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Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal

Carolyn S. Gordon, Juan Pablo Rossetti (2003)

Annales de l'Institut Fourier

Let M be a 2 m -dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on m -forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the same Hodge spectrum on 1- forms, and hyperbolic surfaces, mutually isospectral on 1-forms, with different injectivity radii. The Hodge m -spectrum also does not distinguish orbifolds from manifolds....

Inverse spectral results on even dimensional tori

Carolyn S. Gordon, Pierre Guerini, Thomas Kappeler, David L. Webb (2008)

Annales de l’institut Fourier

Given a Hermitian line bundle L over a flat torus M , a connection on L , and a function Q on M , one associates a Schrödinger operator acting on sections of L ; its spectrum is denoted S p e c ( Q ; L , ) . Motivated by work of V. Guillemin in dimension two, we consider line bundles over tori of arbitrary even dimension with “translation invariant” connections , and we address the extent to which the spectrum S p e c ( Q ; L , ) determines the potential Q . With a genericity condition, we show that if the connection is invariant under...

Isospectral deformations of the Lagrangian Grassmannians

Jacques Gasqui, Hubert Goldschmidt (2007)

Annales de l’institut Fourier

We study the special Lagrangian Grassmannian S U ( n ) / S O ( n ) , with n 3 , and its reduced space, the reduced Lagrangian Grassmannian X . The latter is an irreducible symmetric space of rank n - 1 and is the quotient of the Grassmannian S U ( n ) / S O ( n ) under the action of a cyclic group of isometries of order n . The main result of this paper asserts that the symmetric space X possesses non-trivial infinitesimal isospectral deformations. Thus we obtain the first example of an irreducible symmetric space of arbitrary rank 2 , which is...

Non-Sunada graphs

Robert Brooks (1999)

Annales de l'institut Fourier

We consider the question of whether there is a converse to the Sunada Theorem in the context of k -regular graphs. We give a weak converse to the Sunada Theorem, which gives a necessary and sufficient condition for two graphs to be isospectral in terms of a Sunada-like condition, and show by example that a strong converse does not hold.

On G -sets and isospectrality

Ori Parzanchevski (2013)

Annales de l’institut Fourier

We study finite G -sets and their tensor product with Riemannian manifolds, and obtain results on isospectral quotients and covers. In particular, we show the following: If M is a compact connected Riemannian manifold (or orbifold) whose fundamental group has a finite non-cyclic quotient, then M has isospectral non-isometric covers.

On nonisometric isospectral connected fractal domains.

Brian D. Sleeman, Chen Hua (2000)

Revista Matemática Iberoamericana

A fundamental question raised by M. Kac in 1966 is: Must two isospectral planar domains necessarily be isometric? Following a long history of investigation C. Gordon, D. L. Webb and S. Wolpert in 1992 finally proved that the answer is no. By using the idea of transposition maps one can construct a wide class of planar domains with piecewise continuous boundaries which are isospectral but non-isometric. In this note we study the Kac question in relation to domains with fractal boundaries and by following...

Quantum Equivalent Magnetic Fields that Are Not Classically Equivalent

Carolyn Gordon, William Kirwin, Dorothee Schueth, David Webb (2010)

Annales de l’institut Fourier

We construct pairs of compact Kähler-Einstein manifolds ( M i , g i , ω i ) ( i = 1 , 2 ) of complex dimension n with the following properties: The canonical line bundle L i = n T * M i has Chern class [ ω i / 2 π ] , and for each positive integer k the tensor powers L 1 k and L 2 k are isospectral for the bundle Laplacian associated with the canonical connection, while M 1 and M 2 – and hence T * M 1 and T * M 2 – are not homeomorphic. In the context of geometric quantization, we interpret these examples as magnetic fields which are quantum equivalent but not classically equivalent....

Recovering quantum graphs from their Bloch spectrum

Ralf Rueckriemen (2013)

Annales de l’institut Fourier

We define the Bloch spectrum of a quantum graph to be the map that assigns to each element in the deRham cohomology the spectrum of an associated magnetic Schrödinger operator. We show that the Bloch spectrum determines the Albanese torus, the block structure and the planarity of the graph. It determines a geometric dual of a planar graph. This enables us to show that the Bloch spectrum indentifies and completely determines planar 3 -connected quantum graphs.

Spectral theory of translation surfaces : A short introduction

Luc Hillairet (2009/2010)

Séminaire de théorie spectrale et géométrie

We define translation surfaces and, on these, the Laplace operator that is associated with the Euclidean (singular) metric. This Laplace operator is not essentially self-adjoint and we recall how self-adjoint extensions are chosen. There are essentially two geometrical self-adjoint extensions and we show that they actually share the same spectrum

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