On a second-order matrix differential operator

We consider the second-order matrix differential operator $$N = \left( {\begin{array}{*{20}c} { - \frac{d}{{dx}}\left( {p_0 \frac{d}{{dx}}} \right) + p_1 } \\ r \\ \end{array} \begin{array}{*{20}c} r \\ { - \frac{d}{{dx}}\left( {q_0 \frac{d}{{dx}}} \right) + q_1 } \\ \end{array} } \right)$$ determined by the expression Nφ, [0 ?x < ∞), where \(\phi = \left( {\begin{array}{*{20}c} U \\ V \\ \end{array} } \right)\) . It has been proved that if p₀, q₀, p₁, q₁,r satisfy certain conditions, then N is in the limit point case at ∞. It has been also shown that certain differential operators in the Hilbert space L² of vectors, generated by the operator N, are symmetric and self-adjoint.     

Some problems of the spectral theory of a second-order differential equation with a variable unbounded operator coefficient

A self-adjoint second-order differential expression with an unbounded operator coefficient is considered in the space of vector functions. The domain of definition of the minimum and maximum operators generated by this expression is investigated, it is shown that any generalized resolvent of the minimum operator is an integral operator, and expansion in eigenfunctions is performed.     

On the spectral properties of differential operators with unbounded operator coefficients determined by a linear relation

Necessary and sufficient conditions for the invertibility and the Fredholm property of operators generated by a family of evolution operators and by the boundary conditions determined by a linear relation are obtained.     

On the spectral instability of the Sturm-Liouville operator with a complex potential

Using the Sturm-Liouville operator with a complex potential as an example, we analyze the spectral instability effect for operators that are far from being self-adjoint. We show that the addition of an arbitrarily small compactly supported function with an arbitrarily small support to the potential can substantially change the asymptotics of the spectrum. This fact justifies, in a sense, the necessity of well-known sufficient conditions for the potential under which the spectrum of the operator is localized around some ray. For an operator with a logarithmic growth, we construct a perturbation that preserves the asymptotics of the spectrum but has infinitely many poles inside the main sector.     

On the basis property of eigenfunctions of a singular second-order differential operator

We consider the first boundary value problem for a singular differential operator of second order on an interval with transmission conditions at an interior point of the interval. We show that the system of eigenfunctions corresponding to this problem is complete in the space L ²(0, 1) and forms a Riesz basis in that space.     

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L _p (0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.     

On the spectral properties of an operator pencil

For a broad class of iterative algorithms for solving saddle point problems, the study of the convergence and of the optimal properties can be reduced to an analysis of the eigenvalues of operator pencils of a special form. A new approach to analyzing spectral properties of pencils of this kind is proposed that makes it possible to obtain sharp estimates for the convergence rate.     

Unconditional bases related to a nonclassical second-order differential operator

In the present paper, we introduce the notion of regularity of boundary conditions for a simplest second-order differential equation with a deviating argument. We prove the Riesz basis property for a system of root vectors of the corresponding generalized spectral problem with regular boundary conditions (in the sense of the introduced definition). Examples of irregular boundary conditions to which the theory of Il’in basis property can be applied are given.     

On the convergence to zero of solutions of a second-order differential equation with complex-valued potential

We consider the second-order linear differential equation y" + A(t)y = 0 on the semiaxis with complex-valued potential function. Sufficient conditions for the potential function assuring that all solutions of the equation converge to zero at infinity are obtained. It is shown that the conditions imposed on the potential function are close to the necessary ones. One of the results seems to be new even in the case of real-valued function A(·).     

Spectral properties of a nonlocal second-order difference operator

We consider a spectral problem for a nonlocal difference operator of second derivative with variable coefficients and with a complex parameter in the boundary condition. We study the algebraic and geometric multiplicity of the eigenvalues and the sign of their real part. We obtain conditions on the parameter which ensure that the entire spectrum of the operator lies in the right complex half-plane.     

Boundary value problems for a second-order differential equation with selfadjoint operator coefficients

Translated from Sibirskii Matematicheskii, Vol. 37, No. 6, pp. 1397–1406, November–December, 1996.