What one cannot hear?On drums which sound the same

Rami Band, Ori Parzanchevski, Gilad Ben-Shach

This question was asked by Marc Kac (1966).

Is it possible to have two different drums with the same spectrum (isospectral drums) ?

‘Can one hear the shape of a drum ?’

Marc Kac (1914-1984)

A Drum is an elastic membranewhich is attached to a solid planar frame.

The spectrum is the set of the Laplacian’s eigenvalues, ,(usually with Dirichlet boundary conditions):

A few wavefunctions of the Sinai ‘drum’:

The spectrum of a drum

ffyx

2

2

2

2

0boundaryf

1f 9f 201f , … , , … , , … , , … ,

12

nnk

Isospectral drums

Gordon, Webb and Wolpert (1992):

‘‘One One cannotcannot hear hear the shape of a drum’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).

(a) (b)

ffyx

2

2

2

2

0boundaryf

Isospectral drums A paper-folding

proof (S.J. Chapman – 2000)

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) …

Isospectral drums – A transplantation proof

a graph?’a black hole?’the universe ?’a dataset ?’a network ?’a molecule?’your throat ?’a pore ?’a violin ?’an electrode ?’a drum?’‘Can one hear the shape of 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.

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 then the drums , are isospectral.

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

HGH

arxid

Groups & Drums

Example: The Dihedral group – the symmetry group of the square.

G = { id , a , a2 , a3 , rx , ry , ru , rv }

yy

xx

uuvv

How does the Dihedral group act on the square drum?

Two subgroups of the Dihedral group: H1 = { id , a2 , rx , ry }H2 = { id , a2 , ru , rv }

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:

Example 2 - G has the following 2-dimensional rep. S2:

Restriction: is the following rep. of H1:

Induction: is the following rep. of G :

Groups - Representations

1id 1a 12 a 13 a 1xr 1yr 1ur 1vr

10

01id

01

10a

10

012a

01

103a

10

01xr

10

01yr

01

10ur

01

10vr

11ResQ SG

H

1id 1a 12 a 13 a 1xr 1yr 1ur 1vr

QGH1Ind

10

01id

10

01a

10

012a

10

013a

10

01xr

10

01yr

10

01ur

10

01vr

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 then the drums , are isospectral.

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

HGH

yy

xx

uuvv

An application of the theorem with:

Two subgroups of G:

vuyx rrrraaa ,,,,,,id,G 32

vu

yx

rra

rra

,,id,H

,,id,H2

2

21

We choose representations

such that

21 21IndInd RR G

HGH

(1)(-1)(-1)(1)id:R 21 yx rra

(-1)(1)(-1)(1)id:R 22 vu rra

Consider the following rep. R1 of the subgroup H1:

We construct by inquiring what do we know abouta function f on Γ which transforms according to R1.

Constructing Quotient Graphs

ffrx ffry

Neumann

1RΓ

The construction of a quotient drum is motivated by an encoding scheme.

(1)(-1)(-1)(1)id:R 21 yx rra

Dirichlet

1RΓ

Consider the following rep. R2 of the subgroup H2:

We construct by inquiring what do we know abouta function g on Γ which transforms according to R2.

ggru ggrv

Constructing Quotient Graphs

(-1)(1)(-1)(1)id:R 22 vu rra

2RΓ

Neumann Dirichlet

1RΓ

Consider the following rep. R1 of the subgroup H1:

We construct by inquiring what do we know abouta function f on Γ which transforms according to R1.

ffrx ffry

1RΓ

(1)(-1)(-1)(1)id:R 21 yx rra

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

HGH

1RΓ

2RΓ

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

then the drums , are isospectral.

Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach)The drums , constructed according to the conditions above,possess a transplantation.

1RΓ

2RΓ

Remarks:1. The isospectral theorem is applicable not only for drums, but for general manifolds, graphs, etc.

2. The following isospectral example is a courtesy of Martin Sieber.

Transplantation The transplantation of our example is

11

11T

A

BA-BA+B

1RΓ

2RΓ

Extending our example:

H1 = { e , a2, rx , ry} R1:

H2 = { e , a2, ru , rv} R2:

H3 = { e , a, a2, a3} R3:

Extending the Isospectral pair

1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

321 321IndIndInd RRR G

HGH

GH

1RΓ

2RΓ

fifa

×

i

Extending our example:

H1 = { e , a2, rx , ry} R1:

H2 = { e , a2, ru , rv} R2:

H3 = { e , a, a2, a3} R3:

Extending the Isospectral pair

1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

321 321IndIndInd RRR G

HGH

GH

1RΓ

2RΓ

fifa

×

i 3RΓ

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)

D

N

D

N

We construct the known isospectral drums of Gordon et al.but with new boundary conditions:

L1L1

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.

A few isospectral examples – quantum graphs

The resulting quotient graphs are:

L1L3

L3

L3

L2L2

L2

L1L3

L3

L3

L2L2

L2

L1

L2

L2

L3

L3 L2 L3L2 L3

L1 L2

L3

L3

L2

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).

What one cannot hear?On drums which sound the same