arXiv subjects

James B. Kennedy

Publications and source records attributed to James B. Kennedy.

At least 19 recordsLinked to original sources

arXiv2026

Spectral minimal partitions of combinatorial graphs

This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincaré-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.

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arXiv2026

Spectral minimal partitions of unbounded domains

We study the problem of constructing $k$-spectral minimal partitions of domains in $d$ dimensions, where the energy functional to be minimized is a $p$-norm ($1 \le p \le \infty$) of the infimum of the spectrum of a suitable Schrödinger operator $-Δ+V$, with Dirichlet conditions on the boundary of the partition elements (cells). The main novelty of this paper is that the domains may be unbounded, including of infinite volume. First, we prove a sharp upper bound for the infimal energy among all $k$-partitions by a threshold value which involves the infimum $Σ$ of the essential spectrum of the Schrödinger operator on the whole domain as well as the infimal energy among all $k-1$-partitions. Strictly below such threshold, we develop a concentration-compactness-type argument showing optimal partitions exist, and each cell admits ground states (i.e., the infimum of the spectrum on each cell is a simple isolated eigenvalue). Second, for $p<\infty$, when the energy and the threshold level coincide, we show there may or may not be minimizing partitions. Moreover, even when these exist, they may not have ground states. Third, for $p=\infty$, minimal partitions always exist, even at the threshold level, but these may or may not admit ground states. Moreover, below the threshold, we can always construct a minimizer, which is an equipartition. At the threshold value we show that spectral minimal partitions may not need to be equipartitions. We give a variety of examples of both domains and potentials to illustrate the new phenomena that occur in this setting.

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arXiv2026

Bounds on eigenvalue ratios of quantum graphs

We study ratios of eigenvalues of the Laplacian on compact metric graphs. Our goals are threefold: First, we prove a sharp Ashbaugh--Benguria-type bound for the ratio of the first two eigenvalues on compact trees with Dirichlet conditions at all leaves, concretely showing that the ratio is maximized when the graph is an interval or an equilateral star. This improves a previous Payne--Pólya--Weinberger-type result due to Nicaise [Bull. Sci. Math., II. Sér. 111 (1987), 401--413]. Second, we extend this bound to a set of inequalities for the ratio of any pair of eigenvalues of such compact Dirichlet trees which respect the Weyl asymptotics up to an absolute constant. Third, we show that on non-trees, on which we also allow any mix of Neumann and Dirichlet conditions at the leaves, it is possible to recover bounds on the eigenvalue ratios depending only on the number of independent cycles and the number of Neumann leaves, in addition to the eigenvalue indices. This complements previously known counterexamples to analogues of the Ashbaugh--Benguria bound for general quantum graphs, by showing that the only way the bound can fail is through cycles and Neumann leaves, and by explicitly quantifying the extent to which it can fail.

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arXiv2025

On The Spectrum Of Infinite Quantum Graphs

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new criterion for compactness of the resolvent and apply this to identify a transition from purely discrete to non-empty essential spectrum among a class of infinite metric graphs, a phenomenon that seems to have no known counterpart for Laplacians on Euclidean domains of infinite volume. In the case of discrete spectrum we then prove upper and lower bounds on eigenvalues, thus extending a number of bounds previously only known in the compact setting to infinite graphs. Some of our bounds, for instance in terms of the inradius, are new even on compact graphs.

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arXiv2025

On the hot spots conjecture in higher dimensions

We prove a strong form of the hot spots conjecture for a class of domains in $\mathbb{R}^d$ which are a natural generalization of the lip domains of Atar and Burdzy [J. Amer. Math. Soc. 17 (2004), 243-265] in dimension two, as well as for a class of symmetric domains in $\mathbb{R}^d$ generalizing the domains studied by Jerison and Nadirashvili [J. Amer. Math. Soc. 13 (2000), 741-772]. Our method of proof is based on studying a vector-valued Laplace operator whose spectrum contains the spectrum of the Neumann Laplacian. This proof is essentially variational and does not require tools from stochastic analysis, nor does it use deformation arguments. In particular, it contains a new proof of the main result of Jerison and Nadirashvili.

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arXiv2025

On domain monotonicity of Neumann eigenvalues of convex domains

Inspired by a recent result of Funano's, we provide a sharp quantitative comparison result between the first nontrivial eigenvalues of the Neumann Laplacian on bounded convex domains $Ω_{1} \subset Ω_{2}$ in any dimension $d$ greater than or equal to two, recovering domain monotonicity up to an explicit multiplicative factor. We provide upper and lower bounds for such multiplicative factors for higher-order eigenvalues, and study their behaviour with respect to the dimension and order. We further consider different scenarios where convexity is no longer imposed. In a final section we formulate some related open problems.

