Related papers: The discrete spectrum for complex perturbations of…

We construct discrete versions of $\kappa$-Minkowski space related to a certain compactness of the time coordinate. We show that these models fit into the framework of noncommutative geometry in the sense of spectral triples. The dynamical…

From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period $p \geq 2$ may have at most $p-2$ closed spectral gaps. We discuss the maximal number of closed gaps for…

We study Jacobi matrices on star-like graphs, which are graphs that are given by the pasting of a finite number of half-lines to a compact graph. Specifically, we extend subordinacy theory to this type of graphs, that is, we find a…

We consider a second order difference equation with operator-valued coefficients. More precisely, we study either compact or trace class perturbations of the discrete Laplacian in the Hilbert space of bi-infinite square-summable sequence…

One of the first and therefore most important theorems in perturbation theory claims that for an arbitrary self-adjoint operator A there exists a perturbation B of Hilbert-Schmidt class with arbitrary small operator norm, which destroys…

We solve the inverse problem for Jacobi operators on the half lattice with finitely supported perturbations, in particular, in terms of resonances. Our proof is based on the results for the inverse eigenvalue problem for specific finite…

We discuss a functional model for multi--diagonal selfadjoint operators with almost periodic coefficients that generalizes the well known model for finite band Jacobi matrices. It give us an opportunity to construct examples of almost…

This paper extends our previous works arXiv:1802.07306 [math.NT], arXiv:1808.02382 [math.NT] on determining the spectrum, in the Berkovich sense, of ultrametric linear differential equations. Our previous works focused on equations with…

We derive exact equations that determine the spectra of undirected and directed sparsely connected regular graphs containing loops of arbitrary length. The implications of our results to the structural and dynamical properties of networks…

In this paper, we study random matrix models which are obtained as a non-commutative polynomial in random matrix variables of two kinds: (a) a first kind which have a discrete spectrum in the limit, (b) a second kind which have a joint…

We present a comprehensive treatment of relative oscillation theory for finite Jacobi matrices. We show that the difference of the number of eigenvalues of two Jacobi matrices in an interval equals the number of weighted sign-changes of the…

We give general spectral and eigenvalue perturbation bounds for a selfadjoint operator perturbed in the sense of the pseudo-Friedrichs extension. We also give several generalisations of the aforementioned extension. The spectral bounds for…

For 2D periodic Jacobi operators we obtain the estimate of the Lebesgue measure of the spectrum and estimates of edges of spectral bands.

We characterize the spectrum of one-dimensional Jacobi operators H=aS^{+}+a^{-}S^{-}+b in l^{2}(\Z) with quasi-periodic complex-valued algebro-geometric coefficients (which satisfy one (and hence infinitely many) equation(s) of the…

We study the spectrum of a periodic non-self-adjoint Dirac operator, and its dependence on a semiclassical parameter is also considered. Several bounds on the spectrum are obtained which provide sharp spectral enclosure estimates.…

We present an unified framework to identify spectra of Jacobi matrices. We give applications to long-standing conjecture of Chihara concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent…

In this paper it is considered a spectral density for a class of Jacobi matrices with absolutely continuous spectrum that was examined first by Moszynski. It is shown that the corresponding spectral density is equivalent to the positive…

We characterize possible spectra of rank-one perturbations B of a self-adjoint operator A with discrete spectrum and, in particular, prove that the spectrum of B may include any number of real or non-real eigenvalues of arbitrary algebraic…

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger operators. These include an Oracle Theorem that predicts the potential and rather general…

Hitherto, it is well known that complex PT-symmetric Scarf II has real discrete spectrum in the parametric domain of unbroken PT-symmetry. We reveal new interesting complex, non-PT-symmetric parametric domains of this versatile potential,…