Adam Sawicki, Rami Band, Uzy Smilansky

Scattering from isospectral graphs

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

fkfyx

22

2

2

2

0boundaryf

1f 9f 201f , … , , … ,

1

2

nnk

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).

(a) (b)

fkfyx

22

2

2

2

0boundaryf

Isospectral drums – A transplantation proof

Given an eigenfunction on drum (a), create an eigenfunction with the same eigenvalue on drum (b).

(a) (b)

fkfyx

22

2

2

2

0boundaryf

We can use another basic building block

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

Okada, Shudo, Tasaki and Harayama conjecture (2005) that ‘One can distinguish isospectral drums by measuring their scattering poles’

‘Can one hear the shape of a drum ?’: revisited

What’s next?

Scattering matrices of quantum graphs.

Isospectral construction.

Scattering from isospectral quantum graphs.

Back to drums.

A quantum graph is defined by: A graph A length for each edge A scattering matrix for each vertex

(indicates vertex conditions)

The eigenfunctions of the Laplacianare given by

on each of the edges e of the graph.

They can be described by these coefficients .

L13

L23

L34

L45

L46

1

43

5

2 62

1

3 4

5

6

10Ca

Quantum Graphsfkf 2

eoutee

inee ikxaikxaxf expexp)(

A quantum graph is defined by: A graph A length for each edge A scattering matrix for each vertex

(indicates vertex conditions)

The eigenfunctions of the Laplacianare described by

L13

L23

L34

L45

L46

1

43

5

2 62

1

3 4

5

6

333

6

5

2

1

444

444

444

333

333

)(kU

34

46

45

23

13

34

46

45

23

13

exp

L

L

L

L

L

L

L

L

L

L

ik

akUaa

)(such that10 C

Quantum Graphsfkf 2

)(kU

34

46

45

23

13

34

46

45

23

13

exp

L

L

L

L

L

L

L

L

L

L

ik

L13

L23

L34 L45

L46

1

3

5

26

4

333

6

5

2

1

444

444

444

333

333

Quantum Graphs

A quantum graph is defined by: A graph A length for each edge A scattering matrix for each vertex

(indicates boundary conditions)

The eigenfunctions are described by

Examples of several eigenfunctions of the Laplacian on the graph above:

8f 13f 16f

L13

L23

L34

L45

L46

1

43

5

2 62

1

3 4

5

6akUaa

)(such that10 C

Quantum Graphs - Introduction

L13

L23

L34

L45

L46

1

43

5

26

2

1

3 4

5

6

333

6

5

2

1

4444

4444

4444

3333

3333

4444

3333

)(kU

34

46

45

23

13

34

46

45

23

13

0

0

exp

L

L

L

L

L

L

L

L

L

L

ik

Quantum Graphs – Attaching leads

1111 expexp ikxcikxc outin 2222 expexp ikxcikxc outin

L13

L23

L34

L45

L46

1

43

5

26

2

1

3 4

5

6

UT

TR

in

out

~

Quantum Graphs – Attaching leads

1111 expexp ikxcikxc outin 2222 expexp ikxcikxc outin

10

1

2

1

a

a

c

cout

out

10

1

2

1

a

a

c

cin

in

L13

L23

L34

L45

L46

1

43

5

26

2

1

3 4

5

6

aUcTa

aTcRcin

in

outoutout

~

Quantum Graphs – Attaching leads

1111 expexp ikxcikxc outin 2222 expexp ikxcikxc outin

ininout

out cTUTRc 1~

1

inout TUTRkS1~

1)(

UT

TR

in

out

~

a

c in

a

c out

The poles of S(k) are below the real axis.

Theorem (R. Band , O. Parzanchevski, G. 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.

Theorem (R.Band, O. Parzanchevski, G. Ben-Shach)The graphs , constructed according to the conditions above,possess a transplantation.

Remark – The theorems are applicable for other geometrical objects such as manifolds and drums.

Isospectral theorem

1RΓ

2RΓ

21 21IndInd RR G

HGH

D N

D

N

N

D

2RΓ

1RΓ

1RΓ

2RΓ

Transplantation

The transplantation of our example is

11

11T

D

N

N

D

D N

2RΓ

1RΓ

A

B

A-B A+B

Transplantation & Scattering We may apply the isospectral construction on graphs with leads:

Transplantation & Scattering We may apply the isospectral construction on graphs with leads: The transplantation relates

the function’s values on the leads:

leadsleadsfTg

11

11T

ikxcikxcTikxdikxd outinoutin expexpexpexp

outout

inin

cTd

cTd

inout

inout

dkSd

ckSc

)(

)(

2

1

inout cTkSTc 121

1 )( TkSTkS )()( 2

11

In particular S1(k), S2(k)have the same pole structure.

In addition, we have the scattering relations:

g f

‘Can one hear the shape of a graph ?’: revisited

Definition – The graphs Γ1 , Γ2 are called isoscattering if they can be extended to scattering systems which share the same pole structure.

The isospectral construction can be used to construct isoscattering graphs.

Two isospectral graphs are also isoscattering.The possible extensions of these graphs to isopolar scattering systems depend on the symmetry that was used in the construction.

The isospectral construction applies also to manifolds and drums - Why the scattering results for drums and graphs are different?

The isospectral drums constructionBuser, Conway, Doyle, Semmler (1994)

AS, Rami Band, Uzy Smilansky

Scattering from isospectral graphs