Vol 239, No 3 (2019)

On a star graph consisting of two infinite edges and one small edge, we consider the Schrödinger operators with piecewise-constant potentials on the infinite edges and with a singular potential on the small edge respectively. A δ′-interaction is given at the interior vertex of the graph, and the Dirichlet or Neumann condition is imposed at the boundary vertex of the small edge. We determine the limit boundary conditions, obtain two-term asymptotics for the resolvents in the operator norm and error estimates. The phenomenon of isolated eigenvalues emerging from the threshold of the essential spectrum is discussed. We establish efficient and easily verified sufficient conditions for the existence or absence of such eigenvalues. We establish the holomorphic dependence of the appeared eigenvalues on the edge length and write explicitly the first terms of the Taylor expansions of such eigenvalues.

We study the Hurwitz product (convolution) in the space of formal Laurent series over an arbitrary field of zero characteristic. We obtain the convolution equation which is satisfied by the Euler series. We find the convolution representation for an arbitrary differential operator of infinite order in the space of formal Laurent series and describe translation invariant operators in this space. Using the p-adic topology in the ring of integers, we show that any differential operator of infinite order with integer coefficients is well defined as an operator from ℤ[[z]] to ℤ_p[[z]].

It is known that for the best uniform approximation of real constants c by simple partial fractions ρn of order n on a real segment it is necessary and sufficient to have an alternance of n + 1 points on this segment for the difference ρ_n − c, n ≥ n₀(c). We propose an algorithm for constructing the alternance and simple partial fraction of the best approximation of constants c.

We consider the Cauchy problem for the Euler–Poisson–Darboux equation with boundary conditions. The Bessel operators in the equation can have different parameters. We establish a representation of the solution in the form of the Poisson formula with a special shift generated by the product of cylindrical functions of the first kind and different orders.

We introduce an elastic polarization matrix M for a junction Ξ of isotropic homogeneous half-strips Π¹, . . . ,Π^J. Thematrix M is used to describe the boundary layer phenomenon near nodes of elastic lattices and is formed by coefficients in expansions at infinity of special solutions to the problem of elasticity theory for the body Ξ. It is shown that the 3J × 3J-matrix M is symmetric and degenerate on a subspace of dimension three, but, under the “correct” choice of local coordinates in the half-strips, it corrsponds to a positive definite operator on the orthogonal complement of this subspace.

We consider the Riemann–Hilbert boundary value problem for the Moisil–Theodoresco system in a multiply connected domain bounded by a smooth surface in the three–dimensional space. We obtain a criterion for the Fredholm property of the problem and a formula for its index æ = m − s, where s is the number of connected components of the boundary and m is the order of the first de Rham cogomology group of the domain. The study is based on the integral representation of a general solution to the Moisil–Theodoresco system and an explicit description of its kernel and cokernel.

We study the relative structural stability (relative roughness) of dynamical systems in some subspace of the space of all dynamical systems; moreover, the space of deformations of (dynamical) systems does not coincide with the space of all admissible deformations. We give some examples of relatively rough systems and relatively nonrough systems of different degree of nonroughness arising in rigid body dynamics and oscillation theory.