arXiv subjects

Matthias Hofmann

Publications and source records attributed to Matthias Hofmann.

15 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 Poincar\'e-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.

math.SP↗

arXiv2026

Ambarzumian-type theorems for Hermitian matrices with applications

A foundational result in inverse spectral theory due to Ambarzumian (1929) states that the Neumann Laplacian on an interval is not isospectral to the Neumann Laplacian with an additional non-zero potential. In this note, our aim is to investigate Ambarzumian-type theorems for certain classes of Hermitian matrices, including well-known matrices such as the discrete Laplacian on finite graphs. In addition, using different methods, we establish an Ambarzumian-type theorem for matrices with vanishing diagonal, in particular, the adjacency matrix on finite graphs. In this way, we generalize existing results on Ambarzumian-type theorems to general finite discrete graphs.

math.SP↗

arXiv2025

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 Schr\"odinger operator $-\Delta +V$, 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 $\Sigma$ of the essential spectrum of the Schr\"odinger 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.

math.SP↗

arXiv2025

On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs

We study the structure of eigenfunctions of the Laplacian on quantum graphs, with a particular focus on Morse eigenfunctions via nodal and Neumann domains. Building on Courant-type arguments, we establish upper bounds for the number of nodal points and explore conditions under which Neumann domains of eigenfunctions correspond to minimizers to a class of spectral partition problems often known as spectral minimal partitions. The main focus will be the analysis on tree graphs, where we characterize the spectral energies of such partitions and relate them to the eigenvalues of the Laplacian under genericity assumptions. Notably, we introduce a notion analogous to Courant-sharpness for Neumann counts and demonstrate when spectral minimal partitions coincide with partitions formed by Neumann domains of eigenfunctions.

math.SP↗

arXiv2025

Eigenvalues of the discrete p-Laplacian via graph surgery

We develop a Hellmann--Feynman type perturbation theory for the discrete signed $p$-Laplacian and apply it to a parametrized perturbation by edge cuts. We show that the eigenvalues of the signed $p$-Laplacian can be characterized as citical values of the parameter-dependent eigenvalues of a simpler graph.

math.AP↗

arXiv2025

On the asymptotic behavior of the spectral gap for discrete Schr\"odinger operators

In this note we elaborate on the asymptotic behavior of the spectral gap of a class of discrete Schr\"odinger operators defined on a path graph in the limit of infinite volume. We confirm recent results and generalize them to a larger class of potentials using entirely different methods. Notably, we also resolve a conjecture previously proposed in this context. This then yields new insights into the rate at which the spectral gap tends to zero as the volume increases.

math.SP↗

arXiv2025

On the existence of Nehari ground states for the Nonlinear Schr\"odinger Equation on Discrete Graphs

We study standing waves for the nonlinear Schr\"odinger equation on a discrete graph. We characterize for a self-adjoint realizations of Schr\"odinger operators conditions related with the geometry of the graph that guarantee discreteness of the spectrum and study ground states on the generalized Nehari manifold in order to prove the existence of standing wave solutions in the self-focusing and defocusing case. In this context, we show properties of the solutions, such as integrability. Finally, we discuss decay properties of solutions and the bifurcation of solutions from the trivial solution.

math.AP↗

arXiv2024

Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds

In this article, we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory and the vanishing order of eigenfunctions. We identify the nodal set of an eigenfunction of the Laplacian (with smooth potential) on a compact, two-dmensional Riemannian manifolds, that is on Riemannian surfaces, as an embedded metric graph and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the $k$-th eigenfunction and the sum of vanishing orders at critical points in terms of $k$ and the Euler-Poincar\'e characteristic of the surface.

math.AP↗

arXiv2022

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 Schr\"odinger operator of the form $-\Delta + V$ 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 $\Sigma$ of the essential spectrum of the corresponding Schr\"odinger 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 $\Sigma$, and if it is strictly below $\Sigma$, 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 Schr\"odinger operator-based partitions of unbounded domains in Euclidean space.

math.SP↗

arXiv2022

Grid Tariffs for Peak Demand Reduction: Is there a Price Signal Conflict with Electricity Spot Prices?

The electricity grid is expected to require vast investments due to the decarbonization-by-electrification trend, calling for a change in grid tariff design which provides proper incentives for reducing peak loads. However, price signals from grid tariffs could be distorted from electricity spot prices which also represents a significant of the total consumer electricity bill. This paper attempts to identify whether there is a price signal conflict between grid tariffs and spot prices. Four different grid tariff designs are compared, using a generic demand response model as part of a cost-minimizing linear program to simulate the reduction in peak load. The method is applied to metered electricity demand from 3608 consumers in Oslo, Norway. Results show that new grid tariff designs reduce peak loads by 1-4%, and that reduction in peak load is smaller when consumers are subject to electricity spot prices.

math.OC↗

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.

math.SP↗

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.

math-ph↗

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.

math.SP↗

arXiv2020

An existence theory for nonlinear equations on metric graphs via energy methods

The purpose of this paper is to develop a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces $(\mathcal M, d, μ)$. We apply this theory to functionals defined on metric graphs $\mathcal G$, in particular $L^2$-constrained minimization problems for functionals of the form $$E(u) = \frac{1}{2} a(u,u) - \frac{1}{q}\int_{\mathcal K} |u|^q \, \mathrm dx,$$ where $q>2$, $a(\cdot, \cdot)$ is a suitable symmetric sesquilinear form on some function space on $\mathcal G$ and $\mathcal K \subseteq \mathcal G$ is given. We show how the existence of solutions can be obtained via decomposition methods using spectral properties of the operator $A$ associated with the form $a(\cdot, \cdot)$ and discuss the spectral quantities involved. An example that we consider is the higher-order variant of the stationary NLS (nonlinear Schrödinger) energy functional with potential $V\in L^2+ L^\infty(\mathcal G)$ $$ E^{(k)}(u)= \frac{1}{2} \int_{\mathcal G} |u^{(k)}|^2+ V(x) |u|^2 \, \mathrm dx - \frac{1}{p} \int_{\mathcal K} |u|^q \, \mathrm dx$$ defined on a class of higher-order Sobolev spaces $H^k(\mathcal G)$ that we introduce. When $\mathcal K$ is a bounded subgraph, one has localized nonlinearities, which we treat as a special case. When $k=1$ we also consider metric graphs with infinite edge set as well as magnetic potentials. Then the operator $A$ associated to the linear form is a Schrödinger operator, and in the $L^2$-subcritical case $2<q<6$, we obtain generalizations of existence results for the NLS functional as for instance obtained by Adami, Serra and Tilli [JFA 271 (2016), 201-223], and Cacciapuoti, Finco and Noja [Nonlinearity 30 (2017), 3271-3303], among others.

math.AP↗

arXiv2004

Dynamic sea level changes following changes in the thermohaline circulation

Using the coupled climate model CLIMBER-3a, we investigate changes in sea surface elevation due to a weakening of the thermohaline circulation (THC). In addition to a global sea level rise due to a warming of the deep sea, this leads to a regional dynamic sea level change which follows quasi-instantaneously any change in the ocean circulation. We show that the magnitude of this dynamic effect can locally reach up to ~1m, depending on the initial THC strength. In some regions the rate of change can be up to 20-25 mm/yr. The emerging patterns are discussed with respect to the oceanic circulation changes. Most prominent is a south-north gradient reflecting the changes in geostrophic surface currents. Our results suggest that an analysis of observed sea level change patterns could be useful for monitoring the THC strength.

physics.ao-ph↗