A Criterion for Spectrum Decomposition

We obtain a necessary and sufficient condition for the decomposition of the spectrum of an arbitrary nonsymmetric potential whose least value is attained at finitely many points.     

Conditions of Spectrum Localization for Operators not Close to Self-Adjoint Operators

The Keldysh theorem is generalized to an arbitrary closed operator that is not necessarily close to self-adjoint operators and has a resolvent of Schatten–von Neumann class S _p . Based on this theorem, conditions of spectrum localization are obtained for certain classes of non-self-adjoint differential operators.     

On the uniqueness criterion for solutions of the Sturm-Liouville equation

We consider the Sturm-Liouville equation $ - y'' + qy = \lambda ^2 y $ in an annular domain K from ? and obtain necessary and sufficient conditions on the potential q under which all solutions of the equation ?y″(z) + q(z)y(z) = λ ² y(z), z ∈ γ, where γ is a certain curve, are unique in the domain K for all values of the parameter λ ∈ ?.     

Necessary Conditions for the Localization of the Spectrum of the Sturm-Liouville Problem on a Curve

We consider the Sturm-Liouville operator on a convex smooth curve lying in the complex plane and connecting the points 0 and 1. We prove that if the eigenvalues λ_k with large numbers are localized near a single ray, then this ray is the positive real semiaxis. Moreover, if the eigenvalues λ_k are numbered with algebraic multiplicities taken into account, then λ_k ∼ π · k as k → +∞.__________Translated from Matematicheskie Zametki, vol. 78, no. 1, 2005, pp. 72–84.Original Russian Text Copyright © 2005 by Kh. K. Ishkin.     

On a trivial monodromy criterion for the Sturm-Liouville equation

We obtain a necessary and sufficient condition for the equation $ - y''(z) + q(z)y(z) = \lambda y(z)$ to be monodromy-free; here z ∈ γ and γ is a piecewise smooth curve which is the boundary of a convex bounded domain.     

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.     

Quantum Defect for the Dirac Operator with a Nonanalytic Potential

We evaluate the quantum defects for the continuous and discrete spectra of the radial Dirac operator with the potential V(r) = –A/r + q(r), where A > 0 and .     

On localization of the spectrum of the problem with complex weight

The paper considers problems with complex weight for whose spectrum the classical asymptotic formula is not true. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 49–64, 2006.