Isospectral

This doesn't however reduce completely the interest of the concept, since we can have an isospectral family of matrices of shape A(t) = M(t)−1AM(t) depending on a parameter t in a complicated way.After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon).In particular Vignéras (1980), based on the Selberg trace formula for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory.Sunada noticed that the method of constructing number fields with the same Dedekind zeta function could be adapted to compact manifolds.It led C. Gordon, D. Webb and S. Wolpert to the discovery in 1991 of a counter example to Mark Kac's problem "Can one hear the shape of a drum?"

mathematicslinear operatorsspectrumeigenvaluesmultiplicitymatricescompact operatorHilbert spaceBanach spaceLaplace operatorhearing the shape of a drumcomplexdiagonalizable matricessimilarityparametersolitoninfinitesimalLax pairsPeter LaxRiemannian manifoldsLaplace–Beltrami operatorJohn MilnorErnst WittSelberg trace formulaclass field theoryToshikazu Sunadacovering spaceSelberg zeta functionDedekind zeta functionfinite groupdeck transformationsD. WebbMark KacCan one hear the shape of a drum?Alan ReidSpectral geometryBrooks, RobertBuser, PeterConway, JohnInternational Mathematics Research NoticesBibcodeFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedCompactHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraC*-algebraSpectrum of a C*-algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theory