Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

Jakobson, Dmitry; Polterovich, Iosif

doi:10.1090/s1079-6762-05-00149-6

Cited by 6 publications

(4 citation statements)

References 31 publications

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“…, as λ → +∞, which appears to be an almost impossibly strong condition (cf. [18], [34]). This is because the curve γ opt , which is the boundary of the region in the problem raised in [11] corresponds to ε(λ) = c/λ, 1 Recall that a Zoll manifold is one for which the geodesic flow is periodic with a common minimal period .…”

Section: Introductionmentioning

confidence: 99%

On L p -Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

Bourgain

Shao

Sogge

et al. 2014

Commun. Math. Phys.

173

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We address an interesting question raised by Dos Santos Ferreira, Kenig and Salo [11] about regions Rg ⊂ C for which there can be uniform L 2n n+2 → L 2n n−2 resolvent estimates for ∆g + ζ, ζ ∈ Rg, where ∆g is the Laplace-Beltrami operator with metric g on a given compact boundaryless Riemannian manifold of dimension n ≥ 3. This is related to earlier work of Kenig, Ruiz and the third author [19] for the Euclidean Laplacian, in which case the region is the entire complex plane minus any disc centered at the origin. Presently, we show that for the round metric on the sphere, S n , the resolvent estimates in [11], involving a much smaller region, are essentially optimal. We do this by establishing sharp bounds based on the distance from ζ to the spectrum of ∆ S n . In the other direction, we also show that the bounds in [11] can be sharpened logarithmically for manifolds with nonpositive curvature, and by powers in the case of the torus, T n = R n /Z n , with the flat metric. The latter improves earlier bounds of Shen [22]. The work of [11] and [22] was based on Hadamard parametrices for (∆g + ζ) −1 . Ours is based on the related Hadamard parametrices for cos t −∆g, and it follows ideas in [26] of proving L p -multiplier estimates using small-time wave equation parametrices and the spectral projection estimates from [25]. This approach allows us to adapt arguments in Bérard [3] and Hlawka [15] to obtain the aforementioned improvements over [11] and [22]. Further improvements for the torus are obtained using recent techniques of the first author [5] and his work with Guth [7] based on the multilinear estimates of Bennett, Carbery and Tao [2]. Our approach also allows us to give a natural necessary condition for favorable resolvent estimates that is based on a measurement of the density of the spectrum of −∆g, and, moreover, a necessary and sufficient condition based on natural improved spectral projection estimates for shrinking intervals, as opposed to those in [25] for unit-length intervals. We show that the resolvent estimates are sensitive to clustering within the spectrum, which is not surprising given Sommerfeld's original conjecture [32] about these operators.

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Section: Introductionmentioning

confidence: 99%

On L p -Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

Bourgain

Shao

Sogge

et al. 2014

Commun. Math. Phys.

173

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“…Ideally, one would want to be able to use a variant of (3.5) where the exponential factor is not present for the second term in the right. Lower bounds of Jakobson and Polterovich [20]- [21] show that this error term cannot be O(λ n−1 2), but their bounds do not rule out some improvement over (3.5), which would lead to more favorable estimates.…”

Section: Discussionmentioning

confidence: 99%

Improved critical eigenfunction estimates on manifolds of nonpositive curvature

Sogge

2017

Mathematical Research Letters

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“…In each cyclic subspace, the function is a complete unitary invariant for U restricted to ( ): by this we mean that the function encodes all the spectral data coming from the vectors = U , ∈ Z. For background literature on spectral function and their applications we refer to [1,10,16,[19][20][21]. In summary, the spectral representation theorem is the assertion that commuting unitary operators in Hilbert space may be represented as multiplication operators in an L 2 -Hilbert space.…”

Section: Introductionmentioning

confidence: 99%

“…In each cyclic subspace, the function p ϕ is a complete unitary invariant for U restricted to H(ϕ): by this we mean that the function p ϕ encodes all the spectral data coming from the vectors f k = U k ϕ, k ∈ Z. For background literature on spectral function and their applications we refer to [1,10,16,19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Spectral theory of discrete processes

Jørgensen

Song

2010

Open Physics

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Abstract:We offer a spectral analysis for a class of transfer operators. These transfer operators arise for a wide range of stochastic processes, ranging from random walks on infinite graphs to the processes that govern signals and recursive wavelet algorithms; even spectral theory for fractal measures. In each case, there is an associated class of harmonic functions which we study. And in addition, we study three questions in depth:In specific applications, and for a specific stochastic process, how do we realize the transfer operator T as an operator in a suitable Hilbert space? And how to spectral analyze T once the right Hilbert space has been selected? Finally we characterize the stochastic processes that are governed by a single transfer operator.In our applications, the particular stochastic process will live on an infinite path-space which is realized in turn on a state space S. In the case of random walk on graphs G, S will be the set of vertices of G. The Hilbert space on which the transfer operator T acts will then be an L 2 space on S, or a Hilbert space defined from an energy-quadratic form. This circle of problems is both interesting and non-trivial as it turns out that T may often be an unbounded linear operator in ; but even if it is bounded, it is a non-normal operator, so its spectral theory is not amenable to an analysis with the use of von Neumann's spectral theorem. While we offer a number of applications, we believe that our spectral analysis will have intrinsic interest for the theory of operators in Hilbert space.

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Lower bounds for the spectral function and for the remainder in local Weyl’s law on manifolds

Cited by 6 publications

References 31 publications

On L p -Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

On L p -Resolvent Estimates and the Density of Eigenvalues for Compact Riemannian Manifolds

Improved critical eigenfunction estimates on manifolds of nonpositive curvature

Spectral theory of discrete processes

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