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

Jung, Junehyuk; Zelditch, Steve

doi:10.5802/aif.3329

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Let dVol be the volume measure on M . Since λ is not a Dirichlet eigenvalue of ∆ M on N ′ , it readily follows that (18) w…”

Section: Global Approximation With Decaymentioning

confidence: 99%

“…for all x ∈ N ′ , f being the smooth function f := ∆ M w 1 + λw 1 supported on N \Ω ′ . A standard continuity argument allows us to approximate the integral (18) uniformly on Ω ′ by a Riemann sum of the form…”

Section: Global Approximation With Decaymentioning

confidence: 99%

“…In fact, Jung and Zelditch have constructed [18] a Riemannian 3-manifold for which all the (nonconstant) Laplace eigenfunctions have exactly two nodal domains, and this can be used to show that there are solutions to the Helmholtz equation that cannot be approximated by rescaled eigenfunctions on this manifold in the sense of (3). Also, in [9] we have recently shown that the inverse localization property does not hold on generic flat tori.…”

Section: Introductionmentioning

confidence: 99%

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Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

Enciso¹,

García-Ruiz²,

Peralta‐Salas³

2022

Preprint

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We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ 2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation ∆h + h = 0 on R d , over the natural length scale O(λ −1/2 ) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces H d (κ). As the dimension of the space of bound states of these operators tends to infinity as κ ց 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on H d (κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

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“…Let dVol be the volume measure on M . Since λ is not a Dirichlet eigenvalue of ∆ M on N ′ , it readily follows that (18) w…”

Section: Global Approximation With Decaymentioning

confidence: 99%

Section: Global Approximation With Decaymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

Enciso¹,

García-Ruiz²,

Peralta‐Salas³

2022

Preprint

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show abstract

“…for all n \in \BbbN has turned out to be a (surprisingly) difficult problem and only little is known, see [13,8,9] and references therein (we remark that T. Hoffmann-Ostenhof also asked the question as to whether there exists a Schrödinger operator for which one has \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ \mathrm{ n\rightar\infty \scrN (\varphi n ) < \infty ).…”

Section: Introductionmentioning

confidence: 99%

On the number of nodal domains on a rectangle with a slit

Kerner¹

2021

Methods of Functional Analysis and Topology

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In spectral geometry, one is interested in estimating the number of nodal domains of eigenfunctions of the Laplacian on planar domains. Well-known classical results due to Courant and Pleijel establish upper bounds, implying that the n-th eigenfunction has at most n nodal domains and that indeed only a finite number of eigenfunctions attain this maximal value. Surprisingly, however, a seemingly simpler question remains largely open. Namely, does there always exist a subsequence of eigenfunctions with an unbounded number of nodal domains? It is the aim of this note to investigate this question in the context of a rectangular domain with a slit. В спектральнiй геометрiї цiкавим є оцiнка кiлькостi вузловiих областей власних функцiй лапласiана в плоских областях. Вiдомi класичнi результати Куранта i Плейеля встановлюють верхнi межi, з яких випливає, що n-та власна функцiя має не бiльше нiж n вузлових областей, i лише скiнченна кiлькiсть власних функцiй досягають цього максимального значення. Однак, бiльш просте питання ще залишається вiдкритим. А саме, чи завжди iснує пiдпослiдовнiсть власних функцiй з необмеженою кiлькiсть вузлових областей? Метою цiєї роботи є дослiдження цього питання в контекстi прямокутної областi з прорiзом.

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Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

Enciso,

García-Ruiz,

Peralta-Salas

2024

Journal of Mathematical Physics

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We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation Δh + h = 0 on Rd, over the natural length scale O(λ−1/2) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces Hd(κ). As the dimension of the space of bound states of these operators tends to infinity as κ ↘ 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on Hd(κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

show abstract

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

Abstract: Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org

Cited by 4 publications

References 13 publications

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

On the number of nodal domains on a rectangle with a slit

Inverse localization and global approximation for some Schrödinger operators on hyperbolic spaces

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