Second-Order Differential Operators in the Limit Circle Case

We consider symmetric second-order differential operators with real coefficients such that the corresponding differential equation is in the limit circle case at infinity. Our goal is to construct the theory of self-adjoint realizations of such operators by an analogy with the case of Jacobi operators. We introduce a new object, the quasiresolvent of the maximal operator, and use it to obtain a very explicit formula for the resolvents of all self-adjoint realizations. In particular, this yields a simple representation for the Cauchy-Stieltjes transforms of the spectral measures playing the role of the classical Nevanlinna formula in the theory of Jacobi operators.

Completeness Theorem for Eigenparameter Dependent Dissipative Dirac Operator with General Transfer Conditions

This paper deals with a singular (Weyl’s limit circle case) non-self-adjoint (dissipative) Dirac operator with eigenparameter dependent boundary condition and finite general transfer conditions. Using the equivalence between Lax-Phillips scattering matrix and Sz.-Nagy-Foiaş characteristic function, the completeness of the eigenfunctions and associated functions of this dissipative operator is discussed.

NONLINEAR SINGULAR STURM-LIOUVILLE PROBLEMS WITH IMPULSIVE CONDITIONS

In this paper, we consider a non-linear impulsive Sturm-Liouville problem on semiinfinite intervals in which the limit-circle case holds at infinity for THE Sturm-Liouville expression. We prove the existence and uniqueness theorems for this problem.

Titchmarsh–Weyl theory for q-Dirac systems

In this work, we establish Titchmarsh–Weyl theory for singular [Formula: see text]-Dirac systems. Thus, we extend classical Titchmarsh–Weyl theory for Dirac systems to [Formula: see text]-analogue of this system. We show that it does not occur for the limit-circle case for the [Formula: see text]-Dirac system.

Livšic’s theorem for q-Sturm—Liouville operators

In this paper, we study dissipative q-Sturm—Liouville operators in Weyl’s limit circle case. We describe all maximal dissipative, maximal accretive, self adjoint extensions of q-Sturm—Liouville operators. Using Livšic’s theorems, we prove a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative q-Sturm—Liouville operators.

Spectral problems of nonself-adjoint singular discrete Sturm-Liouville operators

In this study we construct a space of boundary values of the minimal symmetric discrete Sturm-Liouville (or second-order difference) operators with defect index (1, 1) (in limit-circle case at ±∞ and limit-point case at ∓∞), acting in the Hilbert space

An Alternate Proof of the De Branges Theorem on Canonical Systems

The aim of this paper is to show that, in the limit circle case, the defect index of a symmetric relation induced by canonical systems, is constant on ℂ. This provides an alternative proof of the De Branges theorem that the canonical systems with tr H1 imply the limit point case. To this end, we discuss the spectral theory of a linear relation induced by a canonical system.

Weyl-Titchmarsh Theory for Time Scale Symplectic Systems on Half Line

We develop the Weyl-Titchmarsh theory for time scale symplectic systems. We introduce theM(λ)-function, study its properties, construct the corresponding Weyl disk and Weyl circle, and establish their geometric structure including the formulas for their center and matrix radii. Similar properties are then derived for the limiting Weyl disk. We discuss the notions of the system being in the limit point or limit circle case and prove several characterizations of the system in the limit point case and one condition for the limit circle case. We also define the Green function for the associated nonhomogeneous system and use its properties for deriving further results for the original system in the limit point or limit circle case. Our work directly generalizes the corresponding discrete time theory obtained recently by S. Clark and P. Zemánek (2010). It also unifies the results in many other papers on the Weyl-Titchmarsh theory for linear Hamiltonian differential, difference, and dynamic systems when the spectral parameter appears in the second equation. Some of our results are new even in the case of the second-order Sturm-Liouville equations on time scales.