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arXiv2025

Positivity properties of the Dirichlet-to-Neumann operator on graphs

We explore positivity properties of the semigroup generated by the negative of the Dirichlet-to-Neumann operator with real potential $λ$, defined on a subset of the vertices of a quantum graph. We show that for rationally independent edge lengths and suitable graph topologies, this semigroup will alternate between being positive, eventually positive without being positive (that is, positive only for sufficiently large times), and not even eventually positive, as $λ\to \infty$. For other graph topologies, the semigroup will alternate between being positive and not eventually positive. The topological conditions are related to a reduced graph which is a schematic map of the connections between the vertices on which the Dirichlet-to-Neumann operator acts.

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arXiv2024

Cheeger cuts and Robin spectral minimal partitions of metric graphs

We study partition problems based on two ostensibly different kinds of energy functionals defined on $k$-partitions of metric graphs: Cheeger-type functionals whose minimisers are the $k$-Cheeger cuts of the graph, and the corresponding values are the $k$-Cheeger constants of the graph; and functionals built using the first eigenvalue of the Laplacian with positive, i.e. absorbing, Robin (delta) vertex conditions at the boundary of the partition elements. We prove existence of minimising $k$-partitions, $k \geq 2$, for both these functionals. We also show that, for each $k \geq 2$, as the Robin parameter $α\to 0$, up to a renormalisation the spectral minimal Robin energy converges to the $k$-Cheeger constant. Moreover, up to a subsequence, the Robin spectral minimal $k$-partitions converge in a natural sense to a $k$-Cheeger cut of the graph. Finally, we show that as $α\to \infty$ there is convergence in a similar sense to the corresponding Dirichlet minimal energy and partitions. It is strongly expected that similar results hold on general (smooth, bounded) Euclidean domains and manifolds.

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arXiv2024

Towards a theory of eigenvalue asymptotics on infinite metric graphs: the case of diagonal combs

We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete spectrum for the Neumann Laplacian. In this regime, we establish polynomial upper and lower bounds on the $k$-th eigenvalue, revealing that the eigenvalues grow at a rate strictly slower than quadratic. However, once the diagonal combs transition to finite volume, their growth accelerates to a quadratic rate. Our methodology involves employing spectral geometric principles tailored for metric graphs, complemented by deriving estimates for the $k$-th eigenvalue on compact metric graphs.

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arXiv2024

Mean distance on metric graphs

We introduce a natural notion of mean (or average) distance in the context of compact metric graphs, and study its relation to geometric properties of the graph. We show that it exhibits a striking number of parallels to the reciprocal of the spectral gap of the graph Laplacian with standard vertex conditions: it is maximised among all graphs of fixed length by the path graph (interval), or by the loop in the restricted class of doubly connected graphs, and it is minimised among all graphs of fixed length and number of edges by the equilateral flower graph. We also establish bounds for the correctly scaled product of the spectral gap and the square of the mean distance which depend only on combinatorial, and not metric, features of the graph. This raises the open question whether this product admits absolute upper and lower bounds valid on all compact metric graphs.

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arXiv2024

Optimizing the Fundamental Eigenvalue Gap of Quantum Graphs

We study the problem of minimizing or maximizing the fundamental spectral gap of Schrödinger operators on metric graphs with either a convex potential or a ``single-well'' potential on an appropriate specified subset. (In the case of metric trees, such a subset can be the entire graph.) In the convex case we find that the minimizing and maximizing potentials are piecewise linear with only a finite number of points of non-smoothness, but give examples showing that the optimal potentials need not be constant. This is a significant departure from the usual scenarios on intervals and domains where the constant potential is typically minimizing. In the single-well case we show that the optimal potentials are piecewise constant with a finite number of jumps, and in both cases give an explicit estimate on the number of points of non-smoothness, respectively jumps, the minimizing potential can have. Furthermore, we show that, unlike on domains, it is not generally possible to find nontrivial bounds on the fundamental gap in terms of the diameter of the graph alone, within the given classes.

