What one cannot hear? On drums which sound the same Rami Band, Ori Parzanchevski, Gilad Ben-Shach

‘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) What did Kac mean – Can one deduce the shape from the spectrum? Emphasize the importance of inverse problems in physics. Or phrasing this differently: Do we have isospectral graphs ? 2

, … , , … , The spectrum of a drum 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’: In Quantum mechanics we can write the famous Schroedinger equation which reduces to this equation and give as the energies of a free particle in a bounded region. The shape of the drum is exactly the shape of that region. It’s very similar to what happens in an atom, but there we solve the more complicated Schroedinger eq. in 3d. Still we get a discrete spectrum of energies (this time energies, not frequencies). It’s important to remember that the problems we treat here have quantum mechanical meaning, and not only meaning for playing drums. For both cases we solve the time-independent wave equation – namely, finding the eigenvalues of the Laplacian. , … , , … ,

Isospectral drums Gordon, Webb and Wolpert (1992): ‘One cannot hear the shape of a drum’ The answer was given only in 1992 when 3 mathematicians came up with examples of isospectral drums. Two of them (which are also married) are seen in this picture, proudly presenting their drums. Note that the drums are different! Pay attention to their T-shirts – you can see there another version of this isospectral drums. We’ll soon see this pair of drums again. The existence of such a pair shows that Kac’s question has to be answered by simple “no” – ‘One cannot hear the shape of a drum’. Now arises the question – what additional information do we need to know in order to determine uniquely the shape of a drum ? That was when mathematicians Carolyn S. Gordon and David L. Webb, then at Washington University in St. Louis, and Scott Wolpert of the University of Maryland at College Park came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra. 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) We will present a simple proof for the isospectrality of these drums – the transplantation method. Note – here explain carefully and slowely ! Demonstrate it on the triangles A, B, G: 1) dashed boundary 2) dotted boundary 3) solid inner line – here we also need to examine A, C, E.

Isospectral drums A paper-folding proof (S.J. Chapman – 2000) We will present a simple proof for the isospectrality of these drums – the transplantation method. Note – here explain carefully and slowely ! Demonstrate it on the triangles A, B, G: 1) dashed boundary 2) dotted boundary 3) solid inner line – here we also need to examine A, C, E.

Isospectral drums – A transplantation proof We can use another basic building block The answer was given only in 1992 when 3 mathematicians came up with examples of isospectral drums. Two of them (which are also married) are seen in this picture, proudly presenting their drums. Note that the drums are different! Pay attention to their T-shirts – you can see there another version of this isospectral drums. We’ll soon see this pair of drums again. The existence of such a pair shows that Kac’s question has to be answered by simple “no” – ‘One cannot hear the shape of a drum’. Now arises the question – what additional information do we need to know in order to determine uniquely the shape of a drum ? That was when mathematicians Carolyn S. Gordon and David L. Webb, then at Washington University in St. Louis, and Scott Wolpert of the University of Maryland at College Park came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra.

Isospectral drums – A transplantation proof … or a building block which is not a triangle … The answer was given only in 1992 when 3 mathematicians came up with examples of isospectral drums. Two of them (which are also married) are seen in this picture, proudly presenting their drums. Note that the drums are different! Pay attention to their T-shirts – you can see there another version of this isospectral drums. We’ll soon see this pair of drums again. The existence of such a pair shows that Kac’s question has to be answered by simple “no” – ‘One cannot hear the shape of a drum’. Now arises the question – what additional information do we need to know in order to determine uniquely the shape of a drum ? That was when mathematicians Carolyn S. Gordon and David L. Webb, then at Washington University in St. Louis, and Scott Wolpert of the University of Maryland at College Park came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra.

