Lies Beers, Raffaella Mulas
We investigate Fair and Tolerant (FAT) graph colorings, a coloring framework in which each vertex is allowed to share its color with a prescribed fraction of its neighbors, while the remaining neighbors are required to be distributed evenly among the other coloring classes. In particular, we determine the FAT chromatic number for all complete multipartite graphs, and we analyze the behavior of FAT colorings under several graph operations. Although spectral methods form the primary focus, several combinatorial arguments are included to complement the results.
Kai Hippi, Félix Lequen, Søren Mikkelsen, Tuomas Sahlsten, Henrik Ueberschär
Let $-Δ_{\mathbb{H}}+V$ be the Schrödinger operator on $\mathbb{H}$ where $V \in L^p(\mathbb{H}) \cap L^\infty(\mathbb{H})$ for some $p > 0$. If $(X_n)$ is a uniformly discrete sequence of compact hyperbolic surfaces with a uniform spectral gap that Benjamini-Schramm converges to $\mathbb{H}$, we prove quantum mixing for the eigenfunctions of $-Δ_{X_n}+V_n$ in any sufficiently large spectral window $I$, where $V_n$ is the potential on $X_n$ induced by $V$. These apply to large degree lifts of a potential on a base surface such as congruence covers of arithmetic surfaces, with high probability to random hyperbolic surfaces in the Weil-Petersson model of large genus, and to Hartree one-particle operators arising in thermodynamic limit of many-body Bose gas on hyperbolic surfaces. The proof uses the Duhamel formula for the hyperbolic wave equation together with exponential mixing of the geodesic flow on $T^1 X_n$.
Sara Koljančić, Polona Oblak
In this paper, we investigate the non-symmetric Strong Spectral Property (nSSP) from a combinatorial perspective. To zero-nonzero patterns of matrices we associate directed graphs and study when they require or allow the nSSP, providing a framework that avoids verifying the nSSP for individual matrices. A new combinatorial method is introduced and used to recognise several patterns that require the nSSP. It is shown that loop assignments in double paths play a critical role in establishing this property, and we show that an open question regarding irreducible tridiagonal patterns has a negative answer. We also investigate whether the minimum number of arcs in a directed graph on $n$ vertices that requires the nSSP, is equal to $2n-1$, and confirm this minimum for several specific digraph families.
Yannick de Bruijn, Erik Orvehed Hiltunen
Tridiagonal $k$-Toeplitz operators provide a natural framework for modelling one-dimensional $k$-periodic lattice systems. A fundamental connection is obtained between Coburn's lemma for tridiagonal $k$-Toeplitz operators and the existence of edge modes. We reveal that topological edge modes are characterised by the eigenvalues of the leading principal submatrix of the symbol function. A complete analysis of tridiagonal interface operators satisfying global inversion symmetry is then presented. These results are applied to finite one-dimensional $k$-periodic chains of damped resonators that satisfy both local and global inversion symmetry. Additionally, disordered tight-binding interface operators are shown to support a topologically robust zero-energy interface state. Numerical simulations are conducted to illustrate the theoretical findings.
Dario Giandinoto, Boris Shapiro
Twisted Toeplitz matrices constitute a generalization of Toeplitz matrices in the sense that the entries on each diagonal no longer need to be constant, but are given by the values of a continuous function on a partition of $[0,1]$. We study the limiting statistical distribution of the eigenvalues of matrices of the form $R_n(a) = T_n(a) + σ_n X_n$, where $T_n(a)$ is a sequence of non-Hermitian tridiagonal twisted Toeplitz matrices, $X_n$ is a sequence of tridiagonal random matrices whose entries have mean $0$ and finite variance, and $σ_n\to0$. The limiting distribution turns out to be a two-dimensional measure which is in general different from the push-forward of the Lebesgue measure by the symbol. We also explain how the results could extend to banded non-Hermitian twisted Toeplitz matrices.
Luciano Abadías, Alejandro Mahillo, Pedro J. Miana
In this paper we study boundedness and detailed spectral properties for the Cesàro-Hardy operator and some generalizations in $L^p[0,1]$. The study employs $C_0$-semigroup theory, expressing the Cesàro-Hardy operators and their dual operators through subordination with $C_0$-semigroups $T(t)$ and $S(t)$ respectively. The spectral properties of the semigroup's infinitesimal generators are transferred to the Cesàro-Hardy operators using functional calculus methods. Furthermore, some implications for the Invariant Subspace Problem are explored by demonstrating the universality of certain translations related to the semigroup $T(t)$, and providing results on the invariant subspaces of these operators.
Dang-Zheng Liu, Guangyi Zou
The first paper in this series introduced a \emph{short-to-long mixing} condition that captures mean-field GOE/GUE edge universality in the supercritical sparsity regime, for symmetric/Hermitian random matrices with independent entries and a Markov variance profile. This condition reduces the universality problem to the mixing properties of the underlying Markov chains. In this paper, we develop new \emph{short-to-long comparison} conditions that extend the analysis to the subcritical and critical sparsity regimes. Specifically, we prove that two inhomogeneous random matrices exhibit the same universal edge statistics whenever their variance-profile Markov chains are comparable, regardless of the fine details of the matrix entries. To illustrate the power of our Markov chain comparison theorem, we derive the spectral edge statistics for several prototypical models: random band matrices, the Wegner orbital model, and Hankel-profile random matrices. These comparisons uncover a rich landscape of both universal and non-universal phenomena -- shaped by geometric structure, spike patterns, and domains of stable attraction -- features that lie fundamentally beyond the reach of classical random matrix theory.
