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James B. Kennedy

Publications and source records attributed to James B. Kennedy.

33 records · Page 2Linked to original sources
arXiv2020

On the eigenvalues of quantum graph Laplacians with large complex $δ$ couplings

We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as $δ$ conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to $\infty$ in $\mathbb{C}$: for each vertex $v$ with a Robin parameter $α\in \mathbb{C}$ for which $\mathrm{\Re}\,α\to -\infty$ sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like $-α^2/\mathrm{deg}\,v^2$, while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at $v$; if $\mathrm{Re}\,α$ remains bounded from below, then all eigenvalues stay near the Dirichlet spectrum. Our proof is based on an analysis of the corresponding Dirichlet-to-Neumann matrices (Titchmarsh--Weyl M-functions). We also use sharp trace-type inequalities to prove estimates on the numerical range and hence on the spectrum of the operator, which allow us to control both the real and imaginary parts of the eigenvalues in terms of the real and imaginary parts of the Robin parameter(s).

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arXiv2020

A theory of spectral partitions of metric graphs

We introduce an abstract framework for the study of clustering in metric graphs: after suitably metrising the space of graph partitions, we restrict Laplacians to the clusters thus arising and use their spectral gaps to define several notions of partition energies; this is the graph counterpart of the well-known theory of spectral minimal partitions on planar domains and includes the setting in [Band \textit{et al}, Comm.\ Math.\ Phys.\ \textbf{311} (2012), 815--838] as a special case. We focus on the existence of optimisers for a large class of functionals defined on such partitions, but also study their qualitative properties, including stability, regularity, and parameter dependence. We also discuss in detail their interplay with the theory of nodal partitions. Unlike in the case of domains, the one-dimensional setting of metric graphs allows for explicit computation and analytic -- rather than numerical -- results. Not only do we recover the main assertions in the theory of spectral minimal partitions on domains, as studied in [Conti \textit{et al}, Calc.\ Var.\ \textbf{22} (2005), 45--72; Helffer \textit{et al}, Ann.\ Inst.\ Henri Poincaré Anal.\ Non Linéaire \textbf{26} (2009), 101--138], but we can also generalise some of them and answer (the graph counterparts of) a few open questions.

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arXiv2020

Disjointness-preserving operators and isospectral Laplacians

All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of isometric building blocks arranged in different ways, such that the unitary operator intertwining the Laplacians acts as a sum of overlapping "local" isometries mapping the copies to each other. We prove and explore a complementary positive statement: if an operator intertwining two appropriate realisations of the Laplacian on a pair of domains preserves disjoint supports, then under additional assumptions on it generally far weaker than unitarity, the domains are congruent. We show this in particular for the Dirichlet, Neumann and Robin Laplacians on spaces of continuous functions and on $L^2$-spaces.

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arXiv2019

On the eigenvalues of the Robin Laplacian with a complex parameter

We study the spectrum of the Robin Laplacian with a complex Robin parameter $α$ on a bounded Lipschitz domain $Ω$. We start by establishing a number of properties of the corresponding operator, such as generation properties, local analytic dependence of the eigenvalues and eigenspaces on $α\in \mathbb C$, and basis properties of the eigenfunctions. Our focus, however, is on bounds and asymptotics for the eigenvalues as functions of $α$: we start by providing estimates on the numerical range of the associated operator, which lead to new eigenvalue bounds even in the case $α\in \mathbb R$. For the asymptotics of the eigenvalues as $α\to \infty$ in $\mathbb C$, in place of the min-max characterisation of the eigenvalues and Dirichlet-Neumann bracketing techniques commonly used in the real case, we exploit the duality between the eigenvalues of the Robin Laplacian and the eigenvalues of the Dirichlet-to-Neumann map. We use this to show that every Robin eigenvalue either diverges to $\infty$ in $\mathbb C$ or converges to a point in the spectrum of the Dirichlet Laplacian, and also to give a comprehensive treatment of the special cases where $Ω$ is an interval, a hyperrectangle or a ball. This leads to the conjecture that on a general smooth domain in dimension $d\geq 2$ all eigenvalues converge to the Dirichlet spectrum if ${\rm Re}\, α$ remains bounded from below as $α\to \infty$, while if ${\rm Re}\, α\to -\infty$, then there is a family of divergent eigenvalue curves, each of which behaves asymptotically like $-α^2$.

