Browsing by Author "Mutlu, Gokhan"

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A First-Order System of Equations on a Compact Star Graph
(Ankara Univ, FAC Sci, 2025) Mutlu, Gokhan; Ugulu, Ekin
This study concerns a boundary value problem generated by a system of first-order differential equations and some symmetric boundary conditions on a compact star graph. Unlike usual quantum graph Hamiltonians, the Hamiltonian considered in this paper acts on vector-valued functions living on the edges. Appropriate vertex conditions are introduced to ensure the corresponding boundary value problem is symmetric. In particular, coupling and separated conditions are imposed at the central and boundary vertices, respectively. Moreover, the general form of the vertex conditions at the central vertex is discussed. Finally, the characteristic function is constructed for the symmetric boundary value problem generated by the first-order system and specific coupling vertex conditions.
Citation - WoS: 1
Fractional Sturm-Liouville Operators on Compact Star Graphs
(de Gruyter Poland Sp Z O O, 2024) Mutlu, Gokhan; Ugurlu, Ekin
In this article, we examine two problems: a fractional Sturm-Liouville boundary value problem on a compact star graph and a fractional Sturm-Liouville transmission problem on a compact metric graph, where the orders alpha i {\alpha }_{i} of the fractional derivatives on the ith edge lie in ( 0 , 1 ) (0,1) . Our main objective is to introduce quantum graph Hamiltonians incorporating fractional-order derivatives. To this end, we construct a fractional Sturm-Liouville operator on a compact star graph. We impose boundary conditions that reduce to well-known Neumann-Kirchhoff conditions and separated conditions at the central vertex and pendant vertices, respectively, when alpha i -> 1 {\alpha }_{i}\to 1 . We show that the corresponding operator is self-adjoint. Moreover, we investigate a discontinuous boundary value problem involving a fractional Sturm-Liouville operator on a compact metric graph containing a common edge between the central vertices of two star graphs. We construct a new Hilbert space to show that the operator corresponding to this fractional-order transmission problem is self-adjoint. Furthermore, we explain the relations between the self-adjointness of the corresponding operator in the new Hilbert space and in the classical L 