Lyonell Boulton, Connor Evans
We establish a framework to determine the linear completeness of families of non-linear trajectories in Hilbert spaces, which relies on an infinite analytic block Toeplitz operator formulation. By means of this approach, we show the linear completeness in Sobolev spaces of two families of classical functions. One is the moving family of dilated Weierstrass functions. The other is the family of eigenfunctions of the Gross-Pitaevskii equation with trapping potential in an infinite square well. Our results provide a new insight on the formulation of general methods to examine this intriguing concept, bridging classical non-linear analysis and linear approximation theory.
George Bayliss, Jared C. Bronski
The Evans function is an analytic function that encodes information about the intersection of certain subspaces in ODE boundary value problems. As such it is a useful tool for computing the spectrum of boundary value problems arising in the stability of coherent structures. In typical applications one is interested in the roots of the Evans function, but the overall normalization is somewhat arbitrary. We present a natural normalization of the Evans function on compact domains such that the magnitude of the Evans function provides a lower bound on the distance to the nearest point in the spectrum. In other words the magnitude of the Evans function at a point in the resolvent set implies that a ball about the point in question lies in the resolvent set. Thus, when appropriately normalized, not only does the Evans function $E(λ)$ vanish if and only if $λ$ lies in the spectrum of the operator in question, but a non-zero value for the Evans function guarantees that a disk of radius $|E(λ^*)|$ about the point $λ^*$ lies in the resolvent set. We present some calculations for some common sets of boundary conditions on a compact interval, and present some numerical experiments for 2nd and 4th order self-adjoint operators and for a linearized modified Korteweg-De Vries equation.
Jussi Behrndt, Petr Siegl, Nicolas Weber
We study the existence of negative eigenvalues for two-dimensional Schrödinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a century ago, Simon was the first to describe the unique negative eigenvalue emerging from the threshold of the essential spectrum of one- and two-dimensional Schrödinger operators. The aim of this paper is to extend Simon's results in two dimensions to a broader class of potentials, allowing for both stronger singularities and slower decay at infinity, at the cost of losing uniqueness of weakly coupled eigenvalues.
Emily B. Dryden, Carolyn Gordon, Javier Moreno, Julie Rowlett, Carlos Villegas-Blas
The extent to which the geometry of an object is determined by some associated spectral data is a longstanding problem. We investigate this problem in the context of the Steklov spectrum, focusing on convex polygons. We prove that almost all triangles are uniquely determined by their Steklov spectra within the class of all triangles; further results depending on the types of angles in the triangles are given. We examine three special classes of convex quadrilaterals--rectangles, parallelograms, and kites--and obtain results ranging from unique spectral determination to determination up to three possibilities. For regular $n$-gons, we are again able to prove spectral determination within certain classes of polygons. More generally, we investigate the extent to which the Steklov spectrum distinguishes convex polygons from simply-connected domains with smooth boundary; that is, does the Steklov spectrum detect corners? We prove that triangles and quadrilaterals are spectrally distinguished from such smoothly bounded domains; moreover, we show that having the same Steklov spectrum as such a domain imposes substantial restrictions on the edge lengths of higher-order $n$-gons. Throughout, our main tool is the characteristic polynomial developed in works by Stanislav Krymski, Michael Levitin, Leonid Parnovski, Iosif Polterovich, and David A. Sher.
Sergey A. Denisov, Giorgio Young
We study the wave operators for a Jacobi matrix whose spectral measure satisfies the Szegö condition. We prove existence and completeness of wave operators under a mild additional assumption on the Verblunsky coefficients of the associated measure on the unit circle.
Rupert L. Frank, Long Meng, Phan Thành Nam, Heinz Siedentop
We study the spectral properties of minimizers of the Müller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a Müller minimizer $γ_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $γ_*$. In particular, in the atomic case $V=Z|x|^{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z^{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schrödinger ground states, the analysis in Müller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.
Rupert L. Frank, Simon Larson
We review our recent results on the problem of optimizing Riesz means of Laplace eigenvalues among convex sets of given measure in the regime where the cut-off parameter in the definition of the Riesz means tends to infinity. We show that for a certain range of Riesz exponents, the optimizing sets converge to a ball. We also present some new results where we optimize over disjoint unions of convex sets.
Kalpesh M. Popat, Kunal R. Shingala
The graph $G_σ$ is obtained from graph $G$ by attaching self loops on $σ$ vertices. The energy $ E(G_σ)$ of the graph $G_σ$ with order $n$ and eigenvalues $λ_1,λ_2,\dots,λ_n$ is defined as $ E(G_σ)= \displaystyle \sum_{i=1}^n\left|λ_i-\dfracσ{n}\right| $. It has been proved that if $σ=0\; or\; n$ then $ E(G)=E(G_σ) $. The obvious question arise: Are there any graph such that $E(G)=E(G_σ)$ and 0$<σ<n$? We have found an affirmative answer of this question and contributed a graph family which satisfies this property.
