthatwhatmat-
tersisthewaythebuildingblocks(theelementaryrectan-
of seven identical rectangular building blocks. The solid
gles in our example) are gluedto eachother, irrespective
linesarehardwallboundaries,the dottedlines arejusta
oftheirshape. Wenowshowhowthepaper-foldingproof
guidetotheeyemarkingthebuildingblocks. Letφbean
generalizes to other shapes. Suppose we denote by 1, 2,
eigenfunction of the left billiard with eigenvalue E. The
and 3 respectivelythe left, right,andbottom edge oftile
goal is to construct an eigenfunction of the right billiard
4intheleftbilliardofFig.2. Toobtainthewholebilliard
with the same eigenvalue, that is a function which:
one unfolds tile 4 with respect to its side number 3, get-
tingtile7. Thentile7isunfoldedwithrespecttoitsside
verifies the Helmholtz equation (1);
• number 2, yielding tile 6, and so on. The unfolding rules
vanishes on the boundary of the billiard; can be summed up in a graph specifying the way we un-
•
fold the building block. The graphs in Fig.4 correspond
has a continuous normal derivative inside the bil- to the unfoldingsyieldingthe billiardsofFig.2whenap-
•
liard. plied to a rectangularbuilding block. The vertices ofthe
graphrepresentthe building blocks,and the edges ofthe
The idea is to define a function ψ on the rightbilliard as
graphare “colored”accordingto the unfolding rule,that
a superposition of translations of the function φ. Since
is, depending on which of its sides the building block is
the Helmholtz equation (1) satisfied by φ is linear, any
unfolded. The graphs can alternatively be encoded by
linear combination of translations of φ will be a solution
permutations a(µ),b(µ), 1 µ 3. For instance for the
ofthe Helmholtz equationwiththe same eigenvalueE in ≤ ≤
first graph we have a(1) =(23)(56), a(2) = (12)(67), and
the interior of each building block of the second billiard.
a(3) =(25)(47). In fact, only three sides of the rectangle
Theproblemreducestofindingalinearcombinationthat
are involved in the unfolding. So we can start with any
vanishes on the boundary and has the correct continuity
triangular-shaped building block, and unfold it with re-
properties inside the billiard. The paper-folding method
specttoitssidesjustasthebilliardsinFig.2areobtained
allows to satisfy all these conditions simultaneously.
fromtherectangularbuildingblock. Thisleadstobilliard
pairswhoseisospectralityisgrantedbythepaper-folding
proof given above. For example, starting from the trian-
Take three copies of the left billiard of Fig. 2. Fold each gle in Fig. 4 and following the same unfolding rules, we
copy in a different way, as shown in Fig. 3 (left column). get the pair of isospectral billiards shown in Fig. 4 right.
Thenthethree-timesfoldedbilliardsarestackedontopof Taking a building block in the form of a half-square, we
eachotherasindicatedintherightcolumnofFig.3;note recover the example of Fig. 1 when the same unfolding
that the first shape (folding 1) has been translated on rules are applied.
the left before being stacked, and that the second shape
Thebuildingblockisinfactnotevenrequiredtobeatri-
(folding 2) has been rotated by π in the plane of the
angleorarectangle. Anybuildingblockpossessingthree
figure. Once superposed, these three billiard yield the
edges around which to unfold leads to a different pair of
shape on the bottom right, which is the right billiard of
isospectral billiards. Another interesting example is ob-
Fig. 2.
tainedbytakingaheptagonandunfoldingitwithrespect
Now we make a correspondence between stacking two
tothreeofitssides,followingtheunfoldingrulesofFig.4.
sheetsofpaperandaddingthefunctionsdefinedonthese
This yields the first example produced by Gordon et al.
sheets; moreover, stacking the reverse of a sheet corre-
(1992a,b) (see Fig. 5).
sponds to assigning a minus sign to the function. For
instance, in folding 3, a minus sign is associated in the Chapman (1995) produced more involved examples, fol-
rightcolumnwithtiles3and4,sincetheyarefoldedback, lowing the same procedure. Starting from the build-
and a plus sign is assigned to the other tiles since they ing block of Fig. 6 left, one obtains an example of a
arenotfolded. Thefunctionψ isdefinedbythis “folding pair of chaotic billiards with holes. Similarly Dhar et al.
andstacking”procedure. Forinstanceitisdefinedinthe (2003) constructed chaotic isospectral billiards based on
tile numbered 1 in the right billiard of Fig. 2 by the same idea: scattering circular disks were added in-
side the base triangular shape in a way consistent with
ψ = φ +φ φ . (2)
|tile1 − |tile1 |tile2− |tile5 the unfolding.