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arXiv2019

Surgery principles for the spectral analysis of quantum graphs

We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $δ$-type), which show how various types of changes of a local or localised nature to a graph impact the spectrum of the Laplacian. Many of these principles are entirely new, these include "transplantation" of volume within a graph based on the behaviour of its eigenfunctions, as well as "unfolding" of local cycles and pendants. In other cases we establish sharp generalisations, extensions and refinements of known eigenvalue inequalities resulting from graph modification, such as vertex gluing, adjustment of vertex conditions and introducing new pendant subgraphs. To illustrate our techniques we derive a new eigenvalue estimate which uses the size of the doubly connected part of a metric graph to estimate the spectral gap. This quantitative isoperimetric-type inequality interpolates between two known estimates---one assuming the entire graph is doubly connected and the other making no connectivity assumption (and producing a weaker bound)---and includes them as special cases.

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arXiv2018

Eventually and asymptotically positive semigroups on Banach lattices

We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neumann operator on $L^2$ and the Laplacian with non-local boundary conditions on $L^2$ within the one unified theory. We also introduce and analyse a weaker notion of eventual positivity which we call "asymptotic positivity", where trajectories associated with positive initial data converge to the positive cone in the Banach lattice as $t \to \infty$. This allows us to discuss further examples which do not fall within the above-mentioned framework, among them a network flow with non-positive mass transition and a certain delay differential equation.

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arXiv2017

Edge connectivity and the spectral gap of combinatorial and quantum graphs

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds is the well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the $p$-Laplacian and allow us to identify the minimizers. Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are "asymptotically correct", i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

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arXiv2015

Eventually Positive Semigroups of Linear Operators

We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy problem is positive for large enough time. Characterisations of such semigroups are given by means of resolvent properties of the generator and Perron--Frobenius type spectral conditions. We apply these characterisations to prove eventual positivity of several examples of semigroups including some generated by fourth order elliptic operators and a delay differential equation. We also consider eventually positive semigroups on arbitrary Banach lattices and establish several results for their spectral bound which were previously only known for positive semigroups.

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arXiv2015

Nonlinear semigroups generated by $j$-elliptic functionals

We generalise the theory of energy functionals used in the study of gradient systems to the case where the domain of definition of the functional cannot be embedded into the Hilbert space $H$ on which the associated operator acts, such as when $H$ is a trace space. We show that under weak conditions on the functional $φ$ and the map $j$ from the effective domain of $φ$ to $H$, which in opposition to the classical theory does not have to be injective or even continuous, the operator on $H$ naturally associated with the pair $(φ,j)$ nevertheless generates a nonlinear semigroup of contractions on $H$. We show that this operator, which we call the $j$-subgradient of $φ$, is the (classical) subgradient of another functional on $H$, and give an extensive characterisation of this functional in terms of $φ$ and $j$. In the case where $H$ is an $L^2$-space, we also characterise the positivity, $L^\infty$-contractivity and existence of order-preserving extrapolations to $L^q$ of the semigroup in terms of $φ$ and $j$. This theory is illustrated through numerous examples, including the $p$-Dirichlet-to-Neumann operator, general Robin-type parabolic boundary value problems for the $p$-Laplacian on very rough domains, and certain coupled parabolic-elliptic systems.

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arXiv2015

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.

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arXiv2012

Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

We consider the problem of minimising the $n^{th}-$eigenvalue of the Robin Laplacian in $\mathbb{R}^{N}$. Although for $n=1,2$ and a positive boundary parameter $α$ it is known that the minimisers do not depend on $α$, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on $α$. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with $n^{1/N}$, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as $n$ goes to infinity. Numerical results then support the conjecture that for each $n$ there exists a positive value of $α_{n}$ such that the $n^{\rm th}$ eigenvalue is minimised by $n$ disks for all $0<α<α_{n}$ and, combined with analytic estimates, that this value is expected to grow with $n^{1/N}$.

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