In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.
The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices.
In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the shape of a drum.
In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have...
wave equation in two isospectral drums Isospectral Drums by Toby Driscoll at the University of Delaware Some planar isospectral domains by Peter Buser...
number. Two graphs are called cospectral or isospectral if the adjacency matrices of the graphs are isospectral, that is, if the adjacency matrices have...
eigenvalues is not enough to determine the isometry class of a manifold (see isospectral). A general and systematic method due to Toshikazu Sunada gave rise to...
spectrum of L are independent of t. The matrices/operators L are said to be isospectral as t {\displaystyle t} varies. The core observation is that the matrices...
one of only three bipartite convex enneahedra. The smallest pair of isospectral polyhedral graphs are enneahedra with eight vertices each. Slicing a...
are almost always large. Arithmetic groups can be used to construct isospectral manifolds. This was first realised by Marie-France Vignéras and numerous...
eigenvalues but not be isomorphic. Such linear operators are said to be isospectral. If A is the adjacency matrix of the directed or undirected graph G,...
pointed out by M. F. Vignéras and used by her to construct examples of isospectral compact hyperbolic surfaces. The precise statement is as follows: If...
are consequently examples of compact Riemannian manifolds which are isospectral but not isometric (see Hearing the shape of a drum.) III. The modular...
linear graphs to this central vertex. Two finite starlike trees are isospectral, i.e. their graph Laplacians have the same spectra, if and only if they...
by a result of Buser, the length spectrum of a Riemann surface is an isospectral invariant, essentially by the trace formula. A compact hyperbolic surface...
isomorphic, including arithmetically equivalent number fields and isospectral graphs and isospectral Riemannian manifolds. The simple group G = SL3(F2) of order...
a Dirac operator on a Riemannian manifold with a spin structure. The isospectral problem for the Dirac spectrum asks whether two Riemannian spin manifolds...
Peter Lax famously interpreted the inverse scattering method in terms of isospectral deformations and Lax pairs. The inverse scattering method has had an...
numerous contributions, the most famous one is a general construction of isospectral manifolds (1985), which is based on his geometric model of number theory...
Russian-American mathematical analyst Carolyn S. Gordon (born 1950), isospectral geometer who proved that you can't hear the shape of a drum Julia Gordon...
positivity conjecture and the n! conjecture involved showing that the isospectral Hilbert scheme of n points in a plane was Cohen–Macaulay (and even Gorenstein)...
Together with Ben Webb Bunimovich introduced and developed the theory of Isospectral transformations for analysis of multi-dimensional systems and networks...
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