Dan Mangoubi
Let M either be a closed real analytic Riemannian manifold or a closed smooth Riemannian surface. We estimate from below the volume of a nodal domain component in an arbitrary ball provided that this component enters the ball deeply enough.
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The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on the manifold is analytic. We prove that if the eigenfunctions are equidistributed at a small scale, then the weak limits of the hypersurface volume form of their nodal sets are comparable to the volume form on the manifold. In particular, on the negatively curved manifolds with analytic metric and on the tori, we show that in any eigenbasis, there is a full d…
We use the Dong-Sogge-Zelditch formula to obtain a lower bound for the volume of the nodal sets of eigenfunctions. Our result improves the recent results of Sogge-Zelditch and in dimensions n \leq 5 gives a new proof for the lower bounds of Colding-Minicozzi.
We consider two-dimensional Schrödinger operators in bounded domains. We analyze relations between nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions associated with the nodal sets as the nodal domains of Courant-sharp eigenfunctions.
We discuss possible topological configurations of nodal sets, in particular the number of their components, for spherical harmonics on S^2. We also construct a solution of the equation Delta u=u in R^2 that has only two nodal domains. This equation arises in the study of high energy eigenfunctions.
Nodal Deficiency of Neumann Eigenfunctions on a Symmetric Dumbbell Domain
研究对称哑铃域上Neumann特征函数的节点缺陷，揭示其随颈部宽度变化的行为，并提供判断零节点缺陷特征函数的准则。
We give an example of a domain in dimension $N \geq 3$, homeomorphic to a ball and with analytic boundary, for which the second eigenvalue of the Dirichlet Laplacian has an eigenfunction with a closed nodal surface. The domain is constructed via a sequence of perturbations of the domain of S. Fournais [J. Differential Equations \textbf{173} (2001), 145-159].
In this note, we discuss a question posed by T. Hoffmann-Ostenhof concerning the parity of the number of nodal domains for a non-constant eigenfunction of the Laplacian on flat tori. We present two results. We first show that on the torus $(\mathbb{R}/2π\mathbb{Z})^{2}$, a non-constant eigenfunction has an even number of nodal domains. We then consider the torus $(\mathbb{R}/2π\mathbb{Z})\times(\mathbb{R}/2ρπ\mathbb{Z})\,$, with $ρ=\frac{1}{\sqrt{3}}\,$, and construct on it an eigenfunction with three nodal domains.
We show the existence of simply-connected unbounded planar domains for which the second nodal line of the Dirichlet Laplacian does not touch the boundary.
We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. The proof is a combination of Pleijel's original approach and an estimate due to Toth and Zelditch for the number of boundary zeros of Neumann eigenfunctions.
This article concerns the number of nodal domains of eigenfunctions of the Laplacian on special Riemannian $3$-manifolds, namely nontrivial principal $S^1$ bundles $P \to X$ over Riemann surfaces equipped with certain $S^1$ invariant metrics, the Kaluza-Klein metrics. We prove for generic Kaluza-Klein metrics that any Laplacian eigenfunction has exactly two nodal domains unless it is invariant under the $S^1$ action. We also construct an explicit orthonormal eigenbasis on the flat $3$-torus $\mathbb{T}^3$ for which every non-constant eigenfunction belonging to the basis has two nodal domains.