Yunfeng Zhang
We present a basic pointwise bound for the irreducible characters of $\mathrm{SU}(3)$ and, as an application, derive new $L^p$ bounds for these characters. Our approach is based on the descent of characters to singular sets and the cancellation in this formula.
A. Iosevich, C. Park
We establish uncertainty principles on compact Riemannian manifolds without boundary in the setting of Laplace-Beltrami operators, including the case of real-valued singular potentials. We replace the classical homogeneity assumption by a quantitative spectral condition and obtain corresponding stability versions of uncertainty inequalities. In particular, we prove that \[ (1-ε-ε')^2 \leq \frac{|E|}{|M|}\cdot \# X_S \cdot \sup_{x\in E} \frac{A_S(x)}{\frac{\# X_S}{|M|}}, \] which recovers the classical bound in the homogeneous case, quantifies its deterioration in the presence of spectral inhomogeneity, and is shown to be sharp in general. In {\it dimension one}, we show that the homogeneity condition holds automatically, and we complement this rigidity by incorporating Fourier-ratio complexity bounds, yielding a quantitative relationship between spectral complexity and spatial support. In higher dimensions, we derive analogous results using pointwise Weyl laws and the eigenfunction restriction estimates on submanifolds.
Ben Cardoen, Jeremy Budd, Enrico Amico, Ghassan Hamarneh, Fabian Spill
We characterize which local matrix structures saturate Weyl's eigenvalue perturbation bound for graph Laplacians under geometrically constrained vertex displacements. Geometric graphs with heavy-tailed vertex noise arise across sensor networks, biological imaging, and spatial omics, yet tractable predictions for noise-induced spectral error remain limited. We study geometric graphs abstracted from biophysical systems, incorporating clearance, planarity, and identifiability constraints that govern physically realizable embeddings. Within this constrained setting, we identify witness motifs, small subgraphs in maximally noise-sensitive geometric configurations, that dominate weighted-degree and graph Laplacian spectral perturbations under tempered power-law vertex displacements. This motif decomposition reduces global spectral sensitivity to a finite catalog of local extremal structures and identifies configurations that attain Weyl-tight bounds. We then lift these constrained-graph results to general straight-line embedded graphs in arbitrary dimension via local repair operations producing a constrained surrogate graph that preserves sensitivity-relevant structure. To quantify noise-induced spectral variation in both strong-oracle and weak-oracle regimes, we introduce stochastic co-spectrality (SC) and the stochastic spectral separation index (S3I), which characterize when observed spectral distances are noise-driven and when noise parameters are separable. Together, these results provide a principled pathway from local geometric noise to global spectral error in graph Laplacian matrices, enabling estimation of spectral fragility from graph structure without exhaustive eigenvalue computation or restrictive distributional assumptions beyond moment bounds.
Olga Y. Kushel
The concept of matrix $D$-stability, introduced in 1958 by Arrow and McManus is of major importance due to the variety of its applications. However, characterization of matrix $D$-stability for dimensions $n > 4$ is considered as a hard open problem. In this paper, we propose a recursive delete/zero algorithm for testing matrix $D$-stability. The algorithm generates a binary tree of parameter-dependent matrices ${\mathbf A}_s$ and yields recurrence relations for the real and imaginary parts of $\det({\mathbf A}_s)$. These relations lead to a hierarchy of sufficient for $D$-stability conditions, expressed in terms of principal minors. Numerical experiments confirm the practical feasibility of the approach.
Raphael Ponge
Semiclassical analysis and noncommutative geometry are two pillars of quantum theory. It's only recently that bridges between them have been emerging. In this monograph, we combine various techniques from functional analysis and spectral theory to obtain semiclassical Weyl laws and extensions of Connes' integration formula for a large class of noncommutative manifolds (i.e., spectral triples). These results generalize and simplify recent results of McDonald-Sukochev-Zanin. In particular, all the regularity assumptions and restrictions on dimension there are removed in our approach. Moreover, the Tauberian condition used by McDonald-Sukochev-Zanin is replaced by a weaker spectral theoretic condition, called Condition (W). That condition holds in fairly greater generality and significantly open the scope of applicability of the main results. We also give Tauberian conditions that imply Condition (W). These Tauberian conditions are easier to check in practice than the Tauberian condition of McDonald-Sukochev-Zanin and are satisfied in numerous examples. The need for these conditions was highlighted by Alain Connes in an online seminar. The main results of this memoire are illustrated by semiclassical Weyl's laws and integration formulas in the following settings: (i) Dirichlet and Neumann problems on Euclidean domains with smooth boundaries; (ii) closed Riemannian manifolds; (iii) open manifolds with conformally cusp metrics of finite volume; (iv) quantum tori; and (v) sub-Riemannian manifolds.