The procedure above ensures that ψ vanishes on the The central building block of Fig. 6 yields a simple dis-
boundary and has a continuous derivative across the tile connected pair where each billiard consists of a disjoint
boundaries. Indeed, consider for instance the leftmost rectangle and triangle. In this case, isospectrality can
vertical boundary of the right billiard (i.e. the left be checked directly by calculating the eigenvalues, since
edge of tile 1). On this boundary we have φ = 0 theeigenvalueproblemcanbesolvedexactlyfortriangles
tile5
|
(since it is at the boundary of the left billiard), and and half-squares.
6
folding 1 −1+2−5 3−6 −7
−4
5−6+7
3 2+4 1
folding 2
1 2−3
5 6 −4+7
folding 3 −1+2−5 1+3−6 2−3−7 5−6+7
3−4+5 2+4+6 1−4+7
1 2 3 4 1 2 3 4
5 6 7 5 6 7
FIG. 3 Pictorial representation of thepaper-folding method.
Sleeman and Hua (2000) considered a building block untouchedsidesstillallowtheChapmanunfolding(Fig.6
with piecewise fractal boundary: starting from a right). Thisyieldsapairofisospectralbilliardswithfrac-
(π/2,π/3,π/6)basetriangletheycuteachsideintothree tal boundary of dimension ln4/ln3.
piecesandremovethethreetriangularcorners. Alongthe
freshly made cuts a Koch curve is constructed, while the
2
1
3 4 1
7 2
+ = 6 3 FIG.6 Examplesofbuildingblocksyieldingisospectralpairs.
3 5 4
5
4 7 6 5 2 1 2 6
3 1 A separate problem that will not be presented here
7
is to find inhomogeneous vibrating membranes isospec-
1 2 3 5 6 4
7 tral to a homogeneous membrane with the same shape
(see, e.g., (Gottlieb, 2004) for circular membranes).
FIG.4 Graphscorrespondingtoapairofisospectralbilliards: Knowles and McCarthy (2004) used the isospectrality of
Ifwelabelthesidesofthetrianglebyµ=1,2,3,theunfolding the billiards of Fig. 1 to construct a pair of isospectral
rule by symmetry with respect to side µ can be represented circular membranes by a conformal mapping.
byedgesmadeofµbraidsinthegraph. From agivenpairof
graphs, one can construct infinitely many pairs of isospectral
billiards byapplying theunfolding rules to any shape.
B. Transplantation proof
1 The paper-folding proof can be made more formal
be means of the so-called “transplantation” method.
This method was introduced in B´erard (1989); B´erard
2
3 (1992, 1993), and discussed by Buser et al. (1994) and
Okada and Shudo (2001). It will be presented in more
detailin sectionIV. Herewe sketchthe mainideasusing
a simple example.
ConsidertheisospectralpairofFig.2. Letφbeaneigen-
state of the first billiard. Any point in the billiard can
be specified by its coordinates a = (x,y) inside a build-
ing block,and a number i arbitrarilyassociatedwith the
building block (for example 1 i 7 in our example of
≤ ≤
Fig. 2). Thus φ is a function of the variable (a,i). Ac-
FIG. 5 Isospectral billiards. The top left figure is the seven- cording to the paper-folding proof, a building block i of
edged building block. From (Gordon et al., 1992a).
