CyberSec Research

Spectral properties of Toeplitz operators and their finite truncations have long been central in operator theory. In the finite dimensional, non-normal setting, the spectrum is notoriously unstable under perturbations. Random perturbations provide a natural framework for studying this instability and identifying spectral features that emerge in typical noisy situations. This article surveys recent advances on the spectral behavior of (polynomially vanishing) random perturbations of Toeplitz matrices, focusing mostly on the limiting spectral distribution, the distribution of outliers, and localization of eigenvectors, and highlight the major techniques used to study these problems. We complement the survey with new results on the limiting spectral distribution of polynomially vanishing random perturbation of Toeplitz matrices with continuous symbols, on the limiting spectral distribution of finitely banded Toeplitz matrices under exponentially and super-exponentially vanishing random perturbations, and on the complete localization of outlier eigenvectors of randomly perturbed Jordan blocks.

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.

Various threshold effects are investigated on a discrete quasi-1D scattering system. In particular, one of these effects is to add corrections to Levinson's theorem. We explain how these corrections are due to the opening or to the closing of channels of scattering, and how these contributions can be computed as partial winding numbers on newly introduced operators. Embedded thresholds, thresholds associated with changes of spectral multiplicity, and doubly degenerate thresholds are exhibited and analyzed. Most of the investigations are of an analytical nature, but the final equalities rely on a C*-algebraic framework.

Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific probability that, in the annealed case, depends on a necessarily infinite-mean node fitness. The diverging mean determines many properties that are uncommon in models with independent edges, but at the same time widespread in real-world networks. Here we focus on the leading eigenvalues and eigenvectors of the adjacency matrix of the model, where the n nodes are assigned a Pareto($\alpha$)-distributed fitness with 0 < $\alpha$ < 1. We find that the leading eigenvalues are all of order square root of n, alternate in sign and are located at the intersection between the real axis and a logarithmic spiral in the complex plane, which we characterize analytically in terms of the Gamma function. We also calculate the associated eigenvectors, finding that they display complexvalued scaling exponents and log-periodicity, which are signatures of discrete scale invariance. In contrast with the typical finite-rank behaviour of random graphs with finite-mean variables, we find that a growing number of the leading eigenvalues emerges from the bulk, whose edge extends up to order square root of n and therefore reaches the same scale as that of the structural eigenvalues.

In this paper we provide spectral inclusion and mapping theorems for strongly continuous locally equicontinuous semigroups on Hausdorff locally convex spaces. Our results extend the classical spectral inclusion and mapping theorems for strongly continuous semigroups on Banach spaces.

We study Poincar{\'e} series associated to strictly convex bodies in the Euclidean space. These series are Laplace transforms of the distribution of lengths (measured with the Finsler metric associated to one of the bodies) from one convex body to a lattice. Assuming that the convex bodies have analytic boundaries, we prove that the Poincar{\'e} series, originally defined in the right complex half-plane, continues holomorphically to a conical neighborhood of this set, removing a countable set of cuts and points. The latter correspond to the spectrum of a dual elliptic operator. We describe singularities of the Poincar{\'e} series at each of these branching points. One of the steps of the proof consists in showing analytic continuation of the resolvent of multiplication operators by a real-valued analytic Morse function on the sphere as a branched holomorphic function, a result of independent interest.

We prove a Weyl-type theorem for the Kohn Laplacian on sphere quotients as CR manifolds. We show that we can determine the fundamental group from the spectrum of the Kohn Laplacian in dimension three. Furthermore, we prove Sobolev estimates for the complex Green's operator on these quotient manifolds.

This paper focuses on the fractal characteristics of the absolutely continuous spectral measure of the subcritical almost Mathieu operator (AMO) and Diophantine frequency. In particular, we give a complete description of the (classical) multifractal spectrum and a finer description in the logarithmic gauge. The proof combines continued$-$fraction$/$metric Diophantine techniques and refined covering arguments. These results rigorously substantiate (and quantify in a refined gauge) the physicists' intuition that the absolutely continuous component of the spectrum is dominated by energies with trivial scaling index, while also exhibiting nontrivial exceptional sets which are negligible for classical Hausdorff measure but large at the logarithmic scale.

We develop an index theory for variational problems on noncompact quantum graphs. The main results are a spectral flow formula, relating the net change of eigenvalues to the Maslov index of boundary data, and a Morse index theorem, equating the negative directions of the Lagrangian action with the total multiplicity of conjugate instants along the edges. These results extend classical tools in global analysis and symplectic geometry to graph based models, with applications to nonlinear wave equations such as the nonlinear Schroedinger equation. The spectral flow formula is proved by constructing a Lagrangian intersection theory in the Gelfand-Robbin quotients of the second variation of the action. This approach also recovers, in a unified way, the known formulas for heteroclinic, halfclinic, homoclinic, and bounded orbits of (non)autonomous Lagrangian systems.

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.

After introducing a natural notion of continuous fields of locally convex spaces, we establish a new theory of strongly continuous families of possibly unbounded self-adjoint operators over varying Hilbert spaces. This setting allows to treat operator families defined on bundles of Hilbert spaces that are not locally trivial (such as e.g.~the tangent bundle of Wasserstein space), without referring to identification operators at all.

Any self-adjoint extension of a (singular) Sturm-Liouville operator bounded from below uniquely leads to an associated sesquilinear form. This form is characterized in terms of principal and nonprincipal solutions of the Sturm-Liouville operator by using generalized boundary values. We provide these forms in detail in all possible cases (explicitly, when both endpoints are limit circle, when one endpoint is limit circle, and when both endpoints are limit point).

We investigate an open problem arising in iterative image reconstruction. In its general form, the problem is to determine the stability of the parametric family of operators $P(t) = W (I-t B)$ and $R(t) = I-W + (I+tB)^{-1} (2W-I)$, where $W$ is a stochastic matrix and $B$ is a nonzero, nonnegative matrix. We prove that if $W$ is primitive, then there exists $T > 0$ such that the spectral radii $\varrho(P(t))$ and $\varrho(R(t))$ are strictly less than $1$ for all $0 < t < T$. The proof combines standard perturbation theory for eigenvalues and an observation about the analyticity of the spectral radius. This argument, however, does not provide an estimate of $T$. To this end, we compute $T$ explicitly for specific classes of $W$ and $B$. Building on these partial results and supporting numerical evidence, we conjecture that if $B$ is positive semidefinite and satisfies certain technical conditions, then $\varrho(P(t)), \, \varrho(R(t))<1$ for all $0 < t < 2/\varrho(B)$. As an application, we show how these results can be applied to establish the convergence of certain iterative imaging algorithms.

In this paper, we consider the inverse spectral problem of determining the spherically symmetric refractive index in a bounded spherical region of radius $b$. Instead of the usual case of the refractive index $\rho\in W^2_2$, by using singular Sturm-Liouville theory, we {first} discuss the case when the refractive index $\rho$ is a piecewise $ W^1_2$ function. We prove that if $\int_0^b \sqrt{\rho(r)} dr<b$, then $\rho$ is uniquely determined by all special transmission eigenvalues; if $\int_0^b \sqrt{\rho(r)} dr=b$, then all special transmission eigenvalues with some additional information can uniquely determine $\rho$. We also consider the mixed spectral problem and obtain that $\rho$ is uniquely determined from partial information of $\rho$ and the ``almost real subspectrum".

In this paper, we study the Birman-Krein formula for the potential scattering on the product space $\mathbb{R}^n\times M$, where $M$ is a compact Riemannian manifold possibly with boundary, and $\mathbb{R}^N$ is the Euclidean space with $n\geq 3$ being an odd number. We also derive an upper bound for the scattering trace when $M$ is a bounded Euclidean domain.

We show that for any connected graph $G$ with maximum degree $d\ge3$, the spectral gap from $0$ with respect to the adjacency matrix is at most $\sqrt{d-1}$. We further show that the upper bound $\sqrt{d-1}$ is achieved if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$; and for other cases, the upper bound can be improved to $\sqrt{d-2}$. A similar yet more subtle phenomenon involving the normalized Laplacian is also investigated, in which we work on graphs of degrees $\ge d$ rather than $\le d$. We prove that for any graph $G$ with \emph{minimum} degree $d\ge 3$, the spectral gap from the value 1 with respect to the normalized Laplacian is at most $\sqrt{d-1}/d$, with equality if and only if $G$ is the incidence graph of a finite projective plane of order $d-1$. These results are spectral gap analogues to an inequality involving HL-index by Mohar and Tayfeh-Rezaie, as well as an estimate of the energy per vertex by van Dam, Haemers and Koolen. Moreover, we provide a new sharp bound for the convergence rate of some eigenvalues of the Laplacian on the (weighted) neighborhood graphs introduced by Bauer and Jost.

We study anisotropic fractional discrete Laplacians $\Delta_{\mathbb{Z}^d}^{\vec{\mathbf{r}}}$ with exponents $\vec{\mathbf{r}}\in\mathbb{R}^d\setminus\{0\}$ on $\ell^2(\mathbb{Z}^d)$. We establish a Mourre estimate on compact energy intervals away from thresholds. As consequences we derive a Limiting Absorption Principle in weighted spaces, propagation estimates (minimal velocity and local decay), and the existence and completeness of local wave operators for perturbations $H=\Delta_{\mathbb{Z}^d}^{\vec{\mathbf{r}}}+W(Q)$, where $W$ is an anisotropically decaying potential of long--range type. In the stationary scattering framework we construct the on--shell scattering matrix $S(\lambda)$, prove the optical theorem, and, under a standard trace--class assumption on $W$, establish the Birman--Krein formula $\det S(\lambda)=\exp(-2\pi i\,\xi(\lambda))$.

The Gutzwiller trace formula relates the asymptotic spacing of quantum-mechanical energy levels in the semiclassical limit to the dynamics of periodic classical particle trajectories. We generalize this result to the case of non-smooth potentials, for which there is partial reflection of energy from derivative discontinuities of the potential. It is the periodic trajectories of an associated branching dynamics that contribute to the trace asymptotics in this more general setting; we obtain a precise description of their contribution.