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Noema Nicolussi

Publications and source records attributed to Noema Nicolussi.

15 recordsLinked to original sources
arXiv2025

Optimal eigenvalues on a metric graph with densities

We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of operators: discrete graph Laplacians, Kirchhoff Laplacians and Dirichlet-to-Neumann operators on graphs. Our main interest lies in spectral optimization with respect to the underlying measure. In contrast to the setting of domains and manifolds, we prove that a minimal $k$-th eigenvalue exists, whereas the corresponding maximization problem has no meaning. We then establish connections between these optimal eigenvalues and the geometry of the metric graph, including a transparent geometric characterization of the first optimal eigenvalue via the resistance metric, and a Weyl law for the higher optimal eigenvalues.

math.SP↗
arXiv2024

Moduli of hybrid curves I: Variations of canonical measures

The present paper is the first in a series devoted to the study of asymptotic geometry of Riemann surfaces and their moduli spaces. We introduce the moduli space of hybrid curves as a new compactification of the moduli space of curves, refining the one obtained by Deligne and Mumford. This is the moduli space for multiscale geometric objects which mix complex and higher rank tropical and non-Archimedean geometries, reflecting both discrete and continuous features. We define canonical measures on hybrid curves which combine and generalize Arakelov-Bergman measures on Riemann surfaces and Zhang measures on metric graphs. We then show that the universal family of canonically measured hybrid curves over this moduli space varies continuously. This provides a precise link between the non-Archimedean Zhang measure and variations of Arakelov-Bergman measures in families of Riemann surfaces, answering a question which has been open since the pioneering work of Zhang on admissible pairing in the nineties.

math.AG↗
arXiv2023

Capacity of infinite graphs over non-Archimedean ordered fields

In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator.

math.AP↗
arXiv2022

A note on Spectral Analysis of Quantum graphs

We provide an introductory review of some topics in spectral theory of Laplacians on metric graphs. We focus on three different aspects: the trace formula, the self-adjointness problem and connections between Laplacians on metric graphs and discrete graph Laplacians.

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arXiv2022

Moduli of hybrid curves II: Tropical and hybrid Laplacians

The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the Laplace operator and the solutions to the Poisson equation on Riemann surfaces. As the main theorem of this paper, we then obtain a layered description of the asymptotics of Arakelov Green functions on Riemann surfaces close to the boundary of their moduli spaces. This is done in terms of a suitable notion of hybrid Green functions. As a byproduct of our approach, we obtain other results of independent interest. In particular, we introduce higher rank canonical compactifications of fans and polyhedral spaces and use them to define the moduli space of higher rank tropical curves. Moreover, we develop the first steps of a function theory in higher rank non-Archimedean, hybrid, and tame analysis. Furthermore, we establish the convergence of the Laplace operator on metric graphs toward the tropical Laplace operator on limit tropical curves in the corresponding moduli spaces, leading to new perspectives in operator theory on metric graphs. Our result on the Arakelov Green function is inspired by the works of several authors, in particular those of Faltings, de Jong, Wentworth and Wolpert, and solves a long-standing open problem arising from the Arakelov geometry of Riemann surfaces. The hybrid layered behavior close to the boundary of moduli spaces is expected to be a broad phenomenon and will be explored in our forthcoming work.

math.AG↗
arXiv2022

Self-adjoint and Markovian extensions of infinite quantum graphs

We investigate the relationship between one of the classical notions of boundaries for infinite graphs, \emph{graph ends}, and self-adjoint extensions of the minimal Kirchhoff Laplacian on a metric graph. We introduce the notion of \emph{finite volume} for ends of a metric graph and show that finite volume graph ends is the proper notion of a boundary for Markovian extensions of the Kirchhoff Laplacian. In contrast to manifolds and weighted graphs, this provides a transparent geometric characterization of the uniqueness of Markovian extensions, as well as of the self-adjointness of the Gaffney Laplacian -- the underlying metric graph does not have finite volume ends. If however finitely many finite volume ends occur (as is the case of edge graphs of normal, locally finite tessellations or Cayley graphs of amenable finitely generated groups), we provide a complete description of Markovian extensions upon introducing a suitable notion of traces of functions and normal derivatives on the set of graph ends.

