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 graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.
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 existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count). Recently examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by K. Ammann by utilising Godsil-McKay …
We consider the statistics of the number of nodal domains aka nodal counts for eigenfunctions of separable wave equations in arbitrary dimension. We give an explicit expression for the limiting distribution of normalised nodal counts and analyse some of its universal properties. Our results are illustrated by detailed discussion of simple examples and numerical nodal count distributions.
According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs.
The purpose of the present manuscript is to collect known results and present some new ones relating to nodal domains on graphs, with special emphasize on nodal counts. Several methods for counting nodal domains will be presented, and their relevance as a tool in spectral analysis will be discussed.
We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $ϕ$ of the $n$-th eigenfunction of the Schrödinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $ϕ- (n-1)$.
In this thesis, we study Laplacian eigenfunctions on metric graphs, also known as quantum graphs. We restrict the discussion to standard quantum graphs. These are finite connected metric graphs with functions that satisfy Neumann vertex conditions. The first goal of this thesis is the study of the nodal count problem. That is the number of points on which the $n$th eigenfunction vanishes. We provide a probabilistic setting using which we are able to define the nodal count\textquoteright s statistics. We show that the nodal count statistics admit a topological symmetry by which the first Betti …
The nodal domains of eigenvectors of the discrete Schrodinger operator on simple, finite and connected graphs are considered. Courant's well known nodal domain theorem applies in the present case, and sets an upper bound to the number of nodal domains of eigenvectors: Arranging the spectrum as a non decreasing sequence, and denoting by $ν_n$ the number of nodal domains of the $n$'th eigenvector, Courant's theorem guarantees that the nodal deficiency $n-ν_n$ is non negative. (The above applies for generic eigenvectors. Special care should be exercised for eigenvectors with vanishing components.…
An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number $β$. We study the distribution of the nodal surplus values in the countably infinite set of the graph's eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing $β$. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computati…
We study the nodal set of eigenfunctions of the Laplace operator on the right angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventually we use the recursion formula to analyse a large sequence of nodal counts statistically. Our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shape…
We investigate the properties of the zeros of the eigenfunctions on quantum graphs (metric graphs with a Schrödinger-type differential operator). Using tools such as scattering approach and eigenvalue interlacing inequalities we derive several formulas relating the number of the zeros of the n-th eigenfunction to the spectrum of the graph and of some of its subgraphs. In a special case of the so-called dihedral graph we prove an explicit formula that only uses the lengths of the edges, entirely bypassing the information about the graph's eigenvalues. The results are explained from the point of…
Several types of systems were put forward during the past decades to show that there exist {\it isospectral} systems which are {\it metrically} different. One important class consists of Laplace Beltrami operators for pairs of flat tori in $\mathbb{R}^n$ with $n\geq 4$. We propose that the spectral ambiguity can be resolved by comparing the nodal sequences (the numbers of nodal domains of eigenfunctions, arranged by increasing eigenvalues). In the case of isospectral flat tori in four dimensions - where a 4-parameters family of isospectral pairs is known- we provide heuristic arguments support…
Sturm's oscillation theorem states that the n-th eigenfunction of a Sturm-Liouville operator on the interval has n-1 zeros (nodes). This result was generalized for all metric tree graphs and an analogous theorem was proven for discrete tree graphs. We prove the converse theorems for both discrete and metric graphs. Namely, if for all n, the n-th eigenfunction of the graph has n-1 zeros then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues. In the course of the proof we show that it is not possible f…
We study the nodal count of the so-called bi-dendral graphs and show that it exhibits an anomaly: the nodal surplus is never equal to 0 or $β$, the first Betti number of the graph. According to the nodal-magnetic theorem, this means that bands of the magnetic spectrum (dispersion relation) of such graphs do not have maxima or minima at the "usual" symmetry points of the fundamental domain of the reciprocal space of magnetic parameters. In search of the missing extrema we prove a necessary condition for a smooth critical point to happen inside the reciprocal fundamental domain. Using this condi…
We establish a connection between the stability of an eigenvalue under a magnetic perturbation and the number of zeros of the corresponding eigenfunction. Namely, we consider an eigenfunction of discrete Laplacian on a graph and count the number of edges where the eigenfunction changes sign (has a "zero"). It is known that the $n$-th eigenfunction has $n-1+s$ such zeros, where the "nodal surplus" $s$ is an integer between 0 and the number of cycles on the graph. We then perturb the Laplacian by a weak magnetic field and view the $n$-th eigenvalue as a function of the perturbation. It is shown …
Given a Dirichlet eigenfunction of a 2D quantum billiard, the boundary domain count is the number of intersections of the nodal lines with the boundary. We study the integer sequence defined by these numbers, sorted according to the energies of the eigenfunctions. Based on a variant of Berry's random wave model, we derive a semi-classical trace formula for the sequence of boundary domain counts. The formula consists of a Weyl-like smooth part, and an oscillating part which depends on classical periodic orbits and their geometry. The predictions of this trace formula are supported by numerical …
We compute the asymptotic expectation of the number of {\em open} nodal lines for random waves on smooth planar domains. We find that for both the long energy window $[0,λ]$ and the short one $[λ,λ+1]$ the expected number of open nodal lines is proportional to $λ$, asymptotically as $λ\to\infty$. Our results are consistent with the predictions in the physics literature made by Blum, Gnutzmann and Smilansky \cite{BGS}.
