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Quantum ergodicity of boundary values of eigenfunctions: A control theory approach

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Nicolas Burq

Analysis of PDEs · Submitted 30 Jan 2003

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.

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arXiv IDmath/0301349 (version 1)
Pith IDTBH3GWC5, Pith's permanent record of this paper and its review
CitationNicolas Burq. "Quantum ergodicity of boundary values of eigenfunctions: A control theory approach." arXiv:math/0301349 (2003). https://pith.science/paper/TBH3GWC5
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,519 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.96 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.

background contribution method result interpretation main claim

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. background 1.00 0.09 0.02 asserted 0.77
s1 We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$. result 0.87 0.95 0.96 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by G\'erard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.

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