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Adam Sawicki, Rami Band, Uzy Smilansky
Scattering from isospectral graphs
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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)
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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
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Isospectral drums
Gordon, Webb and Wolpert (1992):
‘One cannot hear the shape of a drum’
Using Sunada’s construction (1985)
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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
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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
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We can use another basic building block
Isospectral drums – A transplantation proof
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… or even a funny shaped building block …
Isospectral drums – A transplantation proof
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… or cut it in a nasty way (and ruin the connectivity) …
Isospectral drums – A transplantation proof
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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
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What’s next?
Scattering matrices of quantum graphs.
Isospectral construction.
Scattering from isospectral quantum graphs.
Back to drums.
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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)(
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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
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)(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
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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
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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
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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
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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.
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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
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Transplantation
The transplantation of our example is
11
11T
D
N
N
D
D N
2RΓ
1RΓ
A
B
A-B A+B
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Transplantation & Scattering We may apply the isospectral construction on graphs with leads:
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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
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‘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?
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The isospectral drums constructionBuser, Conway, Doyle, Semmler (1994)
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AS, Rami Band, Uzy Smilansky
Scattering from isospectral graphs