In this paper we consider eigenfunctions of the Laplacian on a planar domain with polygonal boundary with Dirichlet, Neumann, or mixed boundary conditions. The main result is a quantitative estimate on the $L^2$ mass of eigenfunctions near a point in terms of the distance to the nearest non-adjacent boundary face. In particular, eigenfunctions cannot concentrate completely at any one single point. The technique of proof is to use the commutator ideas from the recent work of the author \cite{Chr-tri,Chr-simp} on triangles and simplices.
The purpose of this note is to extend to any space dimension the bilinear estimate for eigenfunctions of the Laplace operator on a compact manifold (without boundary) obtained in a previous work in dimension 2. We also give some related trilinear estimates.
We study the critical points of Laplace eigenfunctions on polygonal domains with a focus on the second Neumann eigenfunction. We show that if each convex quadrilaterals has no second Neumann eigenfunction with an interior critical point, then there exists a convex quadrilateral with an unstable critical point. We also show that each critical point of a second-Neumann eigenfunction on a Lip-1 polygon with no orthogonal sides is an acute vertex.
We obtain restrictions on the persistence barcodes of Laplace-Beltrami eigenfunctions and their linear combinations on compact surfaces with Riemannian metrics. Some applications to uniform approximation by linear combinations of Laplace eigenfunctions are also discussed.
We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.
Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_λ$ on a compact, boundary-less Riemannian manifold with $Δe_λ= -λ^2 e_λ$. We study eigenfunction triple products \[ \langle e_λe_μ, e_ν\rangle = \int e_λe_μ\overline{e_ν} \, dV. \] We show the overall $\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $λ,μ,$ and $ν$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $λ, μ,$ and $ν$ fail to satisfy the triangle inequality. As a co…
In this note, we extend the results on eigenfunction concentration in billiards as proved by the third author in \cite{M1}. There, the methods developed in Burq-Zworski \cite{BZ3} to study eigenfunctions for billiards which have rectangular components were applied. Here we take an arbitrary polygonal billiard $B$ and show that eigenfunction mass cannot concentrate away from the vertices; in other words, given any neighbourhood $U$ of the vertices, there is a lower bound $$ \int_U |u|^2 \geq c \int_B |u|^2 $$ for some $c = c(U) > 0$ and any eigenfunction $u$.
We prove a result, announced by F. Nazarov, L. Polterovich and M. Sodin that exhibits a relation between the average local growth of a Laplace eigenfunction on a closed surface and the global size of its nodal set. More precisely, we provide a lower and an upper bound to the Hausdorff measure of the nodal set in terms of the expected value of the growth exponent of an eigenfunction on disks of wavelength like radius. Combined with Yau's conjecture, the result implies that the average local growth of an eigenfunction on such disks is bounded by constants in the semi-classical limit. We also obt…
Let $Ω$ be a bounded domain in $\mathbb{R}^n$ with $C^{1}$ boundary and let $u_λ$ be a Dirichlet Laplace eigenfunction in $Ω$ with eigenvalue $λ$. We show that the $(n-1)$-dimensional Hausdorff measure of the zero set of $u_λ$ does not exceed $C(Ω)\sqrtλ$. This result is new even for the case of domains with $C^\infty$-smooth boundary.
We study the stability properties of nodal sets of Laplace eigenfunctions on compact manifolds under specific small perturbations. We prove that nodal sets are fairly stable if said perturbations are relatively small, more formally, supported at a sub-wavelength scale. We do not need any assumption on the topology of the nodal sets. As an indirect application, we are able to show that a certain "Payne property" concerning the second nodal line remains stable under controlled perturbations of the domain.
In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent results of Bourgain, Buckley and Wigman, we will prove that some deterministic families of eigenfunctions on $\mathbb{T}^2$ satisfy the conclusions of the random wave conjecture. We also show that on an arbitrary domain, a sequence of Laplace eigenfunctions always admits local weak limits. We explain why these local weak limits can be a powerful tool to study the…
- In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations. R{é}sum{é}.
We investigate several aspects of the nodal geometry and topology of Laplace eigenfunctions, with particular emphasis on the low frequency regime. This includes investigations in and around the Payne property, opening angle estimates of nodal domains, saturation of (fundamental) spectral gaps etc., and behaviour of all of the above under small scale perturbations. We aim to highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions, as opposed to asymptotic results.
In this work we extend a recent result by Dyda et. al. [B. Dyda, A. Kuznetsov, M. Kwasnicki, Eigenvalues of the fractional Laplace equation in the unit ball, J. Lond. Math. Soc. (2) 95 (2017), 500-518.] to dimension 3.
- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly damped wave equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori.
We use the sum-of-squares theorem from number theory to construct eigenfunctions of the Laplacian on the $d$-dimensional torus, $d \geq 2$, which vanish to any prescribed order at some point. These functions are then applied to provide a negative answer (in dimension $d \geq 2$) to a question in the context of quantitative unique continuation for spectral projectors of Schrödinger operators, asked by Egidi and Veselić.
