Optimum quantum memory conditions for a spatial frequency resonator grating

The dynamics of the interaction between microcavities connected to a common waveguide in a multiresonator quantum memory circuit is investigated. Optimum conditions are identified for the use of quantum memory and a dynamic picture of the exchange of energy between different microcavities is obtained.     

The Basis Property of the System of Root Functions of the Oblique Derivative Problem

The oblique-derivative spectral problem for the Laplacian in a disk D is studied. The asymptotic properties of its eigenvalues are established, and the system of root functions of this problem is proved to be a basis with brackets in L₂ (D).     

On some criteria for completely regular growth of entire functions of exponential type

This paper sharpens the author’s previous results concerning the completely regular growth of an entire function of exponential type all of whose zeros are simple, forming a sequence Λ = {λ_k} _k=1 ^∞ . For a function with real zeros, we write the growth regularity conditions (on the real axis and on the entire plane) in terms of lower bounds only for the absolute value of the derivative at the points λ_k. We also obtain an analog of Krein’s theorem concerning the functions whose inverse can be expanded in the corresponding series of simple fractions.     

Nontrivial expansions of zero and representation of analytic functions by series of simple fractions

We propose a modification of the previously-known abstract scheme that reduces the problem of expansion of elements of a locally convex space in series over the system of eigenvectors of some linear operator to the question of existence of a nontrivial expansion of zero in this space. We implement this general scheme for the spaces of analytic functions in domains of the extended complex plane and the systems of simple fractions that are the eigenfunctions of the Pommier operator.     

On a Problem of Leont'ev and Representing Systems of Exponentials

We study whether an entire function of exponential type has totally regular growth if its derivative increases sufficiently fast on the zero set of the function itself. In particular, for a function with a trigonometrically convex (or positive) lower indicator, we obtain a solution of a well-known problem of Leont'ev. As an application, we refine some already known results concerning the characterization of exponents of the representing systems of exponentials.     

On the regularity of growth of canonical products with real zeros

We obtain a criterion for totally regular growth of canonical products with real symmetric zeros, which involves a generalization of the notion of condensation index of a sequence of positive numbers of a finite upper density.     

On the growth of entire functions with discretely measurable zeros

We solve the problem of the least possible type of entire functions of order ρ ∈ (0, 1) with positive zeros in a special class specified by certain conditions on the upper and lower averaged ρ-density of zeros.     

Dual characterization of absolutely representing systems in inductive limits of Banach spaces

We establish a dual criterion for absolutely representing systems in the spaces of a sufficiently broad class of inductive limits of sequences of Banach spaces. The result is stated without any additional restrictions on the system. We consider applications to the systems of partial fractions in the spaces of locally analytic functions.