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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 ( )

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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 wavefunctions of the Sinai ‘drum’: The spectrum of a drum, …,, …,, …,, …,

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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)

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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)

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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 Laplacian are given by on each of the edges e of the graph. They can be described by these coefficients. L 13 L 23 L 34 L 45 L 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 vertex conditions) The eigenfunctions of the Laplacian are described by L 13 L 23 L 34 L 45 L Quantum Graphs

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L 13 L 23 L 34 L 45 L 46 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: L 13 L 23 L 34 L 45 L Quantum Graphs - Introduction

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L 13 L 23 L 34 L 45 L Quantum Graphs – Attaching leads

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L 13 L 23 L 34 L 45 L Quantum Graphs – Attaching leads

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L 13 L 23 L 34 L 45 L Quantum Graphs – Attaching leads 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 H 1, H 2 be two subgroups of G with representations R 1, R 2 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 D N D N N D

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Transplantation The transplantation of our example is D N N D D N 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: In particular S 1 (k), S 2 (k) have the same pole structure. In addition, we have the scattering relations: gf

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‘Can one hear the shape of a graph ?’: revisited Definition – The graphs Γ 1, Γ 2 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 construction Buser, Conway, Doyle, Semmler (1994)

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AS, Rami Band, Uzy Smilansky Scattering from isospectral graphs

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