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In this paper, we analyze conditions under which the spectrum of the Sturm–Liouville operator on a certain smooth curve is localized near a countable number of rays. In the case where the potential is piecewise analytical, an asymptotic of eigenvalues is found for each series localized near the corresponding ray. The result obtained allows one to generalize the well-known formula for the asymptotic of the distribution function of the spectrum stated by Davies in the case of a finite number of localization rays.

We establish conditions for the absolute instability of the solutions of linear and nonlinear difference equations with self-adjoint operator coefficients.

Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation $$ \tilde{A}\supset A $$ in some Hilbert space $$ \tilde{\mathrm{\mathfrak{H}}}\supset \mathrm{\mathfrak{H}} $$ is called an (exit space) extension of A. We study the compressions $$ C\left(\tilde{A}\right)={P}_{\mathrm{\mathfrak{H}}}\tilde{A}\upharpoonright \mathrm{\mathfrak{H}} $$ of extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ Our main result is a description of compressions $$ C\left(\tilde{A}\right) $$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ of A with the maximal symmetric compression $$ C\left(\tilde{A}\right) $$ and all extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).

In this paper, we construct finite-difference approximations of optimal-control problems involving non-self-adjoint convection-diffusion elliptic equations with discontinuous coefficients and states and examine the convergence of these approximations. Control functions in these problems are the coefficients of the convective-transfer operator in the equation of state and its right-hand side. We study the well-posedness of problems considered. For finite-difference approximations, we obtain estimates of the exactness by the state and the convergence rate by the functional and prove the weak convergence by the control. In addition, we regularize approximations in the Tikhonov sense.

The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3). Krein’s theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green’s function of the vector Laplace operator and some additional kernel of finite rank.

We discuss basic theoretical principles underlying new numerical methods of calculation of eigenvalues and eigenfunctions of discrete operators semi-bounded from below. We present algorithms of finding spectral characteristics by methods of regularized traces and examples related to certain spectral Sturm–Liouville problems.

We consider the Sturm–Liouville operator T0 on the semi-axis (0,+∞) with the potential eiθq, where 0 < θ < π and q is a real-valued function that may have arbitrarily slow growth at infinity. This operator does not meet any condition of the Keldysh theorem: T0 is non-self-adjoint and its resolvent does not belong to the Neumann–Schatten class for any p < ∞. We find conditions for q and perturbations of V under which the localization or the asymptotics of its spectrum is preserved.

We study spectral properties of the one-dimensional Schrodinger operators \( {\mathrm{H}}_{\mathrm{X},\alpha, \mathrm{q}}:=-\frac{{\mathrm{d}}^2}{\mathrm{d}{x}^2}+\mathrm{q}(x)+{\varSigma_x}_{{}_n}\in X{\alpha}_n\delta \left(x-{x}_n\right) \) with local interactions, d* = 0, and an unbounded potential q being a piecewise constant function, by using the technique of boundary triplets and the corresponding Weyl functions. Under various sufficient conditions for the self-adjointness and discreteness of Jacobi matrices, we obtain the condition of self-adjointness and discreteness for the operator HX,α,q.

The M- and N-transforms acting, respectively, on divergence-free and curl-free vector fields on a Riemannian manifold with boundary are investigated. These transforms arise in studying the inverse problems of electrodynamics and elasticity theory. A divergence-free field is mapped by M to a field that is tangential to equidistants of the boundary. The N-transform maps a curl-free field to a field that is normal to equidistants. In preceding papers, the operators M and N were considered in the case of smooth equidistants, which is realized in a sufficiently small near-boundary layer. This allows one to consider transforms of fields supported in such a layer; it was proved that M and N are unitary in the corresponding spaces with L2-norms. In one of the papers, the case of fields on the whole manifold was considered, but almost all equidistants were assumed to be Lipschitz surfaces. It was proved that M is coisometric (i.e., the adjoint operator is isometric). In the present paper, the same result is obtained for both transforms in the general case with no constraints on equidistants at all.

The purpose of this work is to study the spectral properties of the problem of transmission arising after the linearization of two-phase problems of Stefan and Florin with classical boundary condition on a small time interval. With the help of the operator methods of mathematical physics, a boundary-value problem is reduced to the study of the spectrum of a weakly perturbed compact self-adjoint operator in a Hilbert space. On the basis of the theorems of M. V. Keldysh and V. B. Lidskii, we have established the basis property of the system of eigen- and associated elements by Abel–Lidskii in some Hilbert space. It is proved that the spectrum is discrete with the single limiting point at infinity. It is situated on the positive semiaxis or, except for a finite number of eigenvalues, in the aperture angle e. The growth of the moduli of eigenvalues is estimated, and some asymptotic formulas are obtained.

In this paper, we continue a series of previous articles and present a simple method of proving the correctness and self-adjointness of operators of the form B4 corresponding to some boundary value problems. We also give representations for the unique solutions of these problems. The algorithm is easy to implement via computer algebra systems. In our examples, Derive and Mathematica were used. Bibliography: 13 titles.

