Linear Representations and Isospectrality with Boundary Conditions

Parzanchevski, Ori; Band, Ram

doi:10.1007/s12220-009-9115-6

Cited by 45 publications

(112 citation statements)

References 26 publications

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“…To prove the existence of the degeneracy in the spectrum (Lemma 4.3) we identify the 2-dimensional (co)representation responsible for it and describe the subspace of the Hilbert space that carries this representation. We also relate our results to the proofs of isospectrality, in particular the isospectrality condition of Band-Parzanchevski-Ben-Shach [5,33].…”

Section: Introductionmentioning

confidence: 86%

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Symmetry and Dirac points in graphene spectrum

Berkolaiko

Comech

2018

J. Spectr. Theory

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Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by 2π/3 and inversion, rotation by 2π/3 and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by 2π/3 and vertical reflection.All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by 2π/3. Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.

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Section: Introductionmentioning

confidence: 86%

“…It can also be checked using an algebraic condition of Band, Parzanchevski and Ben-Shach, see [5,Cor. 4.4] or [33,Cor. 4].…”

Section: 3mentioning

confidence: 99%

Symmetry and Dirac points in graphene spectrum

Berkolaiko

Comech

2018

J. Spectr. Theory

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“…Their result gives a room for existence of graphs with different metric and topological properties but the same spectrum. Up to now there is only one method of construction of isospectral graphs [8,9] where the authors extended the well known Sunada's approach. The method is based on the elements of representation theory and its direct corollary ensures the existence of transplantation between isospectral graphs.…”

mentioning

confidence: 99%

Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?

et al. 2012

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The famous question of Kac "can one hear the shape of a drum?" addressing the unique connection between the shape of a planar region and the spectrum of the corresponding Laplace operator, can be legitimately extended to scattering systems. In the modified version, one asks whether the geometry of a vibrating system can be determined by scattering experiments. We present the first experimental approach to this problem in the case of microwave graphs (networks) simulating quantum graphs. Our experimental results strongly indicate a negative answer. To demonstrate this we consider scattering from a pair of isospectral microwave networks consisting of vertices connected by microwave coaxial cables and extended to scattering systems by connecting leads to infinity to form isoscattering networks. We show that the amplitudes and phases of the determinants of the scattering matrices of such networks are the same within the experimental uncertainties. Furthermore, we demonstrate that the scattering matrices of the networks are conjugated by the so-called transplantation relation.

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“…In particular, they prove that all examples produced in terms of their method posses a transplantation. Later on, Herbrich [8] showed the converse: domains which are transplantable can also be constructed using the isospectral theory in [1,14]. Herbrich Corollary 9.…”

Section: Heat Content and Planar Domainsmentioning

confidence: 99%

Isospectrality and heat content

Berg

Dryden

Kappeler

2014

Bulletin of the London Mathematical Society

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We present examples of isospectral operators that do not have the same heat content. Several of these examples are planar polygons that are isospectral for the Laplace operator with Dirichlet boundary conditions. These include examples with infinitely many components. Other planar examples have mixed Dirichlet and Neumann boundary conditions. We also consider Schrödinger operators acting in L 2 [0, 1] with Dirichlet boundary conditions, and show that an abundance of isospectral deformations do not preserve the heat content.Mathematics Subject Classification (2010): 58J53; 58J50; 35P10.

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Linear Representations and Isospectrality with Boundary Conditions

Cited by 45 publications

References 26 publications

Symmetry and Dirac points in graphene spectrum

Symmetry and Dirac points in graphene spectrum

Are Scattering Properties of Graphs Uniquely Connected to Their Shapes?

Isospectrality and heat content

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