2015
DOI: 10.1112/plms/pdv018
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Completeness of boundary traces of eigenfunctions

Abstract: Abstract. In this paper, we study the boundary traces of eigenfunctions on the boundary of a smooth and bounded domain. An identity derived by Bäcker, Fürstburger, Schubert, and Steiner [BFSS], expressing (in some sense) the asymptotic completeness of the set of boundary traces in a frequency window of size O(1), is proved both for Dirichlet and Neumann boundary conditions. We then prove a semiclassical generalization of this identity.

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Cited by 10 publications
(18 citation statements)
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“…We denote by Op h (a) a semi-classical pseudo-differential operator on ∂M . We refer to [Zwo12,HZ04,TZ13,HHHZ] for background.…”
Section: Proof (Sketch)mentioning
confidence: 99%
“…We denote by Op h (a) a semi-classical pseudo-differential operator on ∂M . We refer to [Zwo12,HZ04,TZ13,HHHZ] for background.…”
Section: Proof (Sketch)mentioning
confidence: 99%
“…• (ii) One needs to prove a small scale Kuznecov asymptotic formula in the sense of [Zel92,HHHZ13], to the effect that there exists a subsquence of density one for which β u j k is of order |β|λ − 1 4 j (log λ j ) 1/3 when |β| ≃ ℓ j . 4 Again, one needs to show that there is a subsequence of density one for which this estimate holds simultaneously for all the balls of the cover.…”
Section: 2mentioning
confidence: 99%
“…The next step is to prove a uniform logarithmic scale Kuznecov period bound in the sense of [Zel92,HHHZ13]. It is also a general result, but for the sake of simplicity, and because it is the case relevant to this note, we assume dim M = 2.…”
Section: 2mentioning
confidence: 99%
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“…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%