Browsing by Author "Uğurlu, Ekin"

Now showing 1 - 20 of 54

A new Hamiltonian system
(2020) Uğurlu, Ekin; 238990
This paper aims to share a new first-order differential equation that contains the continuous analogous of the orthogonal polynomials on the unit-circle. We introduce some basic results on the system and solutions of the system. Using nested-circle approach we introduce the possible number of square-integrable solutions of the system. At the end of the paper we share a limit-point criteria for the two-dimensional system of equations.
A new insight to the Hamiltonian systems with a finite number of spectral parameters
(2023) Uğurlu, Ekin; 238990
A new method for dissipative dynamic operator with transmission conditions
(Springer Basel Ag, 2018) Uğurlu, Ekin; Ugurlu, Ekin; Tas, Kenan; Taş, Kenan; 238990; 4971
In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction.
Coordinate-Free Approach for the Characteristic Function of a Fourth-Order Dissipative Operator
(Taylor & Francis inc, 2019) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this article, we investigate some spectral properties of a singular dissipative fourth-order dissipative operator in case at the singular point. For this purpose we construct the characteristic function of both maximal simple dissipative operator and completely non-unitary contraction which is the Cayley transform of the dissipative operator. Using the properties of the characteristic operator-function we obtain the related results of the boundary value problem. Moreover we obtain the selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing eigenfunctions by using coordinate-free approach.
Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator
(Frontiers Media Sa, 2019) Uğurlu, Ekin; Ugurlu, Ekin; Baleanu, Dumitru; Baleanu, Dumitru; 56389; 238990
In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential expression and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal non-selfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.
Dependence Of Eigenvalues Of Some Boundary Value Problems
(2021) Uğurlu, Ekin; Taş, Kenan; 4971; 238990
In this work we deal with a system of two first-order differential equations containing the same eigenvalue parameter. We consider some suitable separated real and complex coupled boundary conditions, and show that the eigenvalues generated by this system are continuous in an eigenvalue branch. Also we introduce the ordinary and Frechet derivatives of these eigenvalues with respect to some elements of the data.
Dirac systems with regular and singular transmission effects
(Tubitak Scientific & Technological Research Council Turkey, 2017) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper, we investigate the spectral properties of singular eigenparameter dependent dissipative problems in Weyl's limit-circle case with finite transmission conditions. In particular, these transmission conditions are assumed to be regular and singular. To analyze these problems we construct suitable Hilbert spaces with special inner products and linear operators associated with these problems. Using the equivalence of the Lax-Phillips scattering function and Sz-Nagy-Foias characteristic functions we prove that all root vectors of these dissipative operators are complete in Hilbert spaces.
Direct approach for the characteristic function of a dissipative operator with distributional potentials
(2020) Uğurlu, Ekin; 238990
The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class C-0. Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Caratheodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established.
Discrete left-definite hamiltonian systems
(2023) Uğurlu, Ekin; 238990
In this paper we consider an even-dimensional discrete Hamiltonian system on the set of nonnegative integers in the left-definite form. Using the inertia indices of the hermitian form related with the solutions of the equation we construct some maximal subspaces of the solution space. After constructing some ellipsoids preserving nesting properties we introduce a lower bound for the number of Dirichlet-summable solutions of the equation. Moreover we introduce a limit-point criterion.
Dissipative operator and its Cayley transform
(Tubitak Scientific & Technological Research Council Turkey, 2017) Uğurlu, Ekin; Ugurlu, Ekin; Tas, Kenan; Taş, Kenan; 238990; 4971
In this paper, we investigate the spectral properties of the maximal dissipative extension of the minimal symmetric differential operator generated by a second order differential expression and dissipative and eigenparameter dependent boundary conditions. For this purpose we use the characteristic function of the maximal dissipative operator and inverse operator. This investigation is done by the characteristic function of the Cayley transform of the maximal dissipative operator, which is a completely nonunitary contraction belonging to the class C-0. Using Solomyak's method we also introduce the self-adjoint dilation of the maximal dissipative operator and incoming/outgoing eigenfunctions of the dilation. Moreover, we investigate other properties of the Cayley transform of the maximal dissipative operator.
Extensions of a Minimal Third-Order Formally Symmetric Operator
(Malaysian Mathematical Sciences Soc, 2020) Ugurlu, Ekin; Uğurlu, Ekin; 238990
In this paper, we consider some regular boundary value problems generated by a third-order differential equation and some boundary conditions. In particular, we construct maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal operator. Further using Lax-Phillips scattering theory and Sz.-Nagy-Foias characteristic function theory we prove a completeness theorem.
