What one cannot hear? Quantum graphs 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 membrane

which is attached to a solid planar frame.

The spectrum is the set of the Laplacian’s eigenvalues, ,

(usually with Dirichlet boundary conditions):

A few eigenfunctions of the Sinai ‘drum’:

The spectrum of a drum

ffyx

2

2

2

2

0boundary

f

, … , , … ,

1nn

Isospectral drums

Gordon, Webb and

Wolpert (1992):

‘One cannot hear

the shape of a drum’

Using Sunada’s

construction (1985)

‘Can one hear the shape of

a pore ?’ the universe ?’ a network ?’ a molecule?’ a dataset ?’ your throat ?’ an electrode ?’ a graph?’ a drum?’ a black hole?’ a violin ?’

How do we produce isospectral examples?

What geometrical \ topological properties we can hear ?

2

1

3 4

5

6

3

4

5

1

2

Metric Graphs - Introduction

A graph Γ consists of a finite set

of vertices V={vi} and a finite set

of edges E={ej}.

A metric graph has a finite

length (Le>0) assigned to each edge.

A function on the graph is a vector of

functions on the edges:

2

1

3 4

5

6

3

4

5

1

2

L13

L23 L34

L45

L46

L15 L45

L34

L23

L12 L25

L35

),,(1 Eee fff ],0[:

jj ee Lf

Quantum Graphs - Introduction

A quantum graph is a metric graph equipped with an operator,

such as the negative Laplacian:

For each vertex v, we impose vertex conditions, such as

Neumann

Continuity

Zero sum of derivatives

Dirichlet

Zero value at the vertex

A quantum graph is defined by specifying:

Metric graph

Operator

Vertex conditions for each vertex

),...,(1 Eee

fff

)()(,21

21 vfvfEeeeev

0)(' vEe

evf

0)( vfEeev

We are interested in the eigenvalues of the Laplacian:

The Spectrum of Quantum Graphs

Examples of several eigenfunctions of

the Laplacian on the graph:

λ8 λ13 λ16

),...,(),...,(11 EE eeee

ffffff

2 √2

D

D

N

N √3 √3

√2

N N

So…

‘Can one hear the shape of a graph?’

One can hear the shape of a simple graph

if the lengths are incommensurate

(Gutkin, Smilansky 2001)

Otherwise,

we do have isospectral graphs:

Roth (1984)

VonBelow (2001)

Band, Shapira, Smilansky (2006)

Kurasov, enerback (2010)

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.

‘Can one hear the shape of a graph ?’

Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach)

Let Γ be a graph which obeys a symmetry group G.

Let H1, H2 be two subgroups of G with representations R1, R2

that satisfy

then the graphs , are isospectral.

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

H

G

H

We may encode these functions by the following quotient graphs:

Example - A string with Dirichlet vertex conditions.

It obeys the symmetry group .

Two representations of Z2 are:

fkf 2D D

1k

2k

3k

4k

D D

D N

Constructing Quotient Graphs

rid ,2

2RΓ

1,1:1 ridR 1,1:2 ridR

1RΓ

a rx id

Groups & Graphs

Example: The Dihedral group –

the symmetry group of the square

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

y

x

u v

How does the Dihedral group act on a square ?

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 representation

Example 2 - G has the following 2-dimensional representation

Induction: take a representation of H1...

...And turn it into a representation of G (which we denote )

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

1id 12 a 1xr 1yr

RG

H1Ind

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 graph which obeys a symmetry group G.

Let H1, H2 be two subgroups of G with representations R1, R2

that satisfy

then the graphs , are isospectral.

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

H

G

H

y

x

u v

An application of the theorem with:

Two subgroups of G:

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

vu

yx

rra

rra

,, id,H

,, id,H

2

2

2

1

We choose representations

R1 of H1 and R2 of H2

such that

21 21

IndInd RR G

H

G

H

(1),(-1),(-1) (1),id:R 2

1 yx rra

(-1),(1),(-1) (1),id:R 2

2 vu aaa

1RΓ

D

N

N

D

Consider the following rep. R1 of the subgroup H1:

We construct by inquiring what do we know about

a function f on Γ which transforms according to R1.

Constructing Quotient Graphs

ffrx ffry D D D D

N

N

N

N

1RΓ

(1)(-1)(-1)(1)id:R 2

1 yx rra

Neumann Dirichlet

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

2RΓ

Consider the following rep. R2 of the subgroup H2:

We construct by inquiring what do we know about

a function g on Γ which transforms according to R2.

ggru ggrv

D N N

N

D

D

Constructing Quotient Graphs

D

N

N

D

2RΓ

1RΓ

(-1)(1)(-1)(1)id:R 2

2 vu rra

Neumann Dirichlet

Consider the following rep. R1 of the subgroup H1:

We construct by inquiring what do we know about

a function f on Γ which transforms according to R1.

ffrx ffry

1RΓ

(1)(-1)(-1)(1)id:R 2

1 yx rra

Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach)

Let Γ be a graph which obeys a symmetry group G.

Let H1, H2 be two subgroups of G with

representations R1, R2 that satisfy

then the graphs , are isospectral.

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

H

G

H

D N

D

N

N

D

2RΓ

1RΓ

3RΓ

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

D

N

N

D

D N

1RΓ 1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

fifa

×i

2RΓ

321 321IndIndInd RRR G

H

G

H

G

H

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

D

N

N

D

D N

1RΓ 1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

fifa

2RΓ

321 321IndIndInd RRR G

H

G

H

G

H

3RΓ

×i

×i

×i

3R

Γfifa

×i

×i

3RΓ

×i

Extending our example:

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

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

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

D

N

N

D

D N

1RΓ 1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

2RΓ

321 321IndIndInd RRR G

H

G

H

G

H

Extending the Isospectral pair

3RΓ

Extending our example:

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

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

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

D

N

N

D

D N

1RΓ 1e 12 a 1xr 1yr

1e 12 a 1ur 1vr

1e ia 12 a ia 3

2RΓ

321 321IndIndInd RRR G

H

G

H

G

H

×i

×i

fifa

3RΓ

×i

Extending the Isospectral pair

rv Γ is the Cayley graph of G=D4

(with respect to the generators a, rx):

Take the same group and the subgroups:

H1 = { e , a2, rx , ry} with the rep. R1

H2 = { e , a2, ru , rv} with the rep. R2

H3 = { e , a , a2 , a3} with the rep. R3

Arsenal of isospectral examples

1RΓ

2RΓ

e a

a3 a2

rx

ry

ru

L1

L2

L2

L1

The resulting quotient graphs are:

3RΓ

L1

L1

L1

L1

L2

L2

L2

L2

L1

L1

L2

L2

L1

L2

L1

G = D6 = {e, a, a2, a3, a4, a5, rx, ry, rz, ru, rv, rw} with the subgroups:

H1 = { e, a2, a4, rx, ry, rz } with the rep. R1

H2 = { e, a2, a4, ru, rv, rw } with the rep. R2

H3 = { e, a, a2, a3, a4, a5 } with the rep. R3

Arsenal of isospectral examples

1RΓ

2RΓ

The resulting quotient graphs are:

3RΓ

L2

L1 2L2

2L3

2L1 L2

L3 L3

2L2

2L3

2L1

2L3

2L3

2L2

2L2 2L1

L1

L1

2L2

2L2

2L3

2L3

2L1

2L1

L1 L1

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.

Arsenal of isospectral examples

The resulting quotient graphs are:

L1 L3

L3

L3

L2 L2

L2

L1 L3

L3

L3

L2 L2

L2

L1

L2

L2

L3

L3

L2 L3

L2 L3

L1 L2

L3 L3

L2

Why quantum graphs? Why not drums?

1RΓ

2RΓ

N D

N

D

However, is not a planar drum: 3R

Γ

Following Martin Sieber

Arsenal of 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)

Isospectral drums

This isospectral quartet can be obtained when

acting with the group D4xD4 on the following

torus:

Arsenal of isospectral examples

‘One cannot hear the shape of a drum’

Gordon, Webb and Wolpert (1992)

Isospectral drums

D

N

D

N

We construct the known isospectral drums of Gordon et al.

but with new boundary conditions:

What one cannot hear? On drums\graphs 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).