Let $H(\Om_0)=-Δ+V$ be a Schrödinger operator on a bounded domain $\Om_0\subset \mathbb R^d$ with Dirichlet boundary conditions. Suppose that the $\Om_\ell$ ($\ell \in \{1,...,k\}$) are some pairwise disjoint subsets of $\Om_0$ and that $H(\Om_\ell)$ are the corresponding Schrödinger operators again with Dirichlet boundary conditions. We investigate the relations between the spectrum of $H(\Om_0)$ and the spectra of the $H(\Om_\ell)$. In particular, we derive some inequalities for the associated spectral counting functions which can be interpreted as generalizations of Courant's nodal Theorem. For the case that equality is achieved we prove converse results. In particular, we use potential theoretic methods to relate the $\Om_\ell$ to the nodal domains of some eigenfunction of $H(Ω_0)$.
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.
According to Courant's theorem, an eigenfunction as\-sociated with the $n$-th eigenvalue $λ\_n$ has at most $n$ nodal domains. A footnote in the book of Courant and Hilbert, states that the same assertion is true for any linear combination of eigenfunctions associated with eigenvalues less than or equal to $λ\_n$. We call this assertion the \emph{Extended Courant Property}.\smallskipIn this paper, we propose simple and explicit examples for which the extended Courant property is false: convex domains in $\R^n$ (hypercube and equilateral triangle), domains with cracks in $\mathbb{R}^2$, on the …
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.
This paper is devoted to the refine analysis of Courant's theorem for the Dirichlet Laplacian. Many papers (and some of them quite recent) have investigated in which cases this inequality in Courant's theorem is an equality: Pleijel, Helffer--Hoffmann-Ostenhof--Terracini, Helffer--Hoffmann-Ostenhof, Bérard-Helffer, Helffer--Persson-Sundqvist, Léna, Leydold. All these results were devoted to $(2D)$-cases in open sets in $\mathbb R^2$ or in surfaces like $\mathbb S^2$ or $\mathbb T^2$. The aim of the current paper is to look for analogous results for domains in $\mathbb{R}^3$ and, as $Å.$Pleijel…
We address the question of determining the eigenvalues $λ\_n$ (listed in nondecreasing order, with multiplicities) for which Courant's nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with $n$ nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results …
The question of determining for which eigenvalues there exists an eigenfunction which has the same number of nodal domains as the label of the associated eigenvalue (Courant-sharp property) was motivated by the analysis of minimal spectral partitions. In previous works, many examples have been analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots . A natural toy model for further investigations is the Möbius strip, a non-orientable surface with Euler characteristic $0$, and particularly the "square" Möbius strip whose eigenvalues have higher multiplicities. In this case…
We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $\R^{2}$ the domains we consider are the isosceles right triangle and the rectangle with edge ratio $\sqrt{2}$ (also known as the A4 paper). In $\R^{n}$ the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\textbackslash{}unfolding. This structure affects the nodal set of the eigenfunctions, which in turn allows to derive necessary conditions for Courant-sharpness. In addition, th…
The paper addresses the the number of nodal domains for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a "nodal deficiency" arises. Examples are known of eigenfunctions with arbitrarily large index $n$ that have just two nodal domains. It was suggested in the…
Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $ν_n$ of the $n$-th eigenfunction satisfies $n\ge ν_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-ν_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space …
Å. Pleijel (1956) has proved that in the case of the Laplacian with Dirichlet condition, the equality in the Courant nodal theorem (Courant sharp situation) can only be true for a finite number of eigenvalues when the dimension is $\geq 2$. Recently Polterovich extended the result to the Neumann problem in two dimensions in the case when the boundary is piecewise analytic. A question coming from the theory of spectral minimal partitions has motivated the analysis of the cases when one has equality in Courant's theorem. In this note, we identify the Courant sharp eigenvalues for the Dirichlet a…
Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. In a previous paper (Documenta Mathematica, 2018, Vol. 23, pp. 1561--1585), we gave simple counterexamples to this property, including convex domains. In the present paper, using some input from numerical computations, we pursue the investigation of the Extended Courant property with two new examples, the equilateral rhombus and the regular hexagon.
This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard--Helffer, Helffer--Persson--Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of …
In this note, we present a natural proof of a recent and surprising result of Gregory Berkolaiko (arXiv 1110.5373) interpreting the "Courant nodal defect" of a Schrödinger operator on a finite graph as a Morse index associated to the deformations of the operator by switching on a magnetic field. This proof is inspired by a nice paper of Miroslav Fiedler published in 1975.
We consider the cases where there is equality in Courant's nodal domain theorem for the Laplacian with a Robin boundary condition on the square. In our previous two papers, we treated the cases where the Robin parameter $h>0$ is large, small respectively. In this paper we investigate the case where $h<0\,$.