Isospectral drums – A transplantation proof … or even a funny shaped building block … The answer was given only in 1992 when 3 mathematicians came up with examples of isospectral drums. Two of them (which are also married) are seen in this picture, proudly presenting their drums. Note that the drums are different! Pay attention to their T-shirts – you can see there another version of this isospectral drums. We’ll soon see this pair of drums again. The existence of such a pair shows that Kac’s question has to be answered by simple “no” – ‘One cannot hear the shape of a drum’. Now arises the question – what additional information do we need to know in order to determine uniquely the shape of a drum ? That was when mathematicians Carolyn S. Gordon and David L. Webb, then at Washington University in St. Louis, and Scott Wolpert of the University of Maryland at College Park came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra.

Isospectral drums – A transplantation proof … or cut it in a nasty way (and ruin the connectivity) … The answer was given only in 1992 when 3 mathematicians came up with examples of isospectral drums. Two of them (which are also married) are seen in this picture, proudly presenting their drums. Note that the drums are different! Pay attention to their T-shirts – you can see there another version of this isospectral drums. We’ll soon see this pair of drums again. The existence of such a pair shows that Kac’s question has to be answered by simple “no” – ‘One cannot hear the shape of a drum’. Now arises the question – what additional information do we need to know in order to determine uniquely the shape of a drum ? That was when mathematicians Carolyn S. Gordon and David L. Webb, then at Washington University in St. Louis, and Scott Wolpert of the University of Maryland at College Park came up with two drums that have equal areas and perimeters but different geometric shapes. They proved that the drums, each a multisided polygon, display identical spectra.

‘Can one hear the shape of a pore ?’ a drum?’ a graph?’ a violin ?’ the universe ?’ a dataset ?’ a network ?’ a molecule?’ your throat ?’ a black hole?’ an electrode ?’ Many examples of isospectral objects (not only drums): Milnor (1964) 16-dim Tori Buser (1986) & Berard (1992) Transplantation Gordon, Web, Wolpert (1992) Drums Buser, Conway, Doyle, Semmler (1994) More Drums Brooks (1988,1999) Manifolds and Discrete Graphs Gutkin, Smilansky (2001) Quantum Graphs Gordon, Perry, Schueth (2005) Manifolds 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. This is a good place to stop and give a motivation. Why do we need to find more isospectral objects if such objects were already found in the past. Mention that isospectrality was explored with regard to various objects. One of the main researchers in this field was Robert Brooks from the Technion.

Isospectral theorem Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach) Let Γ be a drum which obeys a symmetry group G. Let H1, H2 be two subgroups of G with representations R1, R2 that satisfy then the drums , are isospectral.

Groups & Drums Example: The Dihedral group – the symmetry group of the square. G = { id , a , a2 , a3 , rx , ry , ru , rv } How does the Dihedral group act on the square drum? rx a id y u v x Say that we can treat this square as a quantum graph. This helps later in slide 21. Two subgroups of the Dihedral group: H1 = { id , a2 , rx , ry } H2 = { id , a2 , ru , rv }

Groups - Representations 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 rep. S1: Example 2 - G has the following 2-dimensional rep. S2: Restriction: is the following rep. of H1: Induction: is the following rep. of G :

Isospectral theorem Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach) Let Γ be a drum which obeys a symmetry group G. Let H1, H2 be two subgroups of G with representations R1, R2 that satisfy then the drums , are isospectral. An application of the theorem with: y u v Two subgroups of G: x We choose representations such that

Constructing Quotient Graphs 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. Dirichlet Neumann The construction of a quotient drum is motivated by an encoding scheme. Say that we discuss linear representations – representations of the gorup. Add that F belongs to the isotypical part \Phi etc.and remove the word basis. Say that H_1 is isomorphic to Z_2 x Z_2 and show all the irreps of it (then it is easier to choose one of them speak about its isotypical part) Put the isomorphism (below) as an equation and give the meaning of it in terms of dimensions. Add the fact that the inductions are equal. Note that during this process we have built a new graph which we call a quotient graph (with respect to a representation). There is an isomorphism between functions on the quotient graph to functions on the complete graph which transform with respect to this certain matrix representation. Emphasize the fact that the quotient graphs encode the information of the functions that transform with respect to this certain matrix representation.

