Illya M. Karabash

The Similarity Problem for J-nonnegative Sturm-Liouville Operators

Abstract Sufficient conditions for the similarity of the operator A : = 1 r ( x ) ( − d 2 d x 2 + q ( x ) ) with an indefinite weight r ( x ) = ( sgn x ) | r ( x ) | are obtained. These conditions are formulated in terms of Titchmarsh–Weyl m-coefficients. Sufficient conditions for the regularity of the critical points 0 and ∞ of J-nonnegative Sturm–Liouville operators are also obtained. This result is exploited to prove the regularity of 0 for various classes of Sturm–Liouville operators. This implies the similarity of the considered operators to self-adjoint ones. In particular, in the case r ( x ) = sgn x and q ∈ L 1 ( R , ( 1 + | x | ) d x ) , we prove that A is similar to a self-adjoint operator if and only if A is J-nonnegative. The latter condition on q is sharp, i.e., we construct q ∈ ⋂ γ 1 L 1 ( R , ( 1 + | x | ) γ d x ) such that A is J-nonnegative with the singular critical point 0. Hence A is not similar to a self-adjoint operator. For periodic and infinite-zone potentials, we show that J-positivity is sufficient for the similarity of A to a self-adjoint operator. In the case q ≡ 0 , we prove the regularity of the critical point 0 for a wide class of weights r. This yields new results for “forward–backward” diffusion equations.

Spectral properties of singular Sturm-Liouville operators with indefinite weight sgn x

We consider a singular Sturm-Liouville expression with the indefinite weight sgnx. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of 1. Moreover, in this situation, the point 1 is a regular critical point. We construct an operator A = (sgnx)(id 2 /dx 2 + q) with non-real spectrum accumulating to a real point. The obtained results are applied to several classes of Sturm-Liouville operators.

Bohl–Perron-type stability theorems for linear difference equations with infinite delay

The relation between the following two properties of linear difference equations with infinite delay is investigated: (i) exponential stability and (ii) -input -state stability (Perrons property) which means that solutions of the non-homogeneous equation with zero initial data belong to when non-homogeneous terms are in . It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted -space with an exponentially fading weight (the phase space). Our main result states that whenever and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and -input -state stabilities do not depend on the choice of a phase space and parameters p and q. -input -state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.

Abstract Kinetic Equations with Positive Collision Operators

We consider “forward-backward” parabolic equations in the abstract form Jdψ/dx+Lψ = 0, 0 < x ≤∞, where J and L are operators in a Hilbert space H such that J = J* = J −1, L = L* ≥0, and ker L = {0}. The following theorem is proved: if the operator B = JL is similar to a self-adjoint operator, then associated half-range boundary problems have unique solutions. We apply this theorem to corresponding nonhomogeneous equations, to the time-independent Fokker-Plank equation \( \mu \frac{{\partial \psi }} {{\partial x}}(x,\mu ) = b(\mu )\frac{{\partial ^2 \psi }} {{\partial \mu ^2 }}(x,\mu ) \) 0 < x < τ, μ ∈ ℝ, as well as to other parabolic equations of the “forwardbackward” type. The abstract kinetic equation Tdψ/dx = −Aψ(x)+f(x), where T = T* is injective and A satisfies a certain positivity assumption, is also considered. The method is based on the Krein space theory.

On the Nature of Ill-Posedness of the Forward-Backward Heat Equation

Abstract.We study the Cauchy problem with periodic initial data for the forward-backward heat equation defined by a J-self-adjoint linear operator L depending on a small parameter. The problem originates from the lubrication approximation of a viscous fluid film on the inner surface of a rotating cylinder. For a certain range of the parameter we rigorously prove the conjecture, based on numerical evidence, that the complete set of eigenvectors of the operator L does not form a Riesz basis in

A Functional Model, Eigenvalues, and Finite Singular Critical Points for Indefinite Sturm-Liouville Operators

\mathcal{L}^2(-\pi, \pi)

Recovery of periodicities hidden in heavy‐tailed noise

. Our method can be applied to a wide range of evolution problems given by PT-symmetric operators.

Nonlinear Bang–Bang Eigenproblems and Optimization of Resonances in Layered Cavities

The paper is devoted to optimization of resonances for Krein strings with total mass and statical moment constraints. The problem is to design for a given