Can One Hear the Shape of a Drum?

24 Pages·2010·1.36 MB·English

Can One Hear the Shape of a Drum?
Author(s): Mark Kac
Reviewed work(s):
Source: The American Mathematical Monthly, Vol. 73, No. 4, Part 2: Papers in Analysis (Apr.,
1966), pp. 1-23
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2313748 .
Accessed: 15/03/2012 22:10
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to
The American Mathematical Monthly.
http://www.jstor.org
CAN ONE HEAR THE SHAPE OF A DRUM?
MARK KAC, The RockefellerU niversity,N ew York
To George Eugene Uhlenbeck on the occasion of his sixty-fiftbhi rthday
"La Physique ne nous donne pas seulement
l'occasion de resoudre des problemes . .. elle nous
fait presentirl a solution." H. POINCARE.
Before I explain the title and introducet he theme of the lecture I should like
to state that my presentationw ill be more in the nature of a leisurelye xcursion
than of an organized tour. It will not be my purpose to reach a specifiedd es-
tination at a scheduled time. Rather I should like to allow myselfo n many
occasions the luxury of stopping and looking around. So much efforti s being
spent on streamliningm athematics and in renderingi t more efficientt, hat a
solitary transgressiona gainst the trend could perhaps be forgiven.
r
Q
FIG. 1
1. And now to the theme and the title.
It has been known forw ell over a centuryt hat if a membrane 2, held fixed
along its boundary F (see Fig. 1), is set in motioni ts displacement( in the direc-
tion perpendiculart o its original plane)
F(x,y ; t)-F (p; t)
obeys the wave equation
aF
= c2 72F,
at2
where c is a certain constant depending on the physical propertieso f the mem-
brane and on the tension under which the membranei s held.
I shall choose units to make C2== 2
2 PAPERS IN ANALYSIS
Of special interest (both to the mathematician and to the musician) are
solutions of the form
F(p-; t) =: U(p)eiwl,
for,b eing harmonici n time,t hey representt he putret ones the membranei s capa-
ble of producing.T hese special solutions are also known as normal modes.
To find the normal modes we substitute U(-)eiwti nto the wave equation
and see that U must satisfyt he equation 2 V2U+co2U=O with the boundary
condition U=O on the boundary F of Q, correspondingt o the membraneb eing
held fixeda long its boundary.
The meaning of "U =0 on F" should be made clear; for sufficientlysm ooth
boundaries it simplym eans that U(p)->O as p approaches a point of F (fromt he
inside). To show that a membranei s capable of producinga discrete spectrum
of pure tones i.e. that there is a discrete sequence of c's W<_?2 ?Co3_? * for
which nontrivials olutions of
V2U + (2U =0 , U = 0 on ,
exist, was one of the great problems of 19th century mathematical physics.
Poincare struggledw ith it and so did many others.
The solution was finallya chieved in the early years of our centuryb y the
use of the theoryo f integrale quations.
We now know and I shall ask you to believe me ify ou do not, that forr egions
Q bounded by a smooth curve F there is a sequence of numbers X1_X2_ * * -
called eigenvalues such that to each therec orrespondsa function4 '(-), called an
eigenfunctions, uch that
0
21 72V, + Xn4n-
and 41'(-)->O as p->a point of F.
It is customaryt o normalize the 4t's so that
J / (p)-dp= 1.
Note that I use dp to denote the element of integration( in Cartesian co-
ordinates,e .g., dp-dxdy).
2. The focal point of my exposition is the followingp roblem:
Let ?2 and ?2 be two plane regionsb ounded by curves F, and F2 respectively,
and consider the eigenvalue problems:
V2U + XU 0 in Q 22V + AV =0 in Q2
with with
U= on F, V= on F2.
Assume that fore ach n the eigenvalue N,,f or? 1 is equal to the ei-genvalueA P1
CAN ONE HEAR THE SHAPE OF A DRUM? 3
for 02. Question: Are the regions i1 and 02 congruenti n the sense of Euclidean
geometry?
I firsth eard the problem posed this way some ten years ago fromP rofessor
Bochner. Much more recently,w hen I mentionedi t to ProfessorB ers, he said,
almost at once: "You mean, if you had perfectp itch could you findt he shape
of a drum.'"
You can now see that the "drum" of my title is more like a tambourine
(which really is a membrane) and that stripped of picturesque language the
problem is whetherw e can determineQ if we know all the eigenvalues of the
eigenvalue problem
V2U+XU=0 in Q,
U =0 o0n F.
