Spectral Theory

This paper develops a spatially resolved perturbation theory for singular vectors under high-dimensional separable noise and applies it to data-driven matrix recovery. In the asymptotic regime where the matrix dimensions are proportional…

Using numerical certification, we prove the existence of a nontrivial real-valued two dimensional periodic potential whose associated discrete Schr\"odinger operator is Fermi isospectral to the zero potential. This provides a negative…

We establish Anderson localization for long-range quasi-periodic operators with large trigonometric potentials and Diophantine frequencies, the proof is based on a novel dynamical rigidity argument.

We review some recent rigorous results on the semiclassical behavior ($\epsilon\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/\epsilon\}$ where both $A$ and $S$ are differentiable…

In this article, we study quasi-isospectral operators as a generalization of isospectral operators. The paper contains both expository material and original results. We begin by reviewing known results on isospectral potentials on compact…

This paper extends the study of resonance phenomenon initiated by the authors in~\cite{LS} to the case of doubly degenerate embedded eigenvalues (i.e. eigenvalue of multiplicity two). A fundamentally new concept is introduced to resolve the…

We study the point spectrum of a periodic quantum tree equipped with a Schr\"odinger type differential operator with delta-type vertex conditions, using subsets of the compact graph that generates the tree. We prove analogs of existing…

We consider the elliptical range theorems for the Davis--Wielandt shell, the numerical range, and the conformal range in terms of and related to their quadratic representations. The emphasis is on exposing a variety of elementary…

The Friedrichs model~\cite{Friedrichs} is revisited to obtain precise results about the asymptotic behaviour (the so-called Breit-Wigner formula~\cite{Breit}) of a resonance near an embedded eigenvalue and the ``spectral concentration"…

In this paper, we explore the high-frequency properties of eigenfunctions of point perturbations of the Laplacian on a compact Riemannian manifold. These systems cannot be obtained as the quantization of a classical Hamiltonian, as the…

We consider an operator-differential expression of the form $$ \ell y=\frac{d^m}{dx^m}\Big(By^{(n)}+Cy\Big), \quad 0<x<1, $$ where $B$ is a linear bounded invertible operator, while $C$ is some finite-dimensional linear operator relatively…

We adapt two results of Simon and collaborators to the setting of discrete-time unitary dynamics. We show that pure point spectrum precludes ballistic motion, and exhibit a family of examples showing that this is sharp within the class of…

In this paper, we study the spectrality of the non-self-adjoint Dirac operator L(Q) with a complex-valued periodic matrix potential Q. We establish a condition on the off-diagonal elements of the matrix Q under which L(Q) is an…

We establish a new and simple criterion that suffices to generate many spectral gaps for periodic word models. This leads to new examples of ergodic Schr\"odinger operators with Cantor spectra having zero Hausdorff dimension that…

In this paper, we consider the problem of approximating the spectral distribution for a class of random operators over sofic groups. For this purpose, we make use of the concept of locally and empirically converging measures defined by…

This paper extends Remling's Theorem to vector-valued discrete Schrodinger operators, showing that the {\omega} limit points of the matrix potentials, under the shift map, are reflectionless on the absolutely continuous spectrum with full…

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on…

Consider a group $\mathbb{G}$ and construct its power graph, whose vertex set consists of the elements of $\mathbb{G}$. Two distinct vertices (elements) are adjacent in the graph if and only if one element can be expressed as an integral…

We consider the existence of the integrated density of states (IDS) of the magnetic Schr\"{o}dinger operator with a random potential on the Hilbert space \( L^2(\mathbb{R}^d) \), as an analogue of the law of large numbers (LLN) for trace…

In this paper, we show that a random hyperbolic surface in the Brooks-Makover model has a spectral gap greater than $\left(\frac{1}{4}-\left(\frac{1}{n}\right)^{\frac{1}{221}}\right)$, confirming the nearly optimal spectral gap conjecture…