Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

Sogge, Christopher D.; Zelditch, Steve

doi:10.1080/03605302.2017.1349147

Cited by 4 publications

(8 citation statements)

References 38 publications

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“…Theorem 1.8 is recent joint work with X. Han, A. Hassell and H. Hezari [HHHZ13]. It also uses calculations in recent work [HZ12] with H. Hezari.As mentioned above, Theorem 1.4 is joint work with C. D. Sogge [SZ14]. The boundary quantum ergodicity theorem and boundary local Weyl law is joint work with A. Hassell [HZ04] and with H. Christianson and J. Toth [CTZ13].…”

Section: 2mentioning

confidence: 94%

“…The ergodicity condition (i) is known to be satisfied for a non-positively curved surface with concave boundary [KSS89] (see §2). Moreover, in [SZ14] the second condition (ii) is proved for such surfaces, among many other cases. Using the Melrose-Taylor diffractive parametrix on manifolds with concave boundary, the following is proved: The term boundary self-focal point is a dynamical condition on the billiard flow Φ t of (M, g, ∂M ), i.e.…”

Section: Introductionmentioning

confidence: 93%

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Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

Jung

Zelditch

2015

Math. Ann.

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Abstract.It is an open problem in general to prove that there exists a sequence of ∆g-eigenfunctions ϕj k on a Riemannian manifold (M, g) for which the number N (ϕj k ) of nodal domains tends to infinity with the eigenvalue. Our main result is that N (ϕj k ) → ∞ along a subsequence of eigenvalues of density 1 if the (M, g) is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any symmetries.

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Section: 2mentioning

confidence: 94%

Section: Introductionmentioning

confidence: 93%

Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

Jung

Zelditch

2015

Math. Ann.

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“…In [14] Blair obtained estimates for the restriction of eigenfunctions to hypersurfaces in the low regularity setting (which includes the boundary case). Extreme concentration of eigenfunctions on manifolds was studied in [50,49]. See also the related work on Strichartz estimates on manifolds with boundary by Blair, Smith and Sogge [12,13].…”

Section: Analytic Techniquesmentioning

confidence: 99%

Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues

Barnett¹,

Hassell²,

Tacy³

2018

Duke Math. J.

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For smooth bounded domains in R n , we prove upper and lower L 2 bounds on the boundary data of Neumann eigenfunctions, and prove quasiorthogonality of this boundary data in a spectral window. The bounds are tight in the sense that both are independent of eigenvalue; this is achieved by working with an appropriate norm for boundary functions, which includes a 'spectral weight', that is, a function of the boundary Laplacian. This spectral weight is chosen to cancel concentration at the boundary that can happen for 'whispering gallery' type eigenfunctions. These bounds are closely related to wave equation estimates due to Tataru.Using this, we bound the distance from an arbitrary Helmholtz parameter E > 0 to the nearest Neumann eigenvalue, in terms of boundary normalderivative data of a trial function u solving the Helmholtz equation (∆−E)u = 0. This 'inclusion bound' improves over previously known bounds by a factor of E 5/6 . It is analogous to a recently improved inclusion bound in the Dirichlet case, due to the first two authors.Finally, we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of Neumann eigenpairs on smooth planar domains. We show that the new inclusion bound improves the relative accuracy in a computed Neumann eigenvalue (around the 42000th) from 9 digits to 14 digits, with little extra effort.

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“…Hence given an ONB of eigenfunctions of ∆ g on such a manifold, we can find a full density subsequence for which (a) holds. Condition (c), with Σ = ∂X, was proved in [SoZe14] for nonpositively curved manifolds with smooth concave boundary, but in the more general case of ergodic manifolds with piecewise smooth boundary we can use our Theorem 1.5. In the case of manifolds with smooth boundary, condition (b) follows from:…”

Section: Improved Supnorms and The Number Of Nodal Domainsmentioning

confidence: 99%

“…It is known (see [Si70], [ChSi87], [BuChSi90]) that such billiard tables are ergodic. the boundary values of eigenfunctions, the so called "Kuznecov sum formula" for manifolds with smooth boundary [HHHZ15], and sup norm estimates of size o(λ 1/4 ) for the boundary values of eigenfunctions on positively curved manifolds with smooth concave boundary [SoZe14]. For us to prove the above theorem, the first two ingredients are still available except that our boundary can have singular points.…”

Section: Introductionmentioning

confidence: 99%

Quantum Ergodicity and L p Norms of Restrictions of Eigenfunctions

Hezari

2017

Commun. Math. Phys.

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We prove an analogue of Sogge's local L p estimates [So16] for L p norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov [BuGeTz07], Hu [Hu09], and Chen-Sogge [ChSo14]. The improvements are logarithmic on negatively curved manifolds (without boundary) and by o(1) for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get o(1) improvements on L ∞ estimates of Cauchy data away from a shrinking neighborhood of the corners, and as a result using the methods of [GhReSa13, JuZe16a, JuZe16b], we get that the number of nodal domains of 2-dimensional ergodic billiards tends to infinity as λ → ∞. These results work only for a full density subsequence of any given orthonormal basis of eigenfunctions.We also present an extension of the L p estimates of [BuGeTz07, Hu09, ChSo14] for the restrictions of Dirichlet and Neumann eigenfunctions to compact submanifolds of the interior of manifolds with piecewise smooth boundary. This part does not assume ergodicity on the manifolds. 1 See [LuSa95, Yo16, HeRi16b, LeRu16] for parallel results in the arithmetic setting.

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Sup norms of Cauchy data of eigenfunctions on manifolds with concave boundary

Cited by 4 publications

References 38 publications

Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

Number of nodal domains of eigenfunctions on non-positively curved surfaces with concave boundary

Comparable upper and lower bounds for boundary values of Neumann eigenfunctions and tight inclusion of eigenvalues

Quantum Ergodicity and L p Norms of Restrictions of Eigenfunctions

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