3. Before I go any furtherl et me say that as far as I know the problem is
still unsolved. Personally, I believe that one cannot "hear" the shape of a tam-
bourine but I may well be wrong and I am not prepared to bet large sums
either way.
What I propose to do is to see how much about the shape can be inferred
fromt he knowledge of all the eigenvalues, and to impress upon you the multi-
tude of connectionsb etween our problema nd various parts of mathematicsa nd
physics.
It should perhaps be stated at this point that throughoutt he paper only
asymptoticp ropertieso f large eigenvalues will be used. This may represent,o f
course, a serious loss of informationa nd it may perhaps be argued that precise
knowledgeo f all the eigenvalues may be sufficientto determinet he shape of the
membrane. It should also be pointed out, howTevert,h at quite recentlyJ ohn
Milnor constructedt wo noncongruents ixteen dimensional tori whose Laplace-
Betrami operators have exactly the same eigenvalues (see his one page note
"Eigenvalues of the Laplace operator on certain mlanifolds"P roc. Nat. Acad.
Sc., 51(1964) 542).
4. The firstp ertinentr esult is that one can "hear" the area of Q. This is an
old result with a fascinatingh istoryw hich I shall now relate briefly.
At the end of October 1910 the great Dutch physicist H. A. Lorentz was
invited to G6ttingent o deliver the Wolfskehll ectures. Wolfskehl,b y the way,
endowed a prize for proving, or disproving,F ermat's last theorem and stipu-
lated that in case the prize is not awarded the proceeds fromt he principal be
used to invite eminents cientistst o lecture at Gottingen.
Lorentz gave five lecturesu nder the overall title "Alte und neue Fragen der
Physik"-Old and new problemso f physics-and at the end of the fourthl ecture
he spoke as follows (in freet ranslationf romt he original Germ-Ian)": In conclu-
sion there is a mathematical probleimwi hich perhaps will arouse the interesto f
mathematiciansw ho are present. It originatesi n the radiation theoryo f Jeans.
"In an enclosure with a perfectlyr eflectings urface there can forms tanding
4 PAPERS IN ANALYSIS
electromagneticw AAavaensa logous to tones of aino rgan pipe; we shall confineo ur
attention to very high overtones. Jeans asks for the energy in the frequency
intervald v. To this end he calculates the numlbero f overtonesw hich lie between
the frequenciesv and v+dy and multipliest his numberb y the energyw hich be-
longs to the frequenlcyv ,a nd whicha ccordingt o a theoremo f statistical mllechan-
ics is the samnef or all frequencies.
"It is here that therea rises the mathematicalp roblemt o prove that the num-
ber of sufficientlyh igh overtonesw hich lies between v and v+dv is independent
of the shape of the enclosurea nd is simiiplpyr oportionalt o its volumlleF. or many
simnplseh apes for wAhiccha lculations canl be carried out, this theoremhl as been
verifiedi n a Leiden dissertation.T here is no doubt that it holds in general even
for multiplyc onnected regions.S inilar theoremnfso r other vibratings tructures
like inembranes,a ir inasses, etc. should also hold."
If one expresses this conjecture of Lorentz in terms of our imembrane,i t
enmergeisn the form:
V (X)
< 27 r
Here N(X) is the number of eigenvalues less than X, | I the area of Q and
means that
N0 (X) |Q|
a X 27r
X
There is an apocryphalr eportt hat Hilbert predictedt hat the theoremw ould
not be proved in his life time. Well, he was w-rongb y many, many years. For
less than two years later HermiianW eyl, who was present at the Lorentz' lec-
ture and whose interest was aroused by the problem, proved the theorem in
question, i.e. that as X >oc
x.
N1(X)
Weyl used in a masterly way the theory of integral equations, which his
teacher Hilbert developed only a fewy,-e ars before,a nd his proofw as a crown ing
achievement of this beautiful theory. I\lany subsequent developnmentisn the
theoryo f differentiaaln d integrale quations (especially the wNorokf Coturanta nd
his school) can be traced directlyt o Weyl's memoiro n the conjectureo f Lorentz.
5. Let me now consider brieflya differentp hysical problem which too is
closelyr elated to the problenmof the distributiono f eigenvalues of the Laplacian.
It can be taken as a basic postulate of classical statistical mechanics that if
a systemo f 1l particles confinedt o a volume Q is in equilibriumw Titha thermo-
stat of temperatureT the probabilityo f findings pecifiedp articlesa t ri,,r 2,
rm (tn lel nd r -
rMf( wvithinv olume elementsd r1,d r2,. , drAzli) s
CAN ONE HEAR THE SHAPE OF A DRUM? 5
L
.. ...
exp [- kTVd(rrly ** rml)0 dr-, dr,-
* J. exp V(rl ... rm) dr, ... drm
where V(r-1, ,rll) is the interactionpi otential of the particles and k=R/N
with R the "gas constant" and N the Avogadro number.
For identical particles each of mass m obeying the so called Boltzmainii
statistics the correspondinga ssumiptioni n quantum statistical mechanicss eems
much more complicated. One starts with the Schr6dingere quation
Q
2 V - V(ri, ..., r_>) Eq 2- , whereh is the Planck constant)
2m 27
with the boundary condition limn4( ri. , 1) =0, whenever at least one
rk
approaches the boundary of Q. (This boundary condition has the effecto f con-
fining the particles to Q.) Let E1 < E2 ? E3 < be the eigenvalues and
4'114,, . thec orrespondinngo rmalizedei genfunctionTsh. en theb asic postu-
late is that the probabilityo f findings pecifiedp articlesa t ri, r2, , ur( within
dr1* **, d-rm)i s
j
e E,/kT A2(-r+- l * ri)9drl . . . dr31
s=l
Z
e-EsIkT
s=1
There are actually no known particles obeying the Boltzmann statistics. But
don't let this worry you for our purposes this regrettable fact is immilaterial.
Now, let us specialize our discussion to the case of an ideal gas w7hich,b y
definitionm, eans that V( r-, , r 1) _O.
Classically, the probability of findings pecified particles at r1, , r7 is
clearly
dr1 dr3
where I Q | is now the volume of Q.
Quantum mechanically the answer is not nearly so explicit. The Schr6dinger
equation for an ideal gas is
ft2
2m 9 -
and the equation is obviously separable.
If I now consider the three-dimensional (rather than the 3MH-dimenlsional)
eigenvalue problem
6 PAPERS IN ANALYSIS
2V2A(r) = - r
V(r) -> 0 as r the boundary of 9,
it is clear that the Es as well as the , r) are easily expressiblei n termiis
of the X's and corresponding4 1(r)'s.
The formulaf or the probabilityo f findings pecifiedp articles at -ri, rm
turns out to be
exp [-m-- T]'fl(rk)
k=H1 ~ LL r 0_ _ _ _ _ _ _A _ n_ t_ - dr i,
E exp --- ]
Now, as 1->0 (or as T-*> occ) the quantum mechanical result should go over
into the classical one and this immllediatellye ads to the conjecture that as
LmkT_
Ee An(F) i
?1=1 n=1
If instead of a realistic three-dimnensioncaol ntainer Q I consider a two-
dimensional one, the result would still be the same
X
-e_n 2e2-Xr,)- 0
n=1 n=1
except that now I Q| is the area of Q rather than the volume.
Clearly the result is expected to hold only for r in the interioro f Q.
If we believe Weyl's result that (in the two-dimensionalc ase)
I
IV (X) /_ -I 1%),
2er
it followsi mmediatelyb y an Abelian theoremt hat
1 0 1
__Ee-X nT - T O,-- 0
and hence that
1
r2 -XT
E-e XX2 nr) -Ie dX.
n=1 27rT 2ro
Setting A (X) = x <x 4'(r), wNcvaen record the last result as
CAN ONE HEAR THE SHAPE OF A DRUM? 7
de-XrA (X) e-Xrd X, r -> 0.
Since A (X) is nondecreasingw e can apply the Hardy-Littlewood-Karamata
Tauberian theorema nd conclude what everyonew ould be temptedt o conclude,
namely that
2- N
A (X)= + (r)2 , -->o,
fo r every r in the interioro f Q.
Though this asymptotic formula is thus nearly "obvious" on "physical
grounds,"i t was not until 1934 that Carleman succeeded in supplyinga rigorous
proof.
In concluding this section it may be worthwhilet o say a word about the
"strategy" of our approach.
We are primarilyi nterested,o f course, in asymptotic propertieso f X,,f or
large n. This can be approached by the device of studyingt he Dirichlet series
00
E e-Xnt
n=1
for small t. This in turn is most convenientlya pproached throught he series
00 {D00
Ze
CXt,(P) = fe-tdA(x)
n1=1
and thus we are led to the Abelian-Tauberian interplayd escribed above.
6. It would seem that the physical intuition ought not only provide the
mathematicianw ith interestinga nd challengingc onjectures,b ut also show him
the way toward a proof and toward possible generalizations.
The contexto f the theoryo f black body radiation or that of quantum statis-
tical mechanics,h owever,i s too far removed frome lementaryi ntuitiona nd too
full of daring and complex physical extrapolations to be of much use even in
seeking the kind of understandingt hat makes a mathematician comfortable,
let alone in pointingt oward a rigorousp roof.