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arXiv2023

Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap

We derive several upper bounds on the spectral gap of the Laplacian with standard or Dirichlet vertex conditions on compact metric graphs. In particular, we obtain estimates based on the length of a shortest cycle (girth), diameter, total length of the graph, as well as further metric quantities introduced here for the first time, such as the avoidance diameter. Using known results about Ramanujan graphs, a class of expander graphs, we also prove that some of these metric quantities, or combinations thereof, do not to deliver any spectral bounds with the correct scaling.

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arXiv2023

Spectral minimal partitions of unbounded metric graphs

We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-Δ+ V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $Σ$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any $k\in\mathbb{N}$, the infimal energy among all admissible $k$-partitions is bounded from above by $Σ$, and if it is strictly below $Σ$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space.

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arXiv2021

Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs

We prove interlacing inequalities between spectral minimal energies of metric graphs built on Dirichlet and standard Laplacian eigenvalues, as recently introduced in [Kennedy et al, arXiv:2005.01126]. These inequalities, which involve the first Betti number and the number of degree one vertices of the graph, recall both interlacing and other inequalities for the Laplacian eigenvalues of the whole graph, as well as estimates on the difference between the number of nodal and Neumann domains of the whole graph eigenfunctions. To this end we study carefully the principle of cutting a graph, in particular quantifying the size of a cut as a perturbation of the original graph via the notion of its rank. As a corollary we obtain an inequality between these energies and the actual Dirichlet and standard Laplacian eigenvalues, valid for all compact graphs, which complements a version for tree graphs of Friedlander's inequalities between Dirichlet and Neumann eigenvalues of a domain. In some cases this results in better Laplacian eigenvalue estimates than those obtained previously via more direct methods.

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arXiv2021

The Bilaplacian with Robin boundary conditions

We introduce Robin boundary conditions for biharmonic operators, which are a model for elastically supported plates and are closely related to the study of spaces of traces of Sobolev functions. We study the dependence of the operator, its eigenvalues, and eigenfunctions on the Robin parameters. We show in particular that when the parameters go to plus infinity the Robin problem converges to other biharmonic problems, and obtain estimates on the rate of divergence when the parameters go to minus infinity. We also analyse the dependence of the operator on smooth perturbations of the domain, computing the shape derivatives of the eigenvalues and giving a characterisation for critical domains under volume and perimeter constraints. We include a number of open problems arising in the context of our results.

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arXiv2021

On the hot spots of quantum graphs

We undertake a systematic investigation of the maxima and minima of the eigenfunctions associated with the first nontrivial eigenvalue of the Laplacian on a metric graph equipped with standard (continuity--Kirchhoff) vertex conditions. This is inspired by the famous hot spots conjecture for the Laplacian on a Euclidean domain, and the points on the graph where maxima and minima are achieved represent the generically "hottest" and "coldest" spots of the graph. We prove results on both the number and location of the hot spots of a metric graph, and also present a large number of examples, many of which run contrary to what one might naïvely expect. Amongst other results we prove the following: (i) generically, up to arbitrarily small perturbations of the graph, the points where minimum and maximum, respectively, are attained are unique; (ii) the minima and maxima can only be located at the vertices of degree one or inside the doubly connected part of the metric graph; and (iii) for any fixed graph topology, for some choices of edge lengths all minima and maxima will occur only at degree-one vertices, while for others they will only occur in the doubly connected part of the graph.

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arXiv2021

Asymptotics and estimates for spectral minimal partitions of metric graphs

We study properties of spectral minimal partitions of metric graphs within the framework recently introduced in [Kennedy et al, Calc. Var. 60 (2021), 61]. We provide sharp lower and upper estimates for minimal partition energies in different classes of partitions; while the lower bounds are reminiscent of the classic isoperimetric inequalities for metric graphs, the upper bounds are more involved and mirror the combinatorial structure of the metric graph as well. Combining them, we deduce that these spectral minimal energies also satisfy a Weyl-type asymptotic law similar to the well-known one for eigenvalues of quantum graph Laplacians with various vertex conditions. Drawing on two examples we show that in general no second term in the asymptotic expansion for minimal partition energies can exist, but show that various kinds of behaviour are possible. We also study certain aspects of the asymptotic behaviour of the minimal partitions themselves.

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arXiv2020

On Pleijel's nodal domain theorem for quantum graphs

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 $(\frac{ν_n}{n})_{n\in\mathbb N}$, which are shown always to form a finite subset of $(0,1]$. This} extends the previously known result that $ν_n\sim n$ \textit{generically}, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs with rationally dependent edge lengths, one can find eigenfunctions thereon for which ${ν_n}\not\sim {n}$; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

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