In this paper, we present a simple method to prove the correctness and self-adjointness of operators B 3 corresponding to some boundary problems. We also give the unique solutions for these problems. The algorithm is easy to implement via computer algebra systems. In our examples, Derive and Mathematica were used. Bibliography: 10 titles.

The paper gives simple sufficient conditions on the kernel of the self-adjoint integral operator considered ensuring the equiconvergence of expansions in eigenfunctions and associated functions with the ordinary trigonometric Fourier series expansions. The presented arguments correct inaccuracies in [1].

The singular differential operator $$Lf(x) = - {\text{sign }}x\frac{{d^{\text{2}} f(x)}}{{dx^2 }} + p(x)f(x)$$ is studied. It is proved that if the second moment of p is finite and L has no nonreal eigenvalues, then L is similar to a self-adjoint operator. The proof is based on an integral resolvent criterion of similarity applied to a wide class of functions p(x). Bibliography: 20 titles.

The following questions and close problems are studied.(i) Is it true that T is p-nuclear provided that T** is p-nuclear? (ii) Is it true that Tis dually p-nuclear provided that T* is p-nuclear? (iii) Is it true that if T*is compactly factorable in the space lp, then T is (strictly) factorable in the space lp'? Here, T* is the adjoint operator of a bounded operator T:X → Yin Banach spaces X and Y. Bibliography: 30 titles.

Let M be a manifold with boundary δM≠Ф. Let A be a second-order elliptic partial differential operator given on M. Denote by Rλ(x, y), x, y∈M, λe $$\mathbb{C}$$ \σ(A) the Schwartz kernel of (A−λI)−1. Consider the Gel'fand inverse boundary problem of the reconstruction of (M, A) via a given Rλ(x, y), x, y∈δM,λe $$\mathbb{C}$$ . We prove that if the principal symbol of A satisfies some geometric condition (the Bardos-Lebeau-Rauch condition), then these data determine M uniquely, and they determine A to within the group of generalized gauge transformation on M. The above-mentioned geometric condition means, roughly speaking, that any geodesic (in the metric generated by A) leaves M. Bibliography: 29 titles.

An approach to the inverse problem (the so-called BC-method) based on boundary-control theory is developed. A procedure of reconstructing a nonsymmetric matrix-function (a potential) given on a semiaxis by a dynamical response operator is described. The results of numerical tests are presented. Bibliography: 6 titles.

Solvability of mixed boundary-value problems in domains with cylindrical outlets to infinity is investigated for a certain class of non-self-adjoint differential operator-matrices., The structure of these matrices is such that the corresponding asymmetric quadratic forms possess a polynomial property, i.e., they degenerate only on finite-dimensional lineals of vector polynomials. With the help of elementary algebraic operations, this property enables one to indicate some attributes of boundary-value problems, namely, to calculate the operator index, to describe the kernel and co-kernel of an operator, to find an asymptotic behavior of solutions at infinity, etc. Bibliography: 16 titles.

The formally self-adjoint boundary-value problem possesses the polynomial property if the corresponding sesquilinear form degenerates only on a finite-dimensional lineal P of vector-valued polynomials. A problem with the polynomial property is elliptic. The kernel and co-kernel of the corresponding operator are described in terms of P. It is shown that every formally positive operator has the polynomial property. Bibliography: 22 titles.

We construct and justify a numerical algorithm for finding the generalized eigenvalues and eigenfunctions for self-adjoint positive semidefinite linear operators with nonlinear spectral parameter. We give an example of its application to finding the branch points of solutions of a class of nonlinear integral equations of Hammerstein type.

For a non-self-adjoint operator with a characteristic function that has boundary values almost everywhere on the real axis, we consider some problems concerned with local absolutely continuous and singular subspaces. Bibliography: 39 titles.

We prove a theorem on intermediate derivatives and traces for a class of unbounded nonselfadjoint operators in Hilbert space. The results obtained have application in the qualitative theory of operator-differential equations.

A first-order non-self-adjoint system of differential equations with 2n unknown functions and with certain conditions on the coefficients is considered on the entire axis. The system is similar to the well-known Dirac system. An operator ℒ, associated with the considered system, is introduced in the space L2(− ∞, ∞) of 2n-component vector-valued functions. Conditions for the compactness of the resolvent of the operator ℒ are found, yielding sufficient conditions for the discreteness of its spectrum.

Let A be a self-adjoint operator, let (α,β) be a gap in the spectrum of A, and let B=A+V, where, in general, the perturbation operator V is unbounded. We establish some abstract conditions under which the spectrum of B in (α,β) is discrete; does not accumulate to β; is finite. An estimate of the number of eigenvalues of B in (α,β) is obtained. Bibliography: 3 titles.

In this paper, the following assertion is illustrated with some examples. In practically all cases, the problem on natural oscillations for linear holonomic nondissipative mechanical systems reduces to the problem of determining eigenvalues and eigenelements of the corresponding self-adjoint operator acting in the Hilbert space generated by the quadratic form of the kinetic energy. Bibliography: 1 title.