the secondbilliardis constructedfromasuperpositionof
7
three building blocks j obtained by folding the first bil- 2 3 4 7 2
liard. We can code the result of the folding-and-stacking
procedure in a matrix T, as
1 1 0 0 1 0 0 1 6 5
−
1 0 1 0 0 1 0
 − 
0 1 1 0 0 0 1
− −
T = 0 0 0 0 1 1 1 . (3) 2 3 4 7 2
 − 
 0 0 1 1 1 0 0 
 − 
 0 1 0 1 0 1 0 
 
 1 0 0 1 0 0 1 
 − 
 
The paper-foldingproofconsistsinshowingthat onecan 7 3 4 4 3 2
construct an eigenstate ψ of the second billiard as
ψ(a,i)= T φ(a,j), (4)
N ij 1 5 6
j
X
where is some normalization factor. That is, one can
N
”transplant” the eigenfunction of the first billiard to the 7 3 4 4 3 2
second one. The matrix T is called a “transplantation
matrix”. The proof of isospectrality reduces to checking
that ψ given by (3)-(4) vanishes on the boundary and FIG. 7 Thepair73 of isospectral billiards with a rectangular
has a continuous derivative inside the billiard. base shape unfolded to a translation surface (i.e. flat billiard
with opposite sides identified).
Let us first transform the problem into an equiva-
lent one on translation surfaces. Translation surfaces
a 8π angle. An example of a straight line drawn on the
(Gutkin and Judge, 2000), also called planar structures,
first surface is shown on Fig. 7. The eigenvalue problem
are manifolds of zero curvature with a finite number
on these surfaces is equivalent to the problem on the bil-
of singular points (see (Vorobets, 1996) for a more
liards. It is however simpler to handle since the transla-
rigorous mathematical definition). A construction by
tionsurfaceshavenoboundary. Thus,onlythecontinuity
Zemlyakov and Katok(1976)allowstoconstructaplanar
properties of the eigenfunctions have to be checked.
structure on rational polygonal billiards, that is polygo-
Each translation surface is tiled by seven rectangles.
nal billiards whose angles at the vertices are of the form
Again, any point on the surface can be specified by its
α = πm /n , with m ,n positive integers. This planar
i i i i i
coordinates (a,i). Each tile on the translation surface
structure is obtained by “unfolding” the polygon, that is
has six neighboring tiles, attached at its left, upper left,
by gluing to the initial polygon its images obtained by
upper right, right, lower right and lower left edge, and
mirrorreflectionwith respect to eachofits sides, andre-
numbered from 1 to 6 respectively. For instance tile 1 is
peatingthisprocessontheimages. Forpolygonswithan-
surrounded by: tile 5 on its left edge, tile 6 on its right
gles α = πm /n , this process terminates and 2n copies
i i i
edge, tile 3 on its upper left edge, tile 1 itself on its up-
of the initial polygon are required, where n is the gcd of
per right edge (because of the identification of opposite
then . Identifyingparallelsides,onegetsaplanarstruc-
i
sides), tile 3 on its lower left edge and tile 1 on its lower
ture of genus in general greater than 1. This structure
right edge. The way the tiles are glued together can be
has singular points corresponding to vertices of the ini-
specified by permutation matrices A(ν), 1 ν 6, such
mtiail=po1ly.goTnhewhgeerneusthoef athnegletraαnis=latπiomnis/unrifaicsesuthchusthoabt- that Ai(jν) =1 if and only if the edge numb≤er ν≤of i glues
tain6ed is given by (Richens and Berry, 1981) tile i to tile j. For instance for the first translation sur-
face, the matrix specifying which tile is on the right of
n mi 1 which is
g =1+ − . (5)
2 n
i
Xi 0 0 0 0 0 1 0
0 0 1 0 0 0 0
A very simple example of a translation surface is the
 
0 0 0 0 0 0 1
flat torus, obtained by identifying the opposite sides of
a square. Such a translation surface corresponds to four A(2) =0 0 0 1 0 0 0  (6)
 
copies of a square billiard glued together. 1 0 0 0 0 0 0 
 
The billiards of Fig. 2 possess one 2π-angle, two 3π/2- 0 0 0 0 1 0 0 
 
angles and eight π/2-angles each. The translation sur- 0 1 0 0 0 0 0 
 
faces associated to these billiards are obtained by gluing  
together 2n = 4 copies of the billiards, yielding planar (tile 6 is on the right of tile 1, therefore A(1) =1, and so
1,6
surfaces of genus 4. They are shown in Fig. 7. Opposite on). In a similar way, matrices B(ν), 1 ν 6, can be
≤ ≤
sides are identified (e.g. in the first surface, the left edge defined for the second translation surface. Now suppose
of tile 1 is identified with the right edge of tile 5). Each there exists a matrix T such that
surface has four singular points. The symbols and
represent a 6π-angle, while the and symbols◦denot•e ν, 1 ν 6, A(ν)T =TB(ν). (7)
× ∗ ∀ ≤ ≤
8
Then for any given eigenstate φ of the first translation III. FURTHER EXAMPLES IN HIGHER DIMENSIONS
surface we can construct an eigenstate ψ for the second
translationsurface,defined by Eq. (4). Inorderto prove Milnor(1964)showedthatfromtwononisometriclattices
isospectrality we only have to check for continuity prop- of rank 16 in R16 discovered by Witt (1941), one can
erties at each edge. Suppose tiles i and j are neighbors. construct a pair of flat tori that have the same spectrum
This means that there exists a ν, 1 ν 6, such that of eigenvalues (all relevant terms are defined below).