Eigenfunction asymptotics and nodal domain estimates for dumbbell domains
研究哑铃区域Neumann特征函数节点结构，证明颈部变薄时节点域数量增加，第三特征函数达到Courant上界，并利用匹配渐近分析克服奇异极限困难。
On the nodal set conjecture for the $p$-Laplacian in circularly symmetric domains
本文通过移动极化方法，研究了圆对称域中$p$-拉普拉斯算子第二特征函数节点集的性质，验证了类似Pütter在1990年关于狄利克雷拉普拉斯算子节点线的结论。
We consider the zeros on the boundary $\partial Ω$ of a Neumann eigenfunction $ϕ_λ$ of a real analytic plane domain $Ω$. We prove that the number of its boundary zeros is $O (λ)$ where $-Δϕ_λ = λ^2 ϕ_λ$. We also prove that the number of boundary critical points of either a Neumann or Dirichlet eigenfunction is $O(λ)$. It follows that the number of nodal lines of $ϕ_λ$ (components of the nodal set) which touch the boundary is of order $λ$. This upper bound is of the same order of magnitude as the length of the total nodal line, but is the square root of the Courant bound on the number of nodal …
In this paper, we show that equality in Courant's nodal domain theorem can only be reached for a finite number of eigenvalues of the Neumann Laplacian, in the case of an open, bounded and connected set in R n with a C 1,1 boundary. This result is analogous to Pleijel's nodal domain theorem for the Dirichlet Laplacian (1956). It confirms, in all dimensions, a conjecture formulated by Pleijel, which had already been solved by I. Polterovich for a two-dimensional domain with a piecewise-analytic boundary (2009). We also show that the argument and the result extend to a class of Robin boundary con…
We study the volume distribution of nodal domains of random band-limited functions on generic manifolds, and find that in the high energy limit a typical instance obeys a deterministic universal law, independent of the manifold. Some of the basic qualitative properties of this law, such as its support, monotonicity and continuity of the cumulative probability function, are established.
Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/λ^n. We apply this result to prove that each nodal domain contains a ball of radius > C/λ^n.
Recently, Sogge-Zelditch and Colding-Minicozzi gave new power law lower bounds on the size of the nodal sets of eigenfunctions. The purpose of this short note is to point out a third method to obtain a power law lower bound on the volume of the nodal sets. Our method is based on the Donnelly-Fefferman growth bound for eigenfunctions and a growth vs. volume relation we proved in a previous work.
$2$-nodal domain theorems for higher dimensional circle bundles
本文研究主 $S^1$ 丛上等变特征函数实部的节点结构，证明任意维数下一般丛度量恰产生2个节点域，并解释非自由作用时的失效。
We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of the domain. The assertion here is that any eigenfunction associated with the first nontrivial eigenvalue of the Neumann Laplacian on a domain $Ω$ with rotational symmetry of order two (i.e., $x\inΩ$ iff $-x\inΩ$) "should normally" be rotationally antisymmetric. We give both positive and negative results which highlight the heuristic similarity of this asser…
It is an open problem in general to prove that there exists a sequence of $Δ_g$-eigenfunctions $ϕ_{j_k}$ on a Riemannian manifold $(M, g)$ for which the number $N(ϕ_{j_k}) $ of nodal domains tends to infinity with the eigenvalue. Our main result is that $N(ϕ_{j_k}) \to \infty$ along a subsequence of eigenvalues of density $1$ if the $(M, g)$ is a non-positively curved surface with concave boundary, i.e. a generalized Sinai or Lorentz billiard. Unlike the recent closely related work of Ghosh-Reznikov-Sarnak and of the authors on the nodal domain counting problem, the surfaces need not have any …
Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue λ. We give upper and lower bounds on the inner radius of the type C/λ^k. Our proof is based on a local behavior of eigenfunctions discovered by Donnelly and Fefferman, and a Poincaré type inequality proved by Maz'ya. Sharp lower bounds are known only in dimension two. We give an account of this case too.
We give a detailed proof for two discrete analogues of Courant's Nodal Domain Theorem.
Capacity estimates and improved lower bounds for the inner radius of nodal domains
该研究针对闭光滑流形上拉普拉斯特征函数节点域，给出了不同维数下其内半径的下界估计，并证明了$\mathbb{T}^d$上存在内半径更小的节点域序列。
This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$ -Δϕ_{σ_j}=0,\quad\text{ on }Ω,\qquad\qquad \partial_νϕ_{σ_j}=σ_j ϕ_{σ_j}\quad \text{ on }\partialΩ$$ in two-dimensional domains $Ω$. In particular, this paper presents a dense family $\mathcal{A}$ of simply-connected two-dimensional domains with analytic boundaries such that, for each $Ω\in \mathcal{A}$, the nodal set of the eigenfunction $ϕ_{σ_j}$ "is $not$ dense at scale $σ_j^{-1}$". This result addresses a question put forth under "Open Problem 10" in Girouard…
We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio $N$, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often "open" the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the…
Payne nodal set conjecture for the fractional $p$-Laplacian in Steiner symmetric domains
本文研究了分数p-拉普拉斯在斯特拉赫对称域中特征函数和节点解的边界接触性质，通过极化不等式分析证明了正负部分支持接触边界，并推导出节点集的相同性质。
On Pleijel-type nodal domain bounds for the $p$-Laplacian
针对有界域中 $p$-Laplacian 的余属特征值，给出节点域渐近数量的 Pleijel 型上界，并确定第 $k$ 个特征函数节点域数小于 $k$ 的条件。
A slight improvement of Pleijel's estimate on the number of nodal domains is obtained, exploiting a refinement of the Faber-Krahn inequality and packing density of discs.
On Pleijel's nodal domain theorem for the Robin problem
本文改进 Robin 特征值问题的 Pleijel 节域定理，取消 Robin 参数非负限制并优化上界，还给出欧几里得球上常数 Robin 参数的 Pleijel 常数显式值。
The inner radius of nodal domains in high dimensions
本文证明了在d维闭Riemann流形上，Laplacian特征函数的每个结点域均包含一个特定半径的球，且球心位于该域内特征函数绝对值最大值点处。
We prove Polterovich's conjecture concerning the growth of the number of nodal domains for eigenfunctions on a unit square domain, under the assumption that the eigenfunctions do not have any singular points.
Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
本文显式描述了矩形域上可分势Schrödinger算子的谱流，揭示了低能特征函数影响节点亏量的机制。
We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods.
A Counterexample to the Closed Nodal-Line Conjecture for the Third Eigenfunction of the Planar Dirichlet Laplacian
针对平面Dirichlet Laplacian第三本征函数的闭节线猜想，构造出一个第三Dirichlet特征值为单重、节集含两条不相交闭曲线的反例，完成猜想的否定。
We discuss the asymptotic lower bound on the inner radius of nodal domains that arise from Laplacian eigenfunctions $ ϕ_λ$ on a closed Riemannian manifold $ (M,g) $. First, in the real-analytic case we present an improvement of the currently best known bounds, due to Mangoubi (\cite{Man1}). Furthermore, using recent results of Hezari (\cite{Hezari}, \cite{Hezari2}) we obtain $ \log $-type improvements in the case of negative curvature and improved bounds for $ (M,g) $ possessing an ergodic geodesic flow. Second, we discuss the relation between the distribution of the $ L^2 $ norm of an eigenfu…
Eigenfunction localization and nodal geometry on dumbbell domains
本文研究哑铃形区域上第一节点线和热点的位置，利用 Dirichlet 与 Neumann 边界条件下渐近 $L^2$-局部化性质，证明 Neumann 哑铃的第一节点集位于连接通道附近，并讨论 Dirichlet 情形下障碍物的最优放置。
We consider a Laplace eigenfunction $φ_λ$ on a smooth closed Riemannian manifold, that is, satisfying $-Δφ_λ= λφ_λ$. We introduce several observations about the geometry of its vanishing (nodal) set and corresponding nodal domains. First, we give asymptotic upper and lower bounds on the volume of a tubular neighbourhood around the nodal set of $φ_λ$. This extends previous work of Jakobson and Mangoubi in case $ (M, g) $ is real-analytic. A significant ingredient in our discussion are some recent techniques due to Logunov (cf. \cite{L1}). Second, we exhibit some remarks related to the asymptoti…
Nodal Domains on Surfaces under Perturbation: Upper Semicontinuity, Courant-Sharpness, and Boundary Intersections
研究在闭曲面上Schrödinger算子的特征函数节点域数量在光滑扰动下的变化，证明节点域计数的上半连续性，并构造可任意有限水平的Courant尖锐度度量及边界相交控制。
Nodal domain theorems for $p$-Laplacians on signed graphs
本文在带符号图上建立p-拉普拉斯算子的节点域定理，统一了现有图p-拉普拉斯算子与对称矩阵的节点域结果，并由此得到联系变分特征值与多路Cheeger常数的高阶Cheeger不等式，还导出了通常图上的新结论。
We prove that the Hecke--Maass eigenforms for a compact arithmetic triangle group have a growing number of nodal domains as the eigenvalue tends to $+\infty$. More generally the same is proved for eigenfunctions on negatively curved surfaces that are even or odd with respect to a geodesic symmetry and for which Quantum Unique Ergodicity holds.
