What one cannot hear?
On drums which sound the same
Rami Band, Ori Parzanchevski, Gilad Ben-Shach
‘Can one hear the shape of a drum ?’

This question was asked by Marc Kac (1966).
Marc Kac (1914-1984)

Is it possible to have two different drums
with the same spectrum (isospectral drums) ?
The spectrum of a drum


A Drum is an elastic membrane
which is attached to a solid planar frame.
The spectrum is the set of the Laplacian’s eigenvalues,
(usually with Dirichlet boundary conditions):
 2
2 
  2  2  f   f
 x y 

f
boundary
k 
2 
n n 1 ,
0
A few wavefunctions of the Sinai ‘drum’:
,…,
f1
,…,
f9
f 201
Isospectral drums
Gordon, Webb and
Wolpert (1992):
‘One cannot hear
the shape of a drum’
Using Sunada’s
construction (1985)
Isospectral drums – A transplantation proof
Given an eigenfunction on drum (a),
create an eigenfunction with the same eigenvalue on drum (b).
 2
2 
  2  2  f   f
 x y 
(a)
(b)
f
boundary
0
Isospectral drums
A paper-folding proof
(S.J. Chapman – 2000)
Isospectral drums – A transplantation proof
We can use another basic building block
Isospectral drums – A transplantation proof
… or a building block
which is not a triangle …
Isospectral drums – A transplantation proof
… or even a funny shaped building block …
Isospectral drums – A transplantation proof
… or cut it in a nasty way (and ruin the connectivity) …
your
annetwork
molecule?’
graph?’
violin
pore
dataset
electrode
throat
?’hole?’
?’?’?’?’?’
drum?’
‘Can one hear the shape of athe
black
universe

Many examples of isospectral objects (not only drums):
 Milnor (1964)
16-dim Tori
 Buser (1986) & Berard (1992)
Transplantation
 Gordon, Web, Wolpert (1992)
Drums
 Buser, Conway, Doyle, Semmler (1994)
More Drums
 Brooks (1988,1999)
Manifolds and
Discrete Graphs
 Gutkin, Smilansky (2001)
Quantum Graphs
 Gordon, Perry, Schueth (2005)
Manifolds

There are several methods for construction of isospectrality
– the main is due to Sunada (1985).

We present a method based on representation theory arguments
which generalizes Sunada’s method.
Isospectral theorem
Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach)
Let Γ be a drum which obeys a symmetry group G.
Let H1, H2 be two subgroups of G with representations R1, R2
that satisfy Ind GH R1  Ind GH R2
1
2
then the drums Γ R , Γ R 2 are isospectral.
1
Groups & Drums
Example: The Dihedral group – the symmetry group of the square.

G = { id , a , a2 , a3 , rx , ry , ru , rv }
How does the Dihedral group act on the square drum?
rax
id

Two subgroups of the Dihedral group:
H1 = { id , a2 , rx , ry }
H2 = { id , a2 , ru , rv }
v
y
u
x
Groups - Representations

Representation – Given a group G, a representation R is an
assignment of a matrix ρR(g) to each group element
g  G, such that:  g1,g2  G ρR(g1)·ρR(g2)= ρR(g1g2).

Example 1 - G has the following 1-dimensional rep. S1:
id  1

a  1
a 2  1
 0 1

a  
 1 0 
 1 0 

a 2  
 0  1
Q  Res GH1 S1
2
id  1 a  1 a  1

rx   1
ru  1
ry   1
rv  1
Example 2 - G has the following 2-dimensional rep. S2:
1 0

id  
0
1



a 3  1
Restriction:
Induction:
1 0

id  
0
1


Ind GH1 Q
 1 0 

a  
0
1


 0  1

a 3  
1 0 
 1 0 

rx  
0
1


1 0 

ry  
0

1


0
ru  
1
1

0 
 0  1

rv  

1
0


is the following rep. of H1:
a3   1
rx   1
ry   1
ru  1
rv  1
is the following rep. of G :
 1 0

a 2  
 0 1
 1 0 

a 3  
 0 1
 1 0 

rx  
 0  1
 1 0 

ry  
 0  1
1
ru  
0
0

 1
1 0 

rv  
0

1


Isospectral theorem
Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach)
Let Γ be a drum which obeys a symmetry group G.
Let H1, H2 be two subgroups of G with representations R1, R2
that satisfy Ind GH R1  Ind GH R2
1
2
then the drums Γ R , Γ R 2 are isospectral.
1

v

An application of the theorem with:
y
u

G  id, a, a 2 , a3 , rx , ry , ru , rv
Two subgroups of G:
H1  id, a 2 , rx , ry
H2
x

  id,
We choose representations

:  id  (1)
a 2 , ru , rv

r  (-1) 
R1 : id  (1) a 2  (-1) rx  (-1) ry  (1)
R2
such that

a 2  (-1) ru  (1)
v
Ind GH1 R1  Ind GH 2 R2


Constructing Quotient Graphs

Consider the following rep. R1 of the subgroup H1:

