On the asymptotics of the spectrum of a nonsemibounded vector Sturm-Liouville operator

On the half-line, we consider a vector Sturm-Liouville operator with a potential that is unbounded below. Asymptotic formulas for the spectrum are given. These formulas involve the eigenvalues of the matrix potential as well as the “rotational velocities” of the eigenvectors.     

On the Sturm-Liouville operator

We consider the homogeneous Sturm-Liouville differential equation, prove the existence of two linearly independent solutions that form a fundamental system, and obtain some of their properties.     

On a two-point boundary value problem for the Sturm-Liouville operator with a nonclassical asymptotics of the spectrum

We consider a spectral problem generated by a Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We prove the existence of potentials q(x) ∈ L ₂(0, π) such that the multiplicities of the eigenvalues λ _n monotonically tend to infinity as n → ∞.     

Inverse spectral reconstruction problem for the convolution operator perturbed by a one-dimensional operator

We consider a one-dimensional perturbation of the convolution operator. We study the inverse reconstruction problem for the convolution component using the characteristic numbers under the assumption that the perturbation summand is known a priori. The problem is reduced to the solution of the so-called basic nonlinear integral equation with singularity. We prove the global solvability of this nonlinear equation. On the basis of these results, we prove a uniqueness theorem and obtain necessary and sufficient conditions for the solvability of the inverse problem.     

An estimate on the non-real spectrum of a singular indefinite Sturm-Liouville operator

It will be shown with the help of the Birman-Schwinger principle that the non-real spectrum of the singular indefinite Sturm-Liouville operator sgn(·)(−d²/dx² + q) with a real potential q ∈ L¹ ∩ L² is contained in a circle around the origin with radius . (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a transformation operator for a system of Sturm-Liouville equations

We prove the existence of a transformation operator with a condition at infinity that sends a solution of the matrix equation^–y + My=²y (M is a constant Hermitian matrix) into a solution of the matrix equation^–y+Q(x)y+My=²y (the matrix function Q(x) is continuously differentiable for 0 x< and it is Hermitian for each x belonging to [0, )); we study some properties of the kernel of the transformation operator.Translated from Matematicheskii Zametki, Vol. 11, No. 5, pp. 559–567, May, 1972.The authors express their thanks to B. M. Levitan for a discussion.     

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.     

Characterization of the spectrum of the Sturm-Liouville operator with irregular boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation with arbitrary complex-valued potential, q(x), ∈ L ₂(0, π) and irregular boundary conditions. We derive necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.     

Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions

On the interval (0, π), we consider the spectral problem generated by the Sturm-Liouville operator with regular but not strongly regular boundary conditions. For an arbitrary potential q(x) ∈ L ₁ (0, π) [q(x) ∈ L ₂(0, π)], we establish exact asymptotic formulas for the eigenvalues of this problem.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

On inverse problem for singular Sturm-Liouville operator from two spectra

We study an inverse problem with two given spectra for a second-order differential operator with singularity of the type (here, l is a positive integer or zero) at zero point. It is well known that two spectra {λ _n } and {λ _n } uniquely determine the potential function q(r) in the singular Sturm-Liouville equation defined on the interval (0, π]. One of the aims of the paper is to prove the generalized degeneracy of the kernel K(r, s). In particular, we obtain a new proof of the Hochstadt theorem concerning the structure of the difference . Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 1, pp. 132–138, January, 2006.     

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.     

Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator

On the interval (0, τ), we consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L ₂(0, τ) and with regular (but not strengthened-regular) boundary conditions. Under certain additional assumptions, we establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.