≤ ≤
A(ν) = 1. To prove the continuity of ψ between tiles i In this section, we describe a simple criterion for the
ij
and j, we have to show that the quantity construction of nonisometric flat tori with the same
eigenvalues for the Laplace operator, from certain
=ψ(a,i) ψ(a,j) (8) lattices (which was used by Milnor for the particu-
C −
lar case mentioned above), and then we construct,
is equal to zero for all a belonging to the edge between i
for each integer n 17, a pair of lattices of rank
and j. By definition of ν we have A(ν) =1 if and only if ≥
ik n in Rn that match the criterion. Furthermore, we
k =j. Therefore
describe results of S. Wolpert and M. Kneser on the
ψ(a,j)= A(ν)ψ(a,k), (9) moduli space of flat tori. An interesting survey pa-
ik
per focused on the (elementary) construction theory
k
X
ofisospectralmanifoldshasbeengivenbyBrooks(1988).
and is given by
C
=ψ(a,i) A(ν)ψ(a,k). (10)
C − ik
Xk A. Lattices and flat tori
Using Eq. (4), we get
A lattice (that is, a discrete additive subgroup) can be
C =N Tikφ(a,k)−N Ai(kν)Tkk′φ(a,k′). (11) prescribed as AZn with A a fixed matrix. For example,
k k,k′ set
X X
The sum over k on the right-hand side yields a term 1 0
(A(ν)T)ik′. According to the commutation relation (7), A= 1 1 ; (14)
it is equal to (TB(ν)) , which gives (cid:18) (cid:19)
ik′
then the lattice AZ2 consists of the points of the form
= T φ(a,k) B(ν)φ(a,k ) . (12)
C k ik − k′ kk′ ′ ! a(1,1)+b(0,1), a,b∈Z. (15)
X X
Now the continuity of the function φ ensures that all Ann-dimensional(flat)torusT isRnfactoredbyalattice
the terms between parentheses vanish. Thus = 0, L = AZn with A GL(n,R). The torus is thus deter-
C ∈
and continuity of ψ is proved. Continuity of partial mined by identifying points that differ by an element of
derivatives is proved in the same way. the lattice.
If we return to the planar example above, the torus
The proof rests entirely on the fact that we assumed the topologically is a donut — one may see this by cutting
existence of a transplantation matrix T satisfying the out the parallelogram determined by (1,1) and (0,1),
commutation properties (7). It turns out that such a and then gluing opposite sides together.
matrix exists. One can check that given the matrix
With A,B GL(n,R) are associated the lattices AZn
1 0 0 1 0 0 1 and BZn. ∈ The tori Rn/AZn and Rn/BZn, B
0 1 0 0 1 0 1 GL(n,R), are isometric if and only if AZn and BZn ar∈e
 
0 0 1 0 0 1 1
isometricbyleftmultiplicationbyanelementofO(n,R).
T =1 0 0 0 1 1 0 , (13)
  The matrices A and B are associated with the same lat-
0 1 0 1 0 1 0 
  ticeifandonlyiftheyareequivalentbymultiplicationon
0 0 1 1 1 0 0  therightbyanelementofGL(n,Z). SothetoriRn/AZn
1 1 1 0 0 0 0  and Rn/BZn are isometric if and only if A and B are
  equivalent in
the commutation relations (7) are satisfied for all ν,
1 ν 6. Thus the proof of isospectrality is completed.