On the discrete Painlevé equivalence problem, non-conjugate translations and nodal curves
Beijing Institute of Mathematical Sciences and Applications (BIMSA)（北京数学与应用研究所）；Institute of Mathematics, University of Warsaw（华沙大学数学研究所）；Waseda Institute for Advanced Study, Waseda University（早稻田大学高等研究院）
In an attempt to characterize the structure of eigenvectors of random regular graphs, we investigate the correlations between the components of the eigenvectors associated to different vertices. In addition, we provide numerical observations, suggesting that the eigenvectors follow a Gaussian distribution. Following this assumption, we reconstruct some properties of the nodal structure which were observed in numerical simulations, but were not explained so far. We also show that some statistical properties of the nodal pattern cannot be described in terms of a percolation model, as opposed to …
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.…
We present and discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R2 which were introduced recently and are all based on Sunada's construction of isospectral domains. After presenting some of the properties of these graphs, we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.
We consider stationary waves on nonlinear quantum star graphs, i.e. solutions to the stationary (cubic) nonlinear Schrödinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e. the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wide…
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…
This note discusses a relation between the multiplicity m of the second eigenvalue λ2 of a Laplacian on a graph G, tight mappings of G and a discrete analogue of Courant's nodal line theorem. For a certain class of graphs, we show that the m-dimensional eigenspace of λ2 is tight and thus defines a tight mapping of G into an m-dimensional Euclidean space. The tightness of the mapping is shown to set Colin de Verdière's upper bound on the maximal λ2-multiplicity, where chr(γ(G)) is the chromatic number and γ(G) is the genus of G.
We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains $ν_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $L^1$-potentials and a variety of vertex conditions as well as the $p$-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. {Among other things, these results characterise the accumulation points of the sequence $(\frac{ν_n}{n})_{n…
We consider a random Gaussian model of Laplace eigenfunctions on the hemisphere satisfying the Dirichlet boundary conditions along the equator. For this model we find a precise asymptotic law for the corresponding zero density functions, in both short range (around the boundary) and long range (far away from the boundary) regimes. As a corollary, we were able to find a logarithmic negative bias for the total nodal length of this ensemble relatively to the rotation invariant model of random spherical harmonics. Jean Bourgain's research, and his enthusiastic approach to the nodal geometry of Lap…
We study the uniqueness question for two inverse problems on graphs. Both problems consist in finding (possibly complex) edge or nodal based quantities from boundary measurements of solutions to the Dirichlet problem associated with a weighted graph Laplacian plus a diagonal perturbation. The weights can be thought of as a discrete conductivity and the diagonal perturbation as a discrete Schrödinger potential. We use a discrete analogue to the complex geometric optics approach to show that if the linearized problem is solvable about some conductivity (or Schrödinger potential) then the lineari…
Neumann domains of Laplacian eigenfunctions form a natural counterpart of nodal domains. The restriction of an eigenfunction to one of its nodal domains is the first Dirichlet eigenfunction of that domain. This simple observation is fundamental in many works on nodal domains. We consider a similar property for Neumann domains. Namely, given a Laplacian eigenfunction $f$ and its Neumann domain $Ω$, what is the position of $\ .f\ |_Ω$ in the Neumann spectrum of $Ω$? The current paper treats this spectral position problem on the two-dimensional torus. We fully solve it for separable eigenfunction…