In this paper, we prove that the Extended Courant Property fails to be true for certain smooth, strictly convex domains with Neumann boundary condition: there exists a linear combination of a second and a first Neumann eigenfunctions, with three nodal domains. For the proof, we revisit a deformation argument of Jerison and Nadirashvili (J. Amer. Math. Soc. 2000, vol. 13). This argument being interesting in itself, we give full details. In particular, we carefully control the dependence of the constants on the geometry of our Lipschitz domains along the deformations.
This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter $h$). For the square, our first paper focused on the case where $h$ is large and extended results that were obtained by Pleijel, Bérard--Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of $h$, where $h$ is positive and not close to $0$. In this…
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.
Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in this case, the first eigenfunction is constant, and looking at the level sets of $u$ amounts to looking at the nodal sets $\{u-a=0\}$, where $a$ is a real constant. In the first part of the paper, we prove that the Extended Courant property is false for the subequilateral triang…
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…
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…
In this paper, we revisit Courant's nodal domain theorem for the Dirichlet eigenfunctions of a square membrane, and the analyses of A. Stern and Å. Pleijel.
Courant's theorem implies that the number of nodal domains of a Laplace eigenfunction is controlled by the corresponding eigenvalue. Over the years, there have been various attempts to find an appropriate generalization of this statement in different directions. We propose a new take on this problem using ideas from topological data analysis. We show that if one counts the nodal domains in a coarse way, basically ignoring small oscillations, Courant's theorem extends to linear combinations of eigenfunctions, to their products, to other operators, and to higher topological invariants of nodal s…
In this paper, we determine, in the case of the Laplacian on the flat three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality (Courant-sharp situation). Following the strategy of Å. Pleijel (1956), the proof is a combination of an explicit lower bound of the counting function and a Faber-Krahn-type inequality for domains on the torus, deduced as, in the work of P. Bérard and D. Meyer (1982), from an isoperimetric inequality. This inequality relies on the work of L. Hauswirth, J. Perez, P. R…
Given a Laplace eigenfunction on a surface, we study the distribution of its extrema on the nodal domains. It is classically known that the absolute value of the eigenfunction is asymptotically bounded by the 4-th root of the eigenvalue. It turns out that the number of nodal domains where the eigenfunction has an extremum of such order, remains bounded as the eigenvalue tends to infinity. We also observe that certain restrictions on the distribution of nodal extrema and a version of the Courant nodal domain theorem are valid for a rather wide class of functions on surfaces. These restrictions …
Let $Ω\subset \mathbb R^d\,, d\geq 2$, be a bounded open set, and denote by $λ\_j(Ω), j\geq 1$, the eigenvalues of the Dirichlet Laplacian arranged in nondecreasing order, with multiplicities. The weak form of Pleijel's theorem states that the number of eigenvalues $λ\_j(Ω)$, for which there exists an associated eigenfunction with precisely $j$ nodal domains (Courant-sharp eigenvalues), is finite. The purpose of this note is to determine an upper bound for Courant-sharp eigenvalues, expressed in terms of simple geometric invariants of $Ω$. We will see that this is connected with one of the fav…
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 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 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 [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.
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 1956 $Å$. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunctions as the eigenvalues tend to $+\infty$. A similar question occurs naturally for Schr"\odinger operators. The first significant result has been obtained recently by the first author for the harmonic oscillator. The purpose of this paper is to consider more general potentials which are radial.…
We study oscillations in the eigenfunctions for a fractional Schrödinger operator on the real line. An argument in the spirit of Courant's nodal domain theorem applies to an associated local problem in the upper half plane and provides a bound on the number of nodal domains for the extensions of the eigenfunctions. Using the combinatorial properties of noncrossing partitions, we turn the nodal domain bound into an estimate for the number of sign changes in the eigenfunctions. We discuss applications in the periodic setting and the Steklov problem on planar domains.
For the spherical Laplacian on the sphere and for the Dirichlet Laplacian in the square}, Antonie Stern claimed in her PhD thesis (1924) the existence of an infinite sequence of eigenvalues whose corresponding eigenspaces contain an eigenfunction with exactly two nodal domains. These results were given complete proofs respectively by Hans Lewy in 1977, and the authors in 2014 (see also Gauthier-Shalom--Przybytkowski, 2006). In this paper, we obtain similar results for the two dimensional isotropic quantum harmonic oscillator. In the opposite direction, we construct an infinite sequence of regu…
According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturm's theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturm's result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a g…
In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators; more specifically, the GJMS operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We also show that on any manifold of dimension $n\geq 3$, there exist many metrics for which our invariants are nontrivial. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold …
The Maslov index is used to compute the spectra of different boundary value problems for Schrödinger operators on compact manifolds. The main result is a spectral decomposition formula for a manifold $M$ divided into components $Ω_1$ and $Ω_2$ by a separating hypersurface $Σ$. A homotopy argument relates the spectrum of a second-order elliptic operator on $M$ to its Dirichlet and Neumann spectra on $Ω_1$ and $Ω_2$, with the difference given by the Maslov index of a path of Lagrangian subspaces. This Maslov index can be expressed in terms of the Morse indices of the Dirichlet-to-Neumann maps on…
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 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.
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.
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…
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.
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.
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.