We use tools from $n$-dimensional Brownian motion in conjunction with the Feynman-Kac formulation of heat diffusion to study nodal geometry on a compact Riemannian manifold $M$. On one hand we extend a theorem of Lieb and prove that any nodal domain $Ω_λ$ almost fully contains a ball of radius $\sim \frac{1}{\sqrtλ}$. This also gives a slight refinement of a result by Mangoubi, concerning the inradius of nodal domains (\cite{Man2}). On the other hand, we also prove that no nodal domain can be contained in a reasonably thin tubular neighbourhood of unions of finitely many surfaces inside $M$.
Pleijel nodal domain theorem in non-smooth setting
本文在非塌缩 RCD 空间及低正则 Euclidean 区域中证明 Pleijel 定理，建立节点域数渐近上界，并推出 Courant 定理至多有有限个例外。
We obtain lower bounds for the number of nodal domains of Hecke eigenfunctions on the sphere. Assuming the generalized Lindelof hypothesis we prove that the number of nodal domains of any Hecke eigenfunction grows with the eigenvalue of the Laplacian. By a very different method, we show unconditionally that the average number of nodal domains of degree l Hecke eigenfunctions grows significantly faster than the uniform growth obtained under Lindelof.
Å Pleijel has proved that in the case of the Laplacian on the square with Neumann condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues. We identify five Courant sharp eigenvalues for the Neumann Laplacian in the square, and prove that there are no other cases.
We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in a series of works on convex planar domains and rectangles with one curved edge and a large aspect ratio. Here, we study the structure of the nodal curve in greater detail, in that we find precise bounds on its curvature, with uniform estimates up to the two points where it meets the domain at right angles, and show that many of our results hold for relatively small aspect ratios of the side length…
We study the number of nodal domains in balls shrinking slightly above the Planck scale for "generic" toral eigenfunctions. We prove that, up to the natural scaling, the nodal domains count obeys the same asymptotic law as the global number of nodal domains. The proof, on one hand, uses new arithmetic information to refine Bourgain's de-randomisation technique at Planck scale. And on the other hand, it requires a Planck scale version of Yau's conjecture which we believe to be of independent interest.
We prove that the number of nodal domains of a density one subsequence of eigenfunctions grows at least logarithmically with the eigenvalue on negatively curved `real Riemann surfaces'. The geometric model is the same as in prior joint work with Junehyuk Jung (arXiv:1310.2919, to appear in J. Diff. Geom), where the number of nodal domains was shown to tend to infinity, but without a specified rate. The proof of the logarithmic rate uses the new logarithmic scale quantum ergodicity results of Hezari-Riviere (arXiv:1411.4078) and X. Han (arXiv:1410.3911).
