<i>C</i><sup>∞</sup>spectral rigidity of the ellipse

Hezari, Hamid; Zelditch, Steve

doi:10.2140/apde.2012.5.1105

Cited by 32 publications

(73 citation statements)

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“…[39]) will be constant along the deformation. Using [44] one could generalize the results of Hezari and Zelditch [23] for Liouville billiard tables in dimensions 2 and 3. The corresponding results will appear in [45].…”

Section: Further Remarksmentioning

confidence: 69%

Invariants of Isospectral Deformations and Spectral Rigidity

Popov

Topalov

2012

Communications in Partial Differential Equations

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We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace-Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on involving the function in the Robin boundary conditions remain constant under weak isospectral deformations.To this end we construct continuous families of quasimodes associated with . We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to . As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a /2 2 group of symmetries.

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Section: Further Remarksmentioning

confidence: 69%

Invariants of Isospectral Deformations and Spectral Rigidity

Popov

Topalov

2012

Communications in Partial Differential Equations

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“…Such formulas have a long history, going back to Rayleigh [R45] and Hadamard [H08] and are still being developed (see [F92,F81,FO78,G86,GS53,G10,HZ,IKK77,K06,P80] for further results and references).…”

Section: Domain Variation Formulasmentioning

confidence: 99%

Critical Partitions and Nodal Deficiency of Billiard Eigenfunctions

2012

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The paper addresses the nodal count (i.e., the number of nodal domains) for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. The classical Sturm theorem states that in dimension one, the nodal and eigenfunction counts coincide: the nth eigenfunction partitions the interval into n nodal domains. The Courant Nodal Theorem claims that in any dimension, the number of nodal domains ν n of the nth eigenfunction cannot exceed n. However, it follows from an asymptotically stronger upper bound by Pleijel that in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, in most cases a "nodal deficiency" d n = n − ν n arises. One can say that the nature of the nodal deficiency has not been understood. It was suggested in recent years that, rather than starting with eigenfunctions, one can look at partitions of the domain into ν sub-domains, asking which partitions can correspond to eigenfunctions, and what would be the corresponding deficiency. To this end one defines an "energy" of a partition, for example, the maximum of the ground state energies of the subdomains. One notices that if a partition does correspond to an eigenfunction, then the ground state energies of all the nodal domains are the same, i.e., it is an equipartition. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) partitions minimizing the energy functional correspond to the "Courant sharp" eigenfunctions, i.e. to those with zero nodal deficiency. In this paper it is shown that it is beneficial to restrict the domain of the functional to the equipartition, where it becomes smooth. Then, under some genericity conditions, the nodal partitions correspond exactly to the critical points of the functional. Moreover, the nodal deficiency turns out to be equal to the Morse index at the corresponding critical point. This explains, in particular, why the minimal partitions must be Courant sharp.

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“…In [HZ12], the authors microlocalize the wave kernels near periodic orbits and calculate the principal symbol of the composition (13) for Dirichlet and Neumann boundary conditions using the symbol calculus in [DG75]. In contrast to the methods employed in [HZ12], we instead take a more direct approach which avoids the an application of the trace formula in [DG75].…”

Section: A Parametrix For S R and Singularity Expansionmentioning

confidence: 99%

“…Proof. As in [Cha76] and [HZ12], denote by σ 0 the symbol of the restriction to t = 0, σ r the symbol of the boundary restriction operator, and σ B the symbol of rN − K ∈ Ψ 1 . Here, N is an extension of the unit normal vector field on ∂Ω to a tubular neighborhood of the boundary.…”

Section: 2mentioning

confidence: 99%

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Robin Spectral Rigidity of the Ellipse

Vig

2020

J Geom Anal

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In this paper, we investigate C 1 isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. Reparameterizing allows us to show that such smooth deformations must be flat at ε = 0. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard's variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we in fact construct an explicit parametrix for the wave propagator in the interior, microlocally near orbits of rotation number 1/j.

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C^∞spectral rigidity of the ellipse

Abstract: -3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices.APDE peer review and production are managed by EditFLOW ™ from Mathematical Sciences Publishers.PUBLISHED BY mathematical sciences publishers

Cited by 32 publications

References 34 publications

Invariants of Isospectral Deformations and Spectral Rigidity

Invariants of Isospectral Deformations and Spectral Rigidity

Critical Partitions and Nodal Deficiency of Billiard Eigenfunctions

Robin Spectral Rigidity of the Ellipse

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C∞spectral rigidity of the ellipse

Abstract: -3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices.APDE peer review and production are managed by EditFLOW ™ from Mathematical Sciences Publishers.PUBLISHED BY mathematical sciences publishers

Cited by 32 publications

References 34 publications

Invariants of Isospectral Deformations and Spectral Rigidity

Invariants of Isospectral Deformations and Spectral Rigidity

Critical Partitions and Nodal Deficiency of Billiard Eigenfunctions

Robin Spectral Rigidity of the Ellipse

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C^∞spectral rigidity of the ellipse