Fortunately, in a much more elementary context the problem of the dis-
tributiono f eigenvalues of the Laplacian becomes quite tractable. Proofse merge
as natural extensionso f physical intuitiona nd interestingg eneralizationsc ome
withinr each.
7. The physical context in question is that of diffusi on theory,a nother
branch of nineteenthc enturym athematical physics.
Imagine "stuff,"i nitiallyc oncentrateda t p( (xo, yo)), diffusingt hrougha
plane region Q bounded by F. Imagine furthermorteh at the stuffg ets absorbed
( "eaten") at the boundary.
8 PAPERS IN ANALYSIS
The concentrationP Q(p| r; t) of matter at r-((x, y)) at time t obeys the
differentiael quation of diffusion
aPe 1
(a) - =t 2
the boundary condition
(b) PQ(p I r; t) - 0 as r approachesa boundaryp oint,
and the initial condition
(c) PQ(P |rt; ) -*(r-p) as t ->0;
here (-r-p) is the Dirac "delta function,"w ith "value" ooi f -r= p and 0 if r p'.
The boundary condition (b) expressest he fact that the boundary is absorb-
ing and the initial condition (c) the fact that initially all the "stuff"w as con-
centrated at ip.
I have again chosen units so as to make the diffusionc onstant equal to
2.
As is well knownt he concentrationP Q(P I -; t) can be expressedi n termso f the
eigenvalues Xz, and normalized eigenfunctions4 ,(r-) of the problem
'72v +X4/=0 in Q,
0 on P.
In fact, PQ(p Ir ; t)= En-], 6-fl'4' (p)Abl(r).
NOw,f ors mall t, it appearsi ntuitivelycl ear that particleso f the diffusing
stuffw illn oth ave had enought imet o have feltt hei nfluencoef t heb oundaryQ .
As particlesb egint o diffusteh eym ay not be aware,s o to speak,o f the disaster
that awaits themw hent heyr eacht he boundary.
We may thuse xpectt hati n somea pproximatsee nse
PQ(p I r; t) - Po(p I r; t), as t O ,
whereP o(p1r ; t) stills atisfiets he same diffusioenq uation
P
(a') VPO
and the same initialc ondition
(c') Po(p ;t) = P-p), t0,
but is otherwisue nrestricted.
Actuallyt herei s a slighta dditionalr estrictiown ithoutw hicht he solution
is not unique (a remarkablfe actd iscovereds omey earsa go by D. V. Widder).
The restrictioins thatP o > 0 (or moreg enerallyth atP o be boundedf romb elow).
A similarr estrictionfo rP Q is not neededs ince ford iffusionin a bounded
regioni t followsa utomatically.
CAN ONE HEAR THE SHAPE OF A DRUM? 9
An explicit formulaf orP o is, of course, well known. It is
Po(pIr ; t) exp - - ]
where ||- denotes the Euclidean distance between p and r.
I can now state a little morep reciselyt he principleo f "not feelingt he bound-
ary" explained a momenta go.
The statementi s that as t >0
1 F
00 r ~P~2
PPnQp( |r ; t) = 1 -A()nr -,1e exp- 2- 1.o( npr ; t),
n=1 2w1t 1 21-
where - stands here for "is approximatelye qual to." This is a bit vague but
let it go at that for the moment.
If we can trust this formulae ven forp = r we get
00 x'-Ant 2 11
e-e n(r)
27rt
n=l
and if we display still more optimismw e can integratet he above and, making
use of the normalizationc ondition
2
a(nr -) d-r'=: 1,
obtain
Q.
I:e-Xnt,
1=1 27 rt
We recognizei mmediatelyt he formulasd iscussed a while back in connection
with the quantum-statistical-mechanicalt reatment of the ideal gas. If we
apply the Hardy-Littlewood-Karamata theorem,a lluded to before, we obtain
as corollaryt he theoremso f Carleman and Weyl.
To do this, however,w e must be allowed to interpret as meaning "asynmp-
totic to."
8. Now, a little mathematical soul-searching.A ren't we as far froma rigor-
ous treatnmenats we were before? True, diffusioni s more familiar than black
body radiation or quantum statistics. But familiarityg ives comfort,a t best,
and comfortm ay still be (and ofteni s) miles away fromt he rigord emanded by
mathematics.
Let us see then what we can do about tighteningt he loose talk.
First let me dispose of a few minori tems which may cause you worry.
When I writei / = 0 on F or P(p I r; t)-O as r approaches a boundary point of
Q there is always a question of interpretation.