Spectral Theory

We prove that Schr\"odinger operators with meromorphic potentials $(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n$ have purely singular continuous spectrum on the set $\{E:…

We give the description of self-adjoint regular Dirac operators, on $[0, \pi]$, with the same spectra.

In terms of triples of Banach spaces, we define a large class of boundary problems for ordinary differential equations (of arbitrary order) with singular coefficients.

We introduce the periodic Airy-Schr\"odinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a…

We consider functions of Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. It has been known for a long time that the quasi-classical asymptotics for traces of resulting operators strongly depend on the smoothness of the…

This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its…

We solve a spectral and an inverse spectral problem arising in the computation of peakon solutions to the two-component PDE derived by Geng and Xue as a generalization of the Novikov and Degasperis-Procesi equations. Like the spectral…

We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schr\"odinger operators with…

We consider minimal non-negative Jacobi operator with $p\times p-$matrix entries. Using the technique of boundary triplets and the corresponding Weyl functions, we describe the Friedrichs and Krein extensions of the minimal Jacobi operator.…

We study two- and three-dimensional matrix Schr\"odinger operators with $m\in \mathbb N$ point interactions. Using the technique of boundary triplets and the corresponding Weyl functions, we complete and generalize the results obtained by…

In this paper spectral theorems for not necessarily continuous normal and self-adjoint random operators on a complex separable Hilbert space are proved.

Here, we show a method to reconstruct connectivity hypermatrices of a general hypergraph (without any self loop or multiple edge) using tensor. We also study the different spectral properties of these hypermatrices and find that these…

We investigate negative spectra of 1--D Schr\"odinger operators with $\delta$- and $\delta'$-interactions on a discrete set in the framework of a new approach. Namely, using technique of boundary triplets and the corresponding Weyl…

The problem of recovering a signal of finite duration from a piece of its Fourier transform was solved at Bell Labs in the $1960$'s, by exploiting a "miracle": a certain naturally appearing integral operator commutes with an explicit…

Let $\Lambda\subset \mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $\Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-\Delta u=\nu u$ in…

We consider families of non-self-adjoint perturbations of self-adjoint harmonic and anharmonic oscillators. The norms of spectral projections of these operators are found to grow at intermediate rates from arbitrarily slowly to…

Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the area under the curve, and again count…

We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry is embedded into the absolutely continuous spectrum of the other main-diagonal entry.…

We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that…

Recently, several authors have proved inequalities on the spectral radius $\rho$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products and ordinary products of non-negative matrices that define operators on sequence spaces,…