≤ ≤
We return in section IV on this transplantation proof of
O(n,R) GL(n,R)/GL(n,Z).1 (16)
isospectrality. \
Here, O(n,R) is the orthogonalgroup in n dimensions.
Anaturalquestionistoknowhowonecanfindasuitable
matrix T and permutation matrices A(ν), B(ν) verifying
all commutation equations (7). Historically these matri-
ces were obtained by the construction of Sunada triples, 1 LetH,K besubgroupsofthegroupG. Thenthespace ofdouble
aswillbe explainedin sectionVII.3. Infact, itturns out cosets H\G/K consists of the subsets (“double cosets”) of the
form HgK, with g ∈ G. (It is clear that G can be partitioned
thatthematrixT isjusttheincidencematrixofthegraph
in these double cosets, and each such double coset itself can be
associatedwithacertainfiniteprojectivespace(theFano
partitionedinrightcosetsofH,andalsoinleftcosetsofK.) So
plane in our example), as will be explained in detail in inH\G/K,x∼yifandonlyifthereareh∈H andk∈K such
section IV. thathxk=y.
9
The metric structure of Rn projects to T, and We conclude that Rn/L and Rn/L have the same
∗1 ∗2
volume(T)= detA; T carries a Laplace operator spectrum of eigenvalues, while not being isometric. (cid:4)
| |
∆= ∂2/∂x2, (17)
− i
Xi Milnor’s Construction. By using the Witt non-
which is just the projection of the Laplacian of isometric lattices in R16 (Witt, 1941), Milnor (1964)
Rn. The lengths of closed geodesics of T are given by essentiallyusedtheaforementionedcriteriontoconstruct
a foraarbitraryinL, beingtheEuclideannorm. the first example of nonisometric isospectral flat tori.
k k k·k
Let P be a symmetric matrix that defines a quadratic StartingfromthesetwononisometriclatticesL116andL126
form on Rn. The spectrum of P is defined to be the of rank 16 in R16 as described in Witt (1941), one can
sequence (with multiplicities) of values γ = NTPN for in fact construct examples of isospectral flat tori in Rn
N ∈ Zn. The sequence of squares of lengths of closed for all n, n ≥ 16, as follows. The lattices L116 and L126
geodesics of Rn/AZn is the spectrum of ATA = Q; the satisfy the condition of Theorem III.1 (Witt, 1941, p.
sequence of eigenvalues of the Laplacian is the spectrum 324). Now embed R16 in R17 in the canonical way. De-
of4π2(A−1)(A−1)T =4π2Q−1. TheJacobiinversionfor- notethecoordinateaxesofthelatterbyX1,X2,...,X17,
mula yields for positive τ, such that X1,X2,...,X16 = R16. Suppose ℓ = 0 is a
h i 6
vector on the X -axis which has length strictly smaller
17
exp(−4π2τNTQ−1N) than any non-zero vector of L1 (and L2). Define two
N Zn new lattices L17 (of rank 17) generated by L16 and ℓ,
X∈ i i
volume(T) 1 i = 1,2. Since X R16, it follows easily that for
= exp(− MTQM). (18) 17 ⊥
(4πτ)n/2 4τ any r > 0, the ball centered at the origin with radius
MX∈Zn r contains the same number ofelements ofL17 as ofL17.
1 2
This equation therefore relates the eigenvalue spectrum One observes that these lattices are nonisometric. Thus,
of the torus to its length spectrum. We will see in sec- by Theorem III.1, we obtain two nonisometric flat tori
tionV.B.3otherexamplesofthisconnectionbetweenthe R17/L17∗, i = 1,2, which have the same spectrum of
i
spectrum of the Laplacian and the length spectrum. eigenvalues for the Laplace operator.
Inductively, we can now define, in a similar way, the
nonisometric lattices Ln and Ln of rank n, n 17,
B. Construction of examples 1 2 ≥
satisfying the condition of Theorem III.1, and thus
leading to nonisometric flat tori Rn/Lni∗, i= 1,2, which
have the same spectrum of eigenvalues for the Laplace
If L is a lattice of Rn, L denotes its dual lattice, which
∗
operator.
consists of all y Rn for which x,y Z for all x L;
∈ h i ∈ ∈
here, , istheusualscalarproductonRn Rn. Clearly,
h· ·i ×
(L∗)∗ =L,andtwolatticesLandL′ areisometricifand
only if L∗ and L′∗ are. C. The four-parameter family of Conway and Sloane
Recall that two flat tori of the form Rn/L , i 1,2 ,
i
∈ { }
are isometric if and only if the lattices L and L are
1 2 LetΛ be apositive-definite lattice. The theta function of
isometric. The following theorem gives a criterion for
Λ is:
isospectrality of flat tori.