We give an elementary new proof of the hot spots conjecture for L-shaped domains. This result, in addition to a new eigenvalue inequality, allows us to locate the hot spots in Swiss cross translation surfaces. We then prove, in several cases, that first mixed Dirichlet-Neumann eigenfunctions of the Laplacian on L-shaped domains also have no interior critical points. As a combination of these results, we prove the hot spots conjecture for five classes of domains tiled by L-shaped domains, including a class of non-simply connected domains. An interesting feature of the proofs is that we make pos…
We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunc…
We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.
We study the set of critical points of a solution to $Δu = λ\cdot u$ and in particular components of the critical set that have codimension 1. We show, for example, that if a second Neumann eigenfunction of a simply connected polygon $P$ has infinitely many critical points, then $P$ is a rectangle.
In this paper we deal with the problems of the weak localization of the eigenfunction expansions related to Laplace-Beltrami operator on unit sphere. The conditions for weak localization of Fourier-Laplace series are investigated by comparing the Riesz and Cesaro methods of summation for eigenfunction expansions of the Laplace-Beltrami operator. It is shown that the weak localization principle for the integrable functions f(x) at the point x depends not only on behavior of the function around x but on the behavior of the function around diametrically opposite point \overline{x}.
This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Riemannian manifolds. The review exposes the new results for Yau's conjecture in the smooth setting. We try to avoid technical details and emphasize the main ideas of the proof of Nadirashvili's conjecture. We also discuss two-dimensional methods to study zero sets.
Let $Δ_M$ be the Laplace operator on a compact $n$-dimensional Riemannian manifold without boundary. We study the zero sets of its eigenfunctions $u:Δu + λu =0$. In dimension $n=2$ we refine the Donnelly-Fefferman estimate by showing that $H^1(\{u=0 \})\le Cλ^{3/4-β}$, $β\in (0,1/4)$. The proof employs the Donnelli-Fefferman estimate and a combinatorial argument, which also gives a lower (non-sharp) bound in dimension $n=3$: $H^2(\{u=0\})\ge cλ^α$, $α\in (0,1/2)$. The positive constants $c,C$ depend on the manifold, $α$ and $β$ are universal.
We consider mass concentration properties of Laplace eigenfunctions $φ_λ$, that is, smooth functions satisfying the equation $-Δφ_λ= λφ_λ$, on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" par…
Given a compact Riemannian manifold $(M, g)$ without boundary, we estimate the Lebesgue norm of Laplace-Beltrami eigenfunctions when restricted to a wide variety of subsets $Γ$ of $M$. The sets $Γ$ that we consider are Borel measurable, Lebesgue-null but otherwise arbitrary with positive Hausdorff dimension. Our estimates are based on Frostman-type ball growth conditions for measures supported on $Γ$. For large Lebesgue exponents $p$, these estimates provide a natural generalization of $L^p$ bounds for eigenfunctions restricted to submanifolds, previously obtained in \cite{Ho68, Ho71, Sog88, B…
Let $\mathbb{M}$ be a compact $C^\infty$-smooth Riemannian manifold of dimension $n$, $n\geq 3$, and let $φ_λ: Δ_M φ_λ+ λφ_λ= 0$ denote the Laplace eigenfunction on $\mathbb{M}$ corresponding to the eigenvalue $λ$. We show that $$H^{n-1}(\{ φ_λ=0\}) \leq C λ^α,$$ where $α>1/2$ is a constant, which depends on $n$ only, and $C>0$ depends on $\mathbb{M}$ . This result is a consequence of our study of zero sets of harmonic functions on $C^\infty$-smooth Riemannian manifolds. We develop a technique of propagation of smallness for solutions of elliptic PDE that allows us to obtain local bounds from …
We provide a limiting absorption principle for the self-adjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts $Σ$ of) compact hypersurfaces $Γ=\partialΩ$, $Ω\subset{\mathbb{R}}^{n}$. For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Krein-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the fre…
We provide a new approach to studying the Dirichlet-Neumann map for Laplace's equation on a convex polygon using Fokas' unified method for boundary value problems. By exploiting the complex analytic structure inherent in the unified method, we provide new proofs of classical results using mainly complex analytic techniques. The analysis takes place in a Banach space of complex valued, analytic functions and the methodology is based on classical results from complex analysis. Our approach gives way to new numerical treatments of the underlying boundary value problem and the associated Dirichlet…
Let $\De u+\la u=\De v+\la v=0$, where $\De$ is the Laplace--Beltrami operator on a compact connected smooth manifold $M$ and $\la>0$. If $H^1(M)=0$ then there exists $p\in M$ such that $u(p)=v(p)=0$. For homogeneous $M$, $H^1(M)\neq0$ implies the existence of a pair $u,v$ as above that has no common zero.
