Anton Alekseevich Vladimirov

Spectral and oscillatory properties of a linear pencil of fourth-order differential operators

The paper deals with the spectral and oscillatory properties of a linear operator pencilA − λB, where the coefficient A corresponds to the differential expression (py″)″ and the coefficient B corresponds to the differential expression −y″ + cry. In particular, it is shown that all negative eigenvalues of the pencil are simple and, under some additional conditions, the number of zeros of the corresponding eigenfunctions is related to the serial number of the corresponding eigenvalue.

Representation theorems and variational principles for self-adjoint operator matrices

We use the notion of triples D+ → H → D− of Hilbert spaces to develop an analog of the Friedrichs extension procedure for a class of nonsemibounded operator matrices. In addition, we suggest a general approach (stated in the same terms) to the construction of variational principles for the eigenvalues of such matrices.

On the problem of oscillation properties of positive differential operators with singular coefficients

A criterion for a highly singular positive fourth-order operator with separable boundary conditions to have oscillation properties, as well as sufficient conditions for similar higher-order operators to have oscillation properties, are obtained.

Remarks on minorants of Laplacians on a geometric graph

1. By a geometric graph Γ we mean a set ⋃n i=1{i} × [0, i] together with a finite (possibly empty) list of relations of the form (i, xi) = (j, xj), where i, j ∈ 1, . . . , n, xi ∈ {0, i}, and xj ∈ {0, j}, which “glue together” points in this set. Without essential loss of generality, we assume that ∑n i=1 i = 1. Given any (partial) function y : Γ → R, by yi we denote the corresponding maps yi : [0, i] → R defined by yi(x) ≡ y(i, x). By an integrable function defined almost everywhere on Γ we naturally understand a partial map y : Γ → R satisfying the relations yi ∈ L1[0, i]. The integral of such a function is defined as ˆ

Differential operator matrices of mixed order with periodic coefficients

We study spectral properties of 2 x 2 operator matrices H defined in the Hilbert space ℍ = L 2(R) x L 2(R)by linear differential systems of mixed order with periodic coefficients. We prove that the spectrum σ (H) of H has a band and gap structure and consists of two band sequences one of which, when infinite, has a finite accumulation point, and give sufficient conditions for this accumulation to take place.