# Applications of small scale quantum ergodicity in nodal sets

@article{Hezari2016ApplicationsOS, title={Applications of small scale quantum ergodicity in nodal sets}, author={Hamid Hezari}, journal={arXiv: Analysis of PDEs}, year={2016} }

The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(\lambda) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of $r(\lambda)$. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of…

## 16 Citations

### Inner radius of nodal domains of quantum ergodic eigenfunctions

- MathematicsProceedings of the American Mathematical Society
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In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the…

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We prove an analogue of Sogge’s local Lp estimates for Lp norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements…

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In this paper, we investigate the small scale equidistribution properties of randomised sums of Laplacian eigenfunctions (i.e. random waves) on a compact manifold. We prove small scale expectation…

### Quantum Ergodicity and Lp Norms of Restrictions of Eigenfunctions

- MathematicsCommunications in Mathematical Physics
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We prove an analogue of Sogge’s local Lp estimates for Lp norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements…

### On the Lower Bound of the Inner Radius of Nodal Domains

- MathematicsThe Journal of Geometric Analysis
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We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $$ \varphi _{\lambda }$$φλ on a closed Riemannian manifold (M, g) . In the…

### On the geometry of nodal sets and nodal domains

- Mathematics
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In the present work we study and prove results related to the nodal geometry of Laplacian eigenfunctions on closed Riemannian manifolds, as well as solutions to more general classes of elliptic…

### Review of Yau's conjecture on zero sets of Laplace eigenfunctions

- Mathematics
- 2018

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions.
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### Mathematics of Quantum Chaos in 2019

- PhysicsNotices of the American Mathematical Society
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DOI: https://doi.org/10.1090/noti1958 thor [Z17, Z18]. One of the leaders in the field, Nalini Anantharaman, has just given a plenary address on quantum chaos at the 2018 ICM, and interested readers…

### On the Lower Bound of the Inner Radius of Nodal Domains

- Materials ScienceThe Journal of Geometric Analysis
- 2018

We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions φλ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}…

### Distorted Plane Waves on Manifolds of Nonpositive Curvature

- MathematicsCommunications in Mathematical Physics
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We will consider the high frequency behaviour of distorted plane waves on manifolds of nonpositive curvature which are Euclidean or hyperbolic near infinity, under the assumption that the curvature…

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