Spectral Problems Of Jacobi Operators In Limit-Circle Case

This paper investigates the minimal symmetric operator bounded from below and generated by the real infinite Jacobi matrix in the Weyl-Hamburger limitcircle case. It is shown that the inverse operator and resolvents of the selfadjoint, maximal dissipative and maximal accumulative extensions of this operator are nuclear (or trace class) operators. Besides, we prove that the resolvents of the maximal dissipative operators generated by the infinite Jacobi matrix, which has complex entries, are also nuclear (trace class) operators and that the root vectors of these operators form a complete system in the Hilbert space

Resolvent operator of singular Dirac system with transmission conditions

This paper is concerned with the resolvent operator of one dimensional singular Dirac operator with transmission conditions. We study the Titchmarsh-Weyl function of this problem. Later, we construct a Green function and a spectral function for regular and singular problems. With the help of these functions, we obtain an expansion into a Fourier series of resolvent in regular case. Furthermore, we give integral representations in terms of the spectral function for the resolvent of this operator with transmission conditions in singular case. Finally, we obtain a formula for the Titchmarsh-Weyl function in terms of the spectral function of the singular Dirac system

Dilation, Model, Scattering and Spectral Problems of Second-Order Matrix Difference Operator

In the Hilbert space ℓ 2 Ω (Z; E) (Z := {0,±1,±2, ...}, dim E = N < ∞), the maximal dissipative singular second-order matrix difference operators that the extensions of a minimal symmetric operator with maximal deficiency indices (2N, 2N) (in limit-circle cases at ±∞) are considered. The maximal dissipative operators with general boundary conditions are investigated. For the dissipative operator, a self-adjoint dilation and is its incoming and outgoing spectral representations are constructed. These constructions make it possible to determine the scattering matrix of the dilation. Also a functional model of the dissipative operator is constructed. Then its characteristic function in terms of the scattering matrix of the dilation is set. Finally, a theorem on the completeness of the system of root vectors of the dissipative operator is proved

q-fractional Dirac type systems

This paper is devoted to study a regular q-fractional Dirac type system. We investigate the properties of the eigenvalues and the eigenfunctions of this system. By using a fixed point theorem we give a sufficient condition on eigenvalues for the existence and uniqueness of the associated eigenfunctions

REGULAR FRACTIONAL DIRAC TYPE SYSTEMS

In this paper, we study one dimensional fractional Dirac type systems which includes the right-sided Caputo and the left-sided Riemann-Liouvile fractional derivatives of same order α,α∈(0,1). We investigate the properties of the eigenvalues and the eigenfunctions of this syste

Spectral analysis of singular Sturm-Liouville operators on time scales

In this paper, we consider properties of the spectrum of a Sturm-Liouvilleoperator on time scales. We will prove that the regular symmetricSturm-Liouville operator is semi-bounded from below. We will also give someconditions for the self-adjoint operator associated with the singularSturm-Liouville expression to have a discrete spectrum. Finally, we willinvestigate the continuous spectrum of this operator

IMPULSIVE STURM-LIOUVILLE PROBLEMS ON TIME SCALES

In this paper, we consider an impulsive Sturm-Lioville problem on Sturmian time scales. We investigate the existence and uniqueness of the solution of this problem. We study some spectral properties and self-adjointness of the boundary-value problem. Later, we construct the Green function for this problem. Finally, an eigenfunction expansion is obtained

A spectral theory for discontinuous Sturm-Liouville problems on the whole line

In this study, we consider the singular discontinuous Sturm-Liouville problem on the whole line with transmission conditions. For this problem the existence of a spectral matrix-valued function is proved. A Parseval equality and an expansion formula are given for such problem

Hahn multiplicative calculus

In this study, Hahn multiplicative calculus was introduced and as an application of this subject, the classical Sturm--Liouville problem was examined under this structure

Spectral problems of non-self-adjoint singular q -Sturm–Liouville problem with an eigenparameter in the boundary condition

In this paper, a non-self-adjoint (dissipative) q-Sturm–Liouville boundary-value problem in the limit-circle case with an eigenparameter in the boundary condition is investigated. The method is based on the use of the dissipative operator whose spectral analysis is sufficient for boundary value problem. A selfadjoint dilation of the dissipative operator together with its incoming and outgoing spectral representations is established and so it becomes possible to determine the scattering function of the dilation. A functional model of the dissipative operator is constructed and its characteristic function in terms of scattering function of dilation is defined. Theorems on the completeness of the system of eigenvectors and the associated vectors of the dissipative operator and the q-Sturm–Liouville boundary value problem are presented