Constructing Quotient Graphs 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. 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. Add that F belongs to the isotypical part \Phi etc.and remove the word basis. Say that H_1 is isomorphic to Z_2 x Z_2 and show all the irreps of it (then it is easier to choose one of them speak about its isotypical part) Put the isomorphism (below) as an equation and give the meaning of it in terms of dimensions. Add the fact that the inductions are equal. Note that during this process we have built a new graph which we call a quotient graph (with respect to a representation). There is an isomorphism between functions on the quotient graph to functions on the complete graph which transform with respect to this certain matrix representation. Emphasize the fact that the quotient graphs encode the information of the functions that transform with respect to this certain matrix representation. Neumann Dirichlet

Isospectral theorem Theorem (R.B., Ori Parzanchevski, Gilad Ben-Shach) Let Γ be a drum which obeys a symmetry group G. Let H1, H2 be two subgroups of G with representations R1, R2 that satisfy then the drums , are isospectral. The drums , constructed according to the conditions above, possess a transplantation. Remarks: The isospectral theorem is applicable not only for drums, but for general manifolds, graphs, etc. The following isospectral example is a courtesy of Martin Sieber.

Transplantation The transplantation of our example is A A+B A-B B We present the transplantation pictuarially on this slide:

Extending the Isospectral pair Extending our example: H1 = { e , a2, rx , ry} R1: H2 = { e , a2, ru , rv} R2: H3 = { e , a, a2, a3} R3: ×i Turn \Psi into F !!! Maybe identify these points and put the A_V, B_V matrices. Denote them by \Gamma / R_3. And to color the FD of R_3 and pink and in the end say why there is a freedom to choose the FD in the 3rd case, but not in the first two cases !!! And to say that the condition (not stated) applies also to the rep R_3

Extending the Isospectral pair Extending our example: H1 = { e , a2, rx , ry} R1: H2 = { e , a2, ru , rv} R2: H3 = { e , a, a2, a3} R3: ×i Turn \Psi into F !!! Maybe identify these points and put the A_V, B_V matrices. Denote them by \Gamma / R_3. And to color the FD of R_3 and pink and in the end say why there is a freedom to choose the FD in the 3rd case, but not in the first two cases !!! And to say that the condition (not stated) applies also to the rep R_3

A few 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) This isospectral quartet can be obtained when acting with the group D4xD4 on the following torus: Here all the boundary conditions are of Kirchhoff type !

A few isospectral examples ‘One cannot hear the shape of a drum’ Gordon, Webb and Wolpert (1992) We construct the known isospectral drums of Gordon et al. but with new boundary conditions: Here all the boundary conditions are of Kirchhoff type ! D D N N

A few isospectral examples – quantum graphs 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. The resulting quotient graphs are: Here all the boundary conditions are of Kirchhoff type ! L2 L3 L2 L2 L3 L1 L2 L3 L3 L2 L2 L3 L1 L1 L2 L2 L3 L3 L2 L1 L3 L1 L1 L2 L3 L3 L2 L3 L2 L3

The relation to Sunada’s construction Let G be a group. Let H1, H2 be two subgroups of G. Then the triple (G, H1, H2) satisfies Sunada’s condition if: where is the cunjugacy class of g in G. By Sunada’s theorem we get that for such a triple (G, H1, H2), , are isospectral. Pesce (94) proved Sunada’s theorem using the observation that Sunada’s condition is equivalent to the following: The relation to the construction method presented so far is via the identification: The formalism introduced here is a general formalism to write There is also a condition on the matrices Av, Bv due to Kostrykin & Schrader which is necessary and sufficient in order for our operator (second derivative) to be self-adjoint.

What one cannot hear? On drums 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).