We study the nodal set of Laplace eigenfunctions on the flat $2d$ torus $\mathbb{T}^2$. We prove an asymptotic law for the nodal length of such eigenfunctions, under some growth assumptions on their Fourier coefficients. Moreover, we show that their nodal set is asymptotically equidistributed on $\mathbb{T}^2$. The proofs are based on Bourgain's de-randomisation technique and the main new ingredient, which might be of independent interest, is the integrability of arbitrarily large powers of the doubling index of Laplace eigenfunctions on $\mathbb{T}^2$, based on the work of Nazarov \cite{N93,N…
Let $u$ be a harmonic function in the unit ball $B(0,1) \subset \mathbb{R}^n$, $n \geq 3$, such that $u(0)=0$. Nadirashvili conjectured that there exists a positive constant $c$, depending on the dimension $n$ only, such that $H^{n-1}(\{u=0 \}\cap B) \geq c$. We prove Nadirashvili's conjecture as well as its counterpart on $C^\infty$-smooth Riemannian manifolds. The latter yields the lower bound in Yau's conjecture. Namely, we show that for any compact $C^\infty$-smooth Riemannian manifold $M$ (without boundary) of dimension $n$ there exists $c>0$ such that for any Laplace eigenfunction $φ_λ$ …
We construct a Riemannian metric on the $ 2 $-dimensional torus, such that for infinitely many eigenvalues of the Laplace-Beltrami operator, a corresponding eigenfunction has infinitely many isolated critical points. A minor modification of our construction implies that each of these eigenfunctions has a level set with infinitely many connected components (i.e., a linear combination of two eigenfunctions may have infinitely many nodal domains).
It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial. If $S$ is an immersed $C^1-$ manifold then $S$ is a Coxeter system of planes. Full description of common nodal sets of the Laplace spectra of convexly supported distributions is given. In equivalent terms, the result describes ruled injectivity sets for the spherical mean trans…
We use probabilistic tools based on Brownian motion and Feynman-Kac formulae to investigate the heat profile for the ground state Dirichlet and second Neumann eigenfunctions. Among other topics, we comment on supremum norm bounds for ground state Dirichlet eigenfunctions and look at the corresponding Neumann problem, namely the comparison of maximum temperatures on the interior and the boundary, the latter being partially motivated by the hot spots problem. We also investigate the proximity/distance of level sets of ground state Dirichlet eigenfunctions, some with analogous statements for Neum…
We prove upper bounds on the $L^p$ norms of eigenfunctions of the discrete Laplacian on regular graphs. We then apply these ideas to study the $L^p$ norms of joint eigenfunctions of the Laplacian and an averaging operator over a finite collection of algebraic rotations of the $2$-sphere. Under mild conditions, such joint eigenfunctions are shown to satisfy for large $p$ the same bounds as those known for Laplace eigenfunctions on a surface of non-positive curvature.
Let $e(x,y,ł)$ be the spectral function and $χ_ł$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,ł)$ as $ł\to\infty$ by H{\" o}rmander (1968) and the one of $\p^\al_x\p^\bt_y e(x,y,ł)|_{x=y}$ as $ł\to\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\|χ_ł\|_{L_2\to L_p}$ ($2\leq p\leq\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\|χ_ł\|_{L_2\to {\rm Sobo…
The paper deals with asymptotic nodal geometry for the Laplace-Beltrami operator on closed surfaces. Given an eigenfunction f corresponding to a large eigenvalue, we study local asymmetry of the distribution of sign(f) with respect to the surface area. It is measured as follows: take any disc centered at the nodal line {f=0}, and pick at random a point in this disc. What is the probability that the function assumes a positive value at the chosen point? We show that this quantity may decay logarithmically as the eigenvalue goes to infinity, but never faster than that. In other words, only a mil…
A fundamental solution of Laplace's equation in three dimensions is expanded in harmonic functions that are separated in parabolic or elliptic cylinder coordinates. There are two expansions in each case which reduce to expansions of the Bessel functions $J_0(kr)$ or $K_0(kr)$, $r^2=(x-x_0)^2+(y-y_0)^2$, in parabolic and elliptic cylinder harmonics. Advantage is taken of the fact that $K_0(kr)$ is a fundamental solution and $J_0(kr)$ is the Riemann function of partial differential equations on the Euclidean plane.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.
Let ϕ be a Dirichlet or Neumann eigenfunction of the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We prove lower bounds for the size of the nodal set {ϕ=0}.
We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three--term asymptotic expansion for the first eigenvalue of d-dimensional ellipsoids.
We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$ and investigate a density property of their zero sets. More precisely, let $f=\sum_{k=1}^m a_k ϕ_{λ_{j_k}}$, where $-Δ_gϕ_λ=λϕ_λ$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\leq C(m)λ_{j_1}^{-1/2}$. The proof is based on a new integral Harnack-type estimate for positive solutions of higher order elliptic PDEs.
Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots …
