Spectral Theory Papers of the Week

We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains.

In particular, $\textbf{S}$ acts on vectors defined over the shrinking boundary layer but behaves asymptotically like a known operator. In this paper, we use a variational argument to determine conditions for which the eigenvalues and eigenvectors of $\textbf{S}$ converge to the eigenvalues and eigenfunctions of a Dirichlet-to-Neumann map, and we provide evidence of this spectral convergence through a numerical example.

In this paper, we study Egorov's theorem for Laplacian eigenfunctions on compact hyperbolic surfaces in Zelditch's quantization.

We consider the optimal control problem of an elliptic PDE with random diffusion under pointwise box constraints on the control.

We prove that every convex body $K\subset\mathbb{R}^n$ admits $S\in\mathrm{SL}_n(\mathbb{R})$ such that $\lambda_1(SK)|K|^{2/n}\le Cn$, where $C$ is universal.

The \emph{neighbourhood second Zagreb matrix} $\NF(G)$ has $(i,j)$-entry $\delta(i)\delta(j)$ if $i\sim j$ and $0$ otherwise; its spectral radius is denoted by $\rNF(G)$. We prove that $\rNF(G)\ge\rho(G)^{5}$, where $\rho(G)$ is the adjacency spectral radius, with equality if and only if $G$ is regular or bipartite semiregular, and we use this to show that the same graphs are exactly the extremal graphs of a Berman--Zhang--type upper bound.

We discuss a model of a soft waveguide described by Schrödinger operator whose potential is a twisted straight channel of non-circular profile.

We study the asymptotic distribution of the number of primitive, closed geodesics on Weil-Petersson random hyperbolic surfaces with genus $g$ and $n(g)$ cusps, under various growth rates of $n(g)$ as $g\to\infty$.

We determine exactly which gap labels are realized by open spectral gaps for the unitary almost Mathieu operator at every irrational frequency and all coupling constants ($0<\lambda_1,\lambda_2\leq1$).

We develop a weighted scalar spectral geometric description of Sacks-Uhlenbeck $\alpha$-harmonic maps into round spheres and identify its convex geometry realization for $\alpha$-harmonic diffeomorphisms onto the round $2$-sphere.

We prove an integral formula for the spectral flow of differentiable loops of unitaries of the form \operatorname{Id}+ Schatten.

We characterize the equality case in the Brunn–Minkowski inequality for the Monge–Ampère eigenvalue of general bounded convex domains.

We review rigorous results concerning the Area Law for the entanglement entropy of free fermions whose one-body Hamiltonian is an ergodic Schrödinger operator with a strongly localised pure point spectrum.

Background: People who train outdoors accumulate higher ultraviolet doses than the general population, yet the instruments used to quantify their photoprotection were developed in high-latitude, English-speaking settings and are typically scored as undifferentiated summative indices whose internal structure is never examined. We aimed to test the internal structure of a Spanish-language sun-exposure and photoprotection questionnaire for physically active Latin American adults and to refine its item pool through a convergent principal component analysis (PCA) and k-means procedure.

We study multiple orthogonal polynomial systems obtained from a Nikishin system of two measures by applying a Christoffel transform to one of the measures at a point $x=x_0$.

We study low-order eigenvalue ratio estimates for the Laplacian with Muckenhoupt $A_2$ weights.

For $1\leq p\leq n/2$, we derive sharp eigenvalue estimates for the Hodge Laplacian on differential $p$-forms under the curvature condition introduced by Petersen and Wink, and study their equality cases.

We prove a generalization of the conjecture of Casazza, Tremain, and Vershynin for infinite collection of equal norm Bessel sequences.

We prove that resonances of the Schrödinger operator with compactly supported potential can contain arbitrary subset of the angle \{z: -\operatorname{Im}z &gt; C \lvert\operatorname{Re}z\rvert\} that satisfies Blaschke condition.