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arXiv2021

Laplacians on infinite graphs: discrete vs continuous

There are two main notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this overview, we will focus on the relationship between them (spectral and parabolic properties). Our main conceptual message is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides.

math.SP↗
arXiv2021

A note on the Gaffney Laplacian on infinite metric graphs

We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide criteria, formulated in terms of finite volume graph ends, for the Gaffney Laplacian to be closed.

math.FA↗
arXiv2021

A Glazman-Povzner-Wienholtz Theorem on graphs

The Glazman-Povzner-Wienholtz theorem states that the completeness of a manifold, when combined with the semiboundedness of the Schrödinger operator $-Δ+ q$ and suitable local regularity assumptions on $q$, guarantees its essential self-adjointness. Our aim is to extend this result to Schrödinger operators on graphs. We first obtain the corresponding theorem for Schrödinger operators on metric graphs, allowing in particular distributional potentials $q\in H^{-1}_{\rm loc}$. Moreover, we exploit recently discovered connections between Schrödinger operators on metric graphs and weighted graphs in order to prove a discrete version of the Glazman-Povzner-Wienholtz theorem.

math.SP↗
arXiv2020

Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flow

This article is concerned with the isospectral problem \[ -f'' + \frac{1}{4} f = zωf + z^2 \upsilon f \] for the periodic conservative Camassa-Holm flow, where $ω$ is a periodic real distribution in $H^{-1}_{\mathrm{loc}}(\mathbb{R})$ and $\upsilon$ is a periodic non-negative Borel measure on $\mathbb{R}$. We develop basic Floquet theory for this problem, derive trace formulas for the associated spectra and establish continuous dependence of these spectra on the coefficients with respect to a weak$^\ast$ topology.

math.SP↗
arXiv2019

Quantum graphs on radially symmetric antitrees

We investigate spectral properties of Kirchhoff Laplacians on radially symmetric antitrees. This class of metric graphs enjoys a rich group of symmetries, which enables us to obtain a decomposition of the corresponding Laplacian into the orthogonal sum of Sturm--Liouville operators. In contrast to the case of radially symmetric trees, the deficiency indices of the Laplacian defined on the minimal domain are at most one and they are equal to one exactly when the corresponding metric antitree has finite total volume. In this case, we provide an explicit description of all self-adjoint extensions including the Friedrichs extension. Furthermore, using the spectral theory of Krein strings, we perform a thorough spectral analysis of this model. In particular, we obtain discreteness and trace class criteria, criterion for the Kirchhoff Laplacian to be uniformly positive and provide spectral gap estimates. We show that the absolutely continuous spectrum is in a certain sense a rare event, however, we also present several classes of antitrees such that the absolutely continuous spectrum of the corresponding Laplacian is $[0,\infty)$.

math.SP↗
arXiv2018

Spectral Estimates for Infinite Quantum Graphs

We investigate the bottom of the spectra of infinite quantum graphs, i.e., Laplace operators on metric graphs having infinitely many edges and vertices. We introduce a new definition of the isoperimetric constant for quantum graphs and then prove the Cheeger-type estimate. Our definition of the isoperimetric constant is purely combinatorial and thus it establishes connections with the combinatorial isoperimetric constant, one of the central objects in spectral graph theory and in the theory of simple random walks on graphs. The latter enables us to prove a number of criteria for quantum graphs to be uniformly positive or to have purely discrete spectrum. We demonstrate our findings by considering trees, antitrees and Cayley graphs of finitely generated groups.

math.SP↗
arXiv2018

Strong Isoperimetric Inequality for Tessellating Quantum Graphs

We investigate isoperimetric constants of infinite tessellating metric graphs. We introduce a curvature-like quantity, which plays the role of a metric graph analogue of discrete curvature notions for combinatorial tessellating graphs. We then prove a lower estimate and a criterium for positivity of the isoperimetric constant.

math.MG↗
arXiv2017

On the Hamiltonian-Krein index for a non-self-adjoint spectral problem

We investigate the instability index of the spectral problem $$ -c^2y'' + b^2y + V(x)y = -\mathrm{i} z y' $$ on the line $\mathbb{R}$, where $V\in L^1_{\rm loc}(\mathbb{R})$ is real valued and $b,c>0$ are constants. This problem arises in the study of stability of solitons for certain nonlinear equations (e.g., the short pulse equation and the generalized Bullough-Dodd equation). We show how to apply the standard approach in the situation under consideration and as a result we provide a formula for the instability index in terms of certain spectral characteristics of the 1-D Schrödinger operator $H_V=-c^2\frac{d^2}{dx^2}+b^2 +V(x)$.

math.SP↗