Equidistribution of toral eigenfunctions along hypersurfaces
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Hamid Hezari [1] ; Gabriel Rivière [2]
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[1] University of California System
University of California System
Estados Unidos
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[2] University of Nantes
University of Nantes
Arrondissement de Nantes, Francia
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 2, 2020, págs. 435-454
- Idioma: inglés
- DOI: 10.4171/rmi/1135
- Enlaces
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Resumen
We prove a new quantum variance estimate for toral eigenfunctions. As an application, we show that, given any orthonormal basis of toral eigenfunctions and any smooth embedded hypersurface with nonvanishing principal curvatures, there exists a density one subsequence of eigenfunctions that equidistribute along the hypersurface. This is an analogue of the Quantum Ergodic Restriction theorems in the case of the flat torus, which in particular verifies the Bourgain–Rudnick’s conjecture on L2-restriction estimates for a density one subsequence of eigenfunctions in any dimension. Using our quantum variance estimates, we also obtain equidistribution of eigenfunctions against measures whose supports have Fourier dimension larger than d−2. In the end, we also describe a few quantitative results specific to dimension 2.
