Vadim Tkachenko

Spectral Properties of Non-Selfadjoint Hill’s Operators with Smooth Potentials

The aim of the present paper is to describe some spectral properties of Hill’s operators (1) with π-periodic complex-valued potentials q(x) belonging to the Sobolev space where n ≥ 0 is a fixed integer. The space W 0 2 is identified with ℒ 2[0, π], no restrictions being imposed at the boundary points 0 and π. A complete spectral parametrization of potentials q ∈ W 0 2 was given in our previous paper [8].

On solvability of linear difference equations in smooth and real analytic vector functions of several variables

We investigate the multidimensional equations ∑j=1q Aj(x)y(x+ej)=f(x),ej ∈ ℝn wherex ∈ ℝn andAj : ℝn →Hom(ℝp,ℝm),f : ℝn → ℝm are given maps. Sufficient conditions for smooth and analytic solvability for anyf ∈ Ck,k ≤ ω are found.

Non-self-adjoint Periodic Dirac Operators

We consider the class \(\mathcal{D}\) of Dirac operators

Equations with Several Transformations of Argument

\begin{array}{*{20}{c}} {L = J\frac{d}{{dx}} + Q(x),} & {x \in \mathbb{R},} \\ \end{array}

Generalized Abel Equation

(1.1) with matrices

Spectral Parametrization of Non-selfadjoint Hill's Operators

J = \left\| {\begin{array}{*{20}{c}} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right\|,Q(x) = \left\| {\begin{array}{*{20}{c}} {p(x)} & {q(x)} \\ {q(x)} & { - p(x)} \\ \end{array} } \right\|,