R1 : id  (1)
a  (-1)
2
rx  (-1)
ry  (1)

We construct Γ R by inquiring what do we know about
a function f on Γ which transforms according to R1.
1
rx f   f
ry f  f
Dirichlet
Neumann
The construction of a

Γ
R1
quotient drum is motivated by an encoding scheme.
Constructing Quotient Graphs

Consider the following rep. R1 of the subgroup H1:

R1 : id  (1)
a 2  (-1)
rx  (-1)
ry  (1)

We construct Γ R by inquiring what do we know about
a function f on Γ which transforms according to R1.
Γ
1
rx f   f

ry f  f
Consider the following rep. R2 of the subgroup H2:

R 2 : id  (1) a 2  (-1) ru  (1) rv  (-1)

We construct Γ R by inquiring what do we know about
a function g on Γ which transforms according to R2.
2
ru g  g
rv g   g
Neumann
Dirichlet

R1
Isospectral theorem
Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach)
Let Γ be a drum which obeys a symmetry group G.
Let H1, H2 be two subgroups of G with
representations R1, R2 that satisfy
Γ
Ind GH1 R1  Ind GH 2 R2
then the drums
Γ
R1 ,
Γ
R 2 are isospectral.
Remarks:
1. The isospectral theorem is applicable not only for
Theorem
Ori
Parzanchevski,
Ben-Shach)
drums,(R.B.,
but for
general
manifolds,Gilad
graphs,
etc.
The drums Γ R , Γ R constructed
2
1
according to the conditions above,
2. The following isospectral example is
possess
a transplantation.
a
courtesy
of Martin Sieber.
Γ
R2
R1
Transplantation

The transplantation of our example is
Γ
Γ
R1
1 1 

T  
1  1
A
B
R2
A+B A-B
Extending the Isospectral pair
Extending our example:
Ind GH1 R1  Ind GH 2 R2  Ind GH3 R3
H1 = { e , a2, rx , ry}
R1: e  1 a 2  1 rx   1 ry  1
H2 = { e , a2, ru , rv}
R2: e  1 a 2  1 ru  1 rv   1
H3 = { e , a, a2, a3}
3
R3: e  1 a  i  a 2   1 a   i 
Γ
Γ

×
a f i f
i
R1
R2
Extending the Isospectral pair
Extending our example:
Ind GH1 R1  Ind GH 2 R2  Ind GH3 R3
H1 = { e , a2, rx , ry}
R1: e  1 a 2  1 rx   1 ry  1
H2 = { e , a2, ru , rv}
R2: e  1 a 2  1 ru  1 rv   1
H3 = { e , a, a2, a3}
3
R3: e  1 a  i  a 2   1 a   i 
Γ
Γ
×
a f i f
i ΓR
R1
R2
3
A few isospectral examples
‘Spectral problems with mixed Dirichlet-Neumann boundary conditions: isospectrality
and beyond’
D. Jacobson, M. Levitin, N. Nadirashvili, I. Polterovich (2004)
‘Isospectral domains with mixed boundary conditions’
M. Levitin, L. Parnovski, I. Polterovich (2005)
This isospectral quartet can be obtained when
acting with the group D4xD4 on the following
torus:
A few isospectral examples
‘One cannot hear the shape of a drum’
Gordon, Webb and Wolpert (1992)
We construct the known isospectral drums of Gordon et al.
but with new boundary conditions:
D
D
N
N
A few isospectral examples – quantum graphs
G = S3 (D3) acts on Γ with no fixed points.
To construct the quotient graph,
we take the same rep. of G,
but use two different bases
for the matrix representation.

The resulting quotient graphs are:
L3
L2
L1
L3
L2
L2
L3
L2
L2
L1
L3
L2
L1
L2
L3
L2
L3
L3
L3
L3
L2
L3
L1
L2
L1
L1
L2
L3
L2
L3
What one cannot hear?
On drums which sound the same
Rami Band, Ori Parzanchevski, Gilad Ben-Shach
R. Band, O. Parzanchevski and G. Ben-Shach,
"The Isospectral Fruits of Representation Theory: Quantum Graphs and Drums ",
J. Phys. A (2009).
O. Parzanchevski and R. Band,
"Linear Representations and Isospectrality with Boundary Conditions ",
Journal of Geometric Analysis (2010).