Self-adjoint quadratic operator pencils and elliptic problems

Functional Analysis and Its Applications -     

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.     

Spectral Properties of Ordinary Differential Operators with Involution

Doklady Mathematics - Let P and Q be ordinary differential operators of order n and m generated by s boundary conditions (where s = max{n, m}) on a bounded interval [a, b]. We study operators of...     

Peculiarities of asphaltene precipitation in <Emphasis Type="Italic">n</Emphasis>-alkane-oil systems

n-Pentane-, n-hexane-, and n-heptane-insoluble asphaltenes obtained via a standard procedure by precipitating from oil solutions in n-pentane, n-hexane, and n-heptane, respectively, as well as n-pentane/n-hexane/n-heptane-insoluble and n-heptane/n-hexane/n-pentane-insoluble asphaltene constituents prepared through successive washing (fractional dissolution) of n-pentane-insoluble asphaltenes with n-hexane and n-heptane and n-heptane-insoluble asphaltenes with n-hexane and n-pentane, respectively, are studied. Asphaltenes and their constituents extracted from three oils distinguished by high contents of asphaltenes, resins, and paraffins, respectively, are investigated by ¹H NMR spectroscopy in carbon tetrachloride solutions. It is established that the mass fractions and the fragment compositions of asphaltenes and their constituents depend on both the type of oil and the procedure of their preparation; i.e., the precipitation from n-alkane-oil systems or the extraction through the successive washing with a series of n-alkanes. The obtained experimental data made it possible to formulate a hypothesis according to which the precipitation of asphaltenes from oils is controlled by not only the dissolving power of a solvent with respect to molecular components of initial oils, but also (and primarily) by the dissolving power of a solvent with respect to supramolecular structures of asphaltenes formed in n-alkane-oil systems.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

On the basis property of root vectors of a perturbed self-adjoint operator

We study perturbations of a self-adjoint operator T with discrete spectrum such that the number of its points on any unit-length interval of the real axis is uniformly bounded. We prove that if ‖Bϕ _n ‖ ≤ const, where ϕ _n is an orthonormal system of eigenvectors of the operator T, then the system of root vectors of the perturbed operator T + B forms a basis with parentheses. We also prove that the eigenvalue-counting functions of T and T + B satisfy the relation |n(r, T) − n(r, T + B)| ≤ const.