Fourth order differential operators with distributional potentials
(Tubitak Scientific & Technological Research Council Turkey, 2020) Ugurlu, Ekin; Uğurlu, Ekin; Bairamov, Elgiz; 238990
In this paper, regular and singular fourth order differential operators with distributional potentials are investigated. In particular, existence and uniqueness of solutions of the fourth order differential equations are proved, deficiency indices theory of the corresponding minimal symmetric operators are studied. These symmetric operators are considered as acting on the single and direct sum Hilbert spaces. The latter one consists of three Hilbert spaces such that a squarely integrable space and two spaces of complex numbers. Moreover all maximal self-adjoint, maximal dissipative and maximal accumulative extensions of the minimal symmetric operators including direct sum operators are given in the single and direct sum Hilbert spaces.
Fractional Differential Equation With a Complex Potential
(2020) Uğurlu, Ekin; Taş, Kenan; Baleanu, Dumitru; 238990; 4971; 56389
In this manuscript, we discuss the square-integrable property of a fractional differential equation having a complex-valued potential function and we show that at least one of the linearly independent solutions of the fractional differential equation must be squarely integrable with respect to some function containing the imaginary parts of the spectral parameter and the potential function.
Investigation of the eigenvalues and root functions of the boundary value problem together with a transmission matrix
(Taylor&Francis LTD, 2019) Uğurlu, Ekin; 238990
In this paper, we consider a singular even-order Hamiltonian system on the union of two intervals together with appropriate boundary and transmission conditions. For investigating the spectral properties of the problem we pass to the inverse operator with an explicit form and we prove some completeness theorems.
Left-definite Hamiltonian systems and corresponding nested circles
(2023) Uğurlu, Ekin; 238990
This work aims to construct the Titchmarsh-Weyl M(λ)−theory for an even-dimensional left-definite Hamiltonian system. For this purpose, we introduce a suitable Lagrange formula and selfadjoint boundary conditions including the spectral parameter λ. Then we obtain circle equations having nesting properties. Using the intersection point belonging to all the circles we share a lower bound for the number of Dirichlet-integrable solutions of the system
Left-definite system of first-order equations together with eigenparameter-dependent boundary conditions
(2024) Uğurlu, Ekin; 238990
This paper provides some information on the eigenvalues and eigenfunctions of some left-definite system of first-order differential equations subject to eigenparameter-dependent boundary conditions. Namely, we show that the pair of solutions of the system of equations satisfying some initial conditions exists and is unique, and this pair is analytic in the spectral parameter of order 1/2. We also introduce Lagrange's formula for the left-definite equation. Using some Prüfer angels, we investigate oscillation of zeros of eigenfunctions and asymptotics equations for the eigenvalues of the problem. Moreover, we share some ordinary and Fréchet derivatives of eigenvalues and eigenfunctions with respect to some elements of data.
On a fifth-order nonselfadjoint boundary value problem
(2021) Uğurlu, Ekin; Taş, Kenan; 238990
In this paper we aim to share a way to impose some nonselfadjoint boundary conditions for the solutions of a formally symmetric fifth-order differential equation. Constructing a dissipative operator related with the problem we obtain some informations on spectral properties of the problem. In particular, using coordinate-free approach we construct characteristic matrix-function related with the contraction which is obtained with the aid of the dissipative operator. © 2021. TÜBİTAK.
On a new class of fractional operators
(2017) Jarad, Fahd; Uğurlu, Ekin; Abdeljawad, Thabet; Baleanu, Dumitru; 56389
This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.
On a new class of fractional operators
(Springeropen, 2017) Jarad, Fahd; Uğurlu, Ekin; Jarad, Fahd; Ugurlu, Ekin; Abdeljawad, Thabet; Abdeljawad, Thabet; Baleanu, Dumitru; Baleanu, Dumitru; 234808; 238990
This manuscript is based on the standard fractional calculus iteration procedure on conformable derivatives. We introduce new fractional integration and differentiation operators. We define spaces and present some theorems related to these operators.
On dilation, scattering and spectral theory for two-interval singular differential operators
(Soc Matematice Romania, 2015) Uğurlu, Ekin; Uğurlu, Ekin; 238990
This paper aims to construct a space of boundary values for minimal symmetric singular impulsive-like Sturm-Liouville (SL) operator in limit-circle case at singular end points a, b and regular inner point c. For this purpose all maximal dissipative, accumulative and self-adjoint extensions of the symmetric operator are described in terms of boundary conditions. We construct a self-adjoint dilation of maximal dissipative operator, a functional model and we determine its characteristic function in terms of the scattering matrix of the dilation. The theorem verifying the completeness of the eigenfunctions and the associated functions of the dissipative SL operator is proved.