Small scale quantum ergodicity in negatively curved manifolds

Han, Xiaolong

doi:10.1088/0951-7715/28/9/3263

Cited by 35 publications

(65 citation statements)

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“…is the Landau-Ramanujan constant. In three-dimensions, n ∈ N 3 if and only if in the representation n = 4 a n 1 with 4 ∤ n 1 , the number n 1 satisfies n 1 ≡ 7 (8) . Moreover, as X → ∞,…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

Small Scale Equidistribution for a Point Scatterer on the Torus

Yesha

2020

Commun. Math. Phys.

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We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two-and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the "new" eigenfunctions holding all the way down to the Planck scale. In three dimensions, small scale equidistribution is established for all of the "new" eigenfunctions at certain scales.

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Section: Statement Of the Main Resultsmentioning

confidence: 99%

Small Scale Equidistribution for a Point Scatterer on the Torus

Yesha

2020

Commun. Math. Phys.

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“…This is the small-scale approach, see [5]. By choosing T to be the projection onto a linear subspace V of L 2 (M ) and then taking into account the equality…”

Section: Summary Of Main Resultsmentioning

confidence: 99%

“…(4.43) 5 The larger symbol class S δ (1 R ) would also do. However, as the diameter of the support of ̺ h can be assumed to be bounded away from 0, the symbol class S(1 R ) is more natural.…”

Section: 3mentioning

confidence: 99%

On the semiclassical functional calculus for h-dependent functions

Küster

2017

Ann Glob Anal Geom

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Abstract. We study the functional calculus for operators of the form f h (P (h)) within the theory of semiclassical pseudodifferential operators, where {f h } h∈(0,1] ⊂ C ∞ c (R) denotes a family of hdependent functions satisfying some regularity conditions, and P (h) is either an appropriate selfadjoint semiclassical pseudodifferential operator in L 2 (R n ) or a Schrödinger operator in L 2 (M ), M being a closed Riemannian manifold of dimension n. The main result is an explicit semiclassical trace formula with remainder estimate that is well-suited for studying the spectrum of P (h) in spectral windows of width of order h δ , where 0 ≤ δ < 1 2 .

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“…In general, this local frequency function is most useful a harmonic functions and is a local measure of its 'degree' as a polynomial like function on B r (a) and controls the local growth rate of u. Some expositions of frequency functions and their applications can be found in [H,Ku95] He denotes N L by β, but we use that notation below for the doubling exponent. Frequency functions may also be defined for eigenfunctions.…”

Section: Introductionmentioning

confidence: 99%

“…In model cases on R n it may be expressed in terms of Bessel functions (see [Zel08]). The general theorem (Theorem 2.3 of [GaL86] (see also [GaL87,Lin91,H] and [Ku95] (Th. 2.3, 2.4)) is:…”

Section: Introductionmentioning

confidence: 99%

Local and global analysis of nodal sets

Zelditch

2017

Surveys in Differential Geometry

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This is a survey of recent results on nodal sets of eigenfunctions of the Laplacian on a compact Riemannian manifold. In part the techniques are 'local', i.e. only assuming eigenfunctions are defined on small balls, and in part the techniques are 'global', i.e. exploiting dynamics of the geodesic flow. The local part begins with a review of doubling indices and freqeuency functions as local measures of fast or slow growth of eigenfunctions. The pattern of boxes with maximal doubling indices plays a central role in the results of Logunov-Malinnokova, giving upper and lower bounds for hypersurface measures of nodal sets in the setting of general C ∞ metrics. The proofs of both their polynomial upper bound and sharp lower bound are sketched. The survey continues with a global proof of the sharp upper bound for real analytic metrics (originally proved by Donnelly-Fefferman with local arguments), using analytic continuation to Grauert tubes. Then it reviews results of Toth-Zelditch giving sharp upper bounds on Hausdorff measures of intersections of nodal sets with real analytic submanifolds in the real analytic setting. Last, it goes over lower bounds of Jung-Zelditch on numbers of nodal domains in the case of C ∞ surfaces of non-positive curvature and concave boundary or on negatively curved 'real' Riemann surfaces, which are based on ergodic properties of the geodesic flow and eigenfunctions, and on estimates of restrictions of eigenfunctions to hypersurfaces. The last section details recent results on restrictions.

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Small scale quantum ergodicity in negatively curved manifolds

Cited by 35 publications

References 50 publications

Small Scale Equidistribution for a Point Scatterer on the Torus

Small Scale Equidistribution for a Point Scatterer on the Torus

On the semiclassical functional calculus for h-dependent functions

Local and global analysis of nodal sets

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