In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space $\mathbb {R}^3$ , in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighborhood of the origin.

We obtain the representation of the backward shift operator on Chebyshev polynomials involving a principal value (PV) integral. Twice the backward shift on the space of square-summable sequences l2 displays chaotic dynamics, thus we provide an explicit form of a chaotic operator on L2 (−1, 1, (1−x2)–1/2) using Cauchy’s PV integral. We explicitly calculate the periodic points of the operator and provide examples of unbounded trajectories, as well as chaotic ones. Histograms and recurrence plots of shifts of random Chebyshev expansions display interesting behaviour over fractal measures.

In this paper, inverse spectral problems for Sturm–Liouville operators on a tree (a graph without cycles) are studied. We show that if the potential on an edge is known a priori, then b – 1 spectral sets uniquely determine the potential functions on a tree with b external edges. Constructive solutions, based on the method of spectral mappings, are provided for the considered inverse problems.

We consider the equation

where and

We assume that this equation is correctly solvable in Lp(ℝ). Under these assumptions, we study the problem of compactness of the resolvent of the maximal continuously invertible Sturm–Liouville operator . Here

In the case p = 2, for the compact operator , we obtain two-sided sharp-by-order estimates of the maximal eigenvalue.

The paper examines a higher-order ordinary differential equation of the form $\mathcal{P}[u]\,{:=} \sum_{j,k=0}^{m}D^{j}a_{jk}D^{k}u \,{=}\, \lambda{u}, x\in[0,b)$, where $D\,{=}\,i({d}/{dx})$, and where the coefficients $a_{jk}$, $j,k\in[0,m]$, with $a_{mm}\,{=}\,1$, satisfy certain regularity conditions and are chosen so that the matrix $(a_{jk})$ is hermitean. It is also assumed that $m\,{>}\,1$. More precisely, it is proved, using Paley–Wiener methods, that the corresponding spectral measure determines the equation up to conjugation by a function of modulus 1. The paper also discusses under which additional conditions the spectral measure uniquely determines the coefficients $a_{jk}$, $j,k\in[0,m]$, $j+k\neq{2m}$, as well as $b$ and the boundary conditions at 0 and at $b$ (if any).