The eigenvectors for graph $1$-Laplacian possess some sort of localization property: On one hand, any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques. The Courant nodal domain theorem for graphs is extended to graph $1$-Laplacian for strong nodal domains, but for weak nodal domains it is false. The notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the …
In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees an optimal lower bound, of the same order as Courant's upper bound. As an application, we will obtain optimal lower bounds for the number of nodal domains of distorted plane waves on some f…
We consider Dirichlet-to-Neumann operators associated to $Δ+q$ on a Lipschitz domain in a smooth manifold, where $q$ is an $L^{\infty}$ potential. We prove a Courant-type bound for the nodal count of the extensions $u_k$ of the $k$th Dirichlet-to-Neumann eigenfunctions $ϕ_k$ to the interior satisfying $(Δ+q)u_k=0$. The classical Courant nodal domain theorem is known to hold for Steklov eigenfunctions, which are the harmonic extension of the Dirichlet-to-Neumann eigenfunctions associated to $Δ$. Our result extends it to a larger family of Dirichlet-to-Neumann operators. Our proof makes use of t…
Local version of Courant's nodal domain theorem
本文针对闭 Riemann 流形上 Laplace 算子特征函数，结合 Remez 型不等式与 Landis 型增长引理，证明了与局部球相交的结点域数量的精确估计，解决了 Donnelly 和 Fefferman 提出的局部界限问题。
In the case of the sphere and the square, Antonie Stern (1925) claimed in her PhD thesis the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with two nodal domains. These two statements were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom--Przybytkowski (2006)). The aim of this paper is to obtain a similar result in the case of the isotropic quantum harmonic oscillator in the two dimensional case.
We present a comprehensive review of the nodal domains and lines of quantum billiards, emphasizing a quantitative comparison of theoretical findings to experiments. The nodal statistics are shown to distinguish not only between regular and chaotic classical dynamics but also between different geometric shapes of the billiard system itself. We discuss, in particular, how a random superposition of plane waves can model chaotic eigenfunctions and highlight the connections of the complex morphology of the nodal lines thereof to percolation theory and Schramm-Loewner evolution. Various approaches t…
We initiate a systematic study of eigenvectors of random graphs. Whereas much is known about eigenvalues of graphs and how they reflect properties of the underlying graph, relatively little is known about the corresponding eigenvectors. Our main focus in this paper is on the nodal domains associated with the different eigenfunctions. In the analogous realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years. Graphical nodal domains turn out to have interesting and unexpected properties. Our main theorem asserts that the…
We consider the nonlinear graph $p$-Laplacian and its set of eigenvalues and associated eigenfunctions of this operator defined by a variational principle. We prove a nodal domain theorem for the graph $p$-Laplacian for any $p\geq 1$. While for $p>1$ the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian ($p=2$), the behavior changes for $p=1$. We show that the bounds are tight for $p\geq 1$ as the bounds are attained by the eigenfunctions of the graph $p$-Laplacian on two graphs. Finally, using the properties of the nodal domains, we prove a h…
We investigate nodal sets of magnetic Schroedinger operators with zero magnetic field, acting on a non simply connected domain in $\r^2$. For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a charactisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole and a fixed electric potential, we show that the first eigenvalue takes its highest value for circulation 1/2.
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 radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Rivière small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get l…
In [3] the authors studied the 4-parameter family of isospectral flat 4-tori T^\pm(a,b,c,d) discovered by Conway and Sloane. With a particular method of counting nodal domains they were able to distinguish these tori (numerically) by computing the corresponding nodal sequences relative to a few explicit tuples (a,b,c,d). In this note we confirm the expectation expressed in [3] by proving analytically that their nodal count distinguishes any 4-tuple of distinct positive real numbers.
In 2001, Davies, Gladwell, Leydold, and Stadler proved discrete nodal domain theorems for eigenfunctions of generalized Laplacians, i.e., symmetric matrices with non-positive off-diagonal entries. In this paper, we establish nodal domain theorems for arbitrary symmetric matrices by exploring the induced signed graph structure. Our concepts of nodal domains for any function on a signed graph are switching invariant. When the induced signed graph is balanced, our definitions and upper bound estimates reduce to existing results for generalized Laplacians. Our approach provides a more conceptual u…
Nonexistence of Courant-type nodal domain bounds for eigenfunctions of the Dirichlet-to-Neumann operator
本文证明在任意维数 n≥3 的带边紧流形上可构造度量，使 Dirichlet-to-Neumann 算子前 K 个非常数特征函数拥有任意多节域，从而否定其 Courant 型节域界猜想。
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