Quantum unique ergodicity and the number of nodal domains of eigenfunctions

Jang, Seung uk; Jung, Junehyuk

doi:10.1090/jams/883

Cited by 7 publications

(5 citation statements)

References 28 publications

(38 reference statements)

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“…By [30, Theorem 1.3], we know that QUE holds for the sequence of eigenfunctions {φ k }. Then by [22,Corollary 3.2], the lemma is proved.…”

Section: Nodal Domainsmentioning

confidence: 92%

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Sup-Norm and Nodal Domains of Dihedral Maass Forms

Huang

2019

Commun. Math. Phys.

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In this paper, we improve the sup-norm bound and the lower bound of the number of nodal domains for dihedral Maass forms, which are a distinguished sequence of Laplacian eigenfunctions on an arithmetic hyperbolic surface. More specifically, let φ be a dihedral Maass form with spectral parameter t φ , then we prove thatwhich is an improvement over the bound t 5/12+ε φ φ 2 given by Iwaniec and Sarnak. As a consequence, we get a better lower bound for the number of nodal domains intersecting a fixed geodesic segment under the Lindelöf Hypothesis. Unconditionally, we prove that the number of nodal domains grows faster than t 1/8−ε φ for any ε > 0 for almost all dihedral Maass forms. This is to be compared with the "local Weyl law", which gives λ 1/4 φ / log λ φ for any negative curvature surface (see [2]). Iwaniec-Sarnak [21] also proved the same bounds (1.1) for the Date: February 26, 2019.

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“…By [30, Theorem 1.3], we know that QUE holds for the sequence of eigenfunctions {φ k }. Then by [22,Corollary 3.2], the lemma is proved.…”

Section: Nodal Domainsmentioning

confidence: 92%

“…, by assuming the Lindelöf Hypothesis for the L-functions L (s, φ). Recently in Jang-Jung [22], they showed lim t φ →∞ N β (φ) = +∞ without any assumptions. However no quantitative lower bound is given in [22].…”

mentioning

confidence: 99%

Sup-Norm and Nodal Domains of Dihedral Maass Forms

Huang

2019

Commun. Math. Phys.

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“…To put the nodal results into context, it is proved in varying degrees of generality in [8,9,11,12,13,14,17,22] that in dimension 2, the number of nodal domains of an orthonormal basis {u j } of Laplace eigenfunctions on certain surfaces with ergodic geodesic flow tends to infinity with the eigenvalue along almost the entire sequence of eigenvalues. By the first item of Theorem 1.5, the same is true for their lifts to the unit tangent bundle SX as invariant eigenfunctions of the Kaluza-Klein metric.…”

Section: Outline Of the Proofmentioning

confidence: 99%

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds

Jung¹,

Zelditch²

2020

Annales De l'Institut Fourier

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“…To end, we point to a significant advance in the recent preprint [JJ15]. Using results in [CTZ13] they first give a localised (i.e.…”

Section: Introductionmentioning

confidence: 97%

Nodal domains of Maass forms, II

Ghosh¹,

Reznikov²,

Sarnak³

2017

American Journal of Mathematics

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In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindelöf hypothesis. That was a consequence of a topological argument and known subconvexity estimates, together with new sharp lower-bound restriction theorems for the Maass forms. This paper deals with the same question for general (compact or not) arithmetic surfaces which have a reflective symmetry. The topological argument is extended and representation theoretic methods are needed for the restriction theorems, together with results of Waldspurger. Various explicit examples are given and studied.

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Quantum unique ergodicity and the number of nodal domains of eigenfunctions

Cited by 7 publications

References 28 publications

Sup-Norm and Nodal Domains of Dihedral Maass Forms

Sup-Norm and Nodal Domains of Dihedral Maass Forms

Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds

Nodal domains of Maass forms, II

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