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Adam Sawicki, Rami Band, Uzy Smilansky

Adam Sawicki, Rami Band, Uzy Smilansky. Scattering from isospectral graphs. ‘Can one hear the shape of a drum ?’. This question was asked by Marc Kac (1966). Is it possible to have two different drums with the same spectrum ( isospectral drums ) ?. Marc Kac (1914-1984).

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Adam Sawicki, Rami Band, Uzy Smilansky

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  1. Adam Sawicki,Rami Band, Uzy Smilansky Scattering from isospectral graphs

  2. ‘Can one hear the shape of a drum ?’ This question was asked by Marc Kac (1966). Is it possible to have two different drums with the same spectrum (isospectral drums) ? Marc Kac (1914-1984)

  3. The spectrum of a drum 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’: , … , , … ,

  4. Isospectral drums Gordon, Webb and Wolpert (1992): ‘One cannot hear the shape of a drum’ Using Sunada’s construction (1985)

  5. (a) (b) Isospectral drums – A transplantation proof Given an eigenfunction on drum (a), create an eigenfunction with the same eigenvalue on drum (b).

  6. (a) (b) Isospectral drums – A transplantation proof Given an eigenfunction on drum (a), create an eigenfunction with the same eigenvalue on drum (b).

  7. Isospectral drums – A transplantation proof We can use another basic building block

  8. Isospectral drums – A transplantation proof … or even a funny shaped building block …

  9. Isospectral drums – A transplantation proof … or cut it in a nasty way (and ruin the connectivity) …

  10. ‘Can one hear the shape of a drum ?’: revisited Okada, Shudo, Tasaki and Harayama conjecture (2005) that ‘One can distinguish isospectral drums by measuring their scattering poles’

  11. What’s next? • Scattering matrices of quantum graphs. • Isospectral construction. • Scattering from isospectral quantum graphs. • Back to drums.

  12. Quantum Graphs 5 1 L13 L45 L34 4 3 L46 L23 2 6 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 .

  13. Quantum Graphs 5 1 L13 L45 L34 4 3 L46 L23 2 6 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

  14. Quantum Graphs L13 L45 L34 L23 L46

  15. Quantum Graphs - Introduction 5 1 L13 L45 L34 4 3 L46 L23 2 6 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:

  16. Quantum Graphs – Attaching leads 5 1 L13 L45 L34 4 3 L46 L23 2 6

  17. Quantum Graphs – Attaching leads 5 1 L13 L45 L34 4 3 L46 L23 2 6

  18. Quantum Graphs – Attaching leads 5 1 L13 L45 L34 4 3 L46 L23 2 6 The poles of S(k) are below the real axis.

  19. N N D D Isospectral theorem 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. N D

  20. Transplantation N N D D The transplantation of our example is A A+B A-B N D B

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

  22. Transplantation & Scattering • We may apply the isospectral construction on graphs with leads: • The transplantation relatesthe function’s values on the leads: g f In addition, we have the scattering relations: In particular S1(k), S2(k)have the same pole structure.

  23. ‘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?

  24. The isospectral drums construction Buser, Conway, Doyle, Semmler (1994)

  25. AS, Rami Band, UzySmilansky Scattering from isospectral graphs

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