Theorem III.1 Let L1 and L2 be two nonisometric lat- ΘΛ(τ)= eiπτkxk2 = qkxk2 = ∞ Nmqm, (19)
tices of rank n in Rn, n 2, and suppose that for each
x Λ x Λ m=0
r >0 in R, the ball of rad≥ius r about the origin contains X∈ X∈ X
the same number of points of L1 and L2. Then the flat whereIm(τ)>0,andNm isthenumberofvectorsx∈Λ
toriRn/L∗1 and Rn/L∗2 arenonisometric while having the of norm m. ΘΛ can be thought of as a formal power
same spectrum for the Laplace operator. series in the indeterminate q, although sometimes one
takesq =eiπτ forfurtherinvestigation,withτ acomplex
Proof. Suppose x 6= 0 is an element of L1 of length α. variable. In that case, ΘΛ(τ) is a holomorphic function
Then there is an α′ < α such that the ball of radius of τ for Im(τ) 0.
≥
α centered at 0 contains all elements of L with length
′ 1
strictlysmallerthanα(sinceL1isdiscrete). Foranyα′ Conway and Sloane (1992) construct a four-parameter
≤
α′′ <α, the ball of radius α′′ centeredat 0 contains that family of pairs of four-dimensional lattices that are
same number of elements. This ball contains as many isospectral (equivalently, that have the same theta se-
elements of L2 as of L1, and since the ball centered at ries (19)). In a similar way as before, such lattice pairs
0 with radius α contains strictly more elements of L1, it yield isospectral flat tori. The main construction of
follows easily that L2 also contains vectors of length α. (Conway and Sloane, 1992) is given by the next result.
Each element z L , i 1,2 , determines an eigen-
i
∈ ∈ { }
function f(x) = e2πhx,zii for the Laplace operator on Theorem III.2 (Conway and Sloane, 1992) Let
Rsont/hLe∗in,uwmitbhercoofrreeisgpeonnvadliunegseleigsesntvhaalnueorλeq=ua(l2tπo)2(h2zπ,rz)i2, e∞, e0, e1, e2 be orthogonal vectors satisfying
is equal to the number of points of L contained in the
i
ball centered at 0 with radius r. e e =a/12, e e =b/12, e e =c/12, e e =d/12,
0 0 1 1 2 2
∞· ∞ · · ·
10
where a,b,c,d > 0, and let [w,x,y,z] denote the vector In this section we have seen that is essentially “easy” to
[wT1he,∞e±n3+,th1xe,e−0la1+t]t,iyvce1e±1s+L=+ze(a2[1.,,bL−,ec1t,,dv±)∞±3s,p=1a]n,[±nv2e±3d,−b=y1,v−[+11,,,1v−,+−1,]v1,+,v±0±,v3=+]. ce3ox0naymsetarpurlescttwo(anfisonnedixschooimbuienttteredirce)ixnias1om9spp6l4eec.strtBaoulKtflaaittc’thsoaqrsui.etaTstkhieoennMainbilnothuoert
0 1 2
and L−(a,b,c,d) spanned by v−,v0−,v1−,v2−∞are isospec- real plane ...
tral. ∞
Some small values of a,b,c,d give examples which IV. TRANSPLANTATION
were first found by Schiemann (1990). Substituting
(a,b,c,d) = (7,13,19,49), one obtains the pair of The aim of this section is to describe the idea of trans-
Earnest and Nipp (1991). plantationinamoremathematicalwaythaninsectionII.
This concept was presumably first introduced by B´erard
(1992, 1993). There is in fact a deep connection between
transplantation theory and the mathematical field of fi-
D. The eigenvalue spectrum as moduli for flat tori nite geometries. First we review some elementary facts
about finite geometries. Application of these tools to
Wenowdiscusssomeinterestingresultsontheeigenvalue transplantation theory sheds light on the reasons for the
spectrum for flat tori. We already saw that there exist existence of isospectrality.
nonisometric isospectral flat tori. A natural question is
now how such tori are distributed.
The following theoremgives aninsight into this question A. Tiling
byconsideringthe caseofacontinuousfamily ofisospec-
tral flat tori. 1. Graphsand billiards bytiling
Theorem III.3 (Wolpert (1978)) Let T be a contin- In this section, we follow Okada and Shudo (2001).
s
uous family of isospectral tori defined for s [0,1]. Then
∈
the tori T , s [0,1], are isometric. Tiling. All known isospectralbilliards can be obtained
s
∈
by unfolding polygonal-shaped tiles. As the unfolding is
An interesting result by M. Kneser is the following (see
done along only three sides of the polygon we can essen-
(Wolpert, 1978) for a proof). It states that, given an
tially consider triangles. We call such examples isospec-
eigenvalue spectrum of some torus, only a finite number
tral Euclidean TI-domains. The knownones arelistedin
of nonisometric tori can be isospectral to it.
AppendixA. Thewaythetilesareunfoldedcanbespeci-
fiedbythreepermutationd d-matricesM(µ),1 µ 3
Theorem III.4 (M. Kneser) The total number of × ≤ ≤
and d N, associatedwith the three sides of the triangle
nonisometric tori with a given eigenvalue spectrum is fi- ∈
and defined in the following way: M(µ) =1 if tiles i and
nite. ij
j are glued by their side µ; M(µ) =1 if the side µ of tile
ii
Thefollowingresultisrathertechnical. Itsmainmessage
i is the boundary of the billiard, and 0 otherwise. The
is that given two tori Rn/AZn and Rn/BZn with the number of tiles is, of course, d. Call the matrices M(µ)
same eigenvalue spectrum, then either these two tori are
“adjacency matrices”.
isometric, or the quadratic forms (ATA) and (BTB) lie One can sum up the action of the M(µ) in a graph with
on a certain subvariety in the space of positive definite
colored edges: each copy of the base tile is associated
quadratic forms. A more precise statement is as follows.
with a vertex, and vertices i and j, i = j, are joined
Dmeantroitceesthbey s℘p(anc,eRo),fapnodsiotibvseerdveefitnhitaet styhme mmaetpric n×n- by an edge of color µ if and only if Mi(6jµ) = 1. In the
same way, in the second member of the pair, the tiles
A GL(n,R) ATA ℘(n,R) (20) are unfolded according to permutation matrices N(µ),
∈ 7→ ∈ 1 µ 3. We call such a colored graph an involution
≤ ≤
determines a bijection from O(n,R) GL(n,R) to graph for reasons to be explained later in this section.
\
℘(n,R). Then the following theorem holds. An example of such graphs is given in Fig. 4. If D is
a Euclidean TI-domain with base tile a triangle, and
Theorem III.5 (Wolpert (1978)) There is a properly M = M(µ) µ 1,2,3 is the set of associated
discontinuous group G acting on ℘(n,R) containing the { k ∈ { }}
n permutation matrices (or, equivalently, the associated
transformation group induced by the GL(n,Z) action coloring), denote by Γ(D,M) the corresponding involu-
tion graph.
S A[ ], (21)
7→ Z
where S ℘(n,R) and GL(n,Z). Given P,S Thefollowingpropositioniseasybutratheruseful(Thas,
∈ Z ∈ ∈
℘(n,R) with the same spectrum, either g(P) = S for 2007b).
some g G , or P,S V , where the latter is a subva-
n n Proposition IV.1 LetDbeaEuclideanTI-domainwith
∈ ∈
riety of ℘(n,R). Moreover, base tile a triangle, and let M = M(µ) µ 1,2,3
{ k ∈ { }}
(i) V = Q ℘(n,R) spec(Q) = spec(R), R be the set of associated permutation matrices. Then the
n
℘(n,R){with∈R=g(Q)kfor all g G , and ∈ matrix
n
6 ∈ }
3
(ii) uVnnioins othfesuibnstpearcseecstoiofnRomf f℘o(rns,oRm)eamnd. a countable ∆ij = Mi(jµ)−Mi(iµ)δij , (22)
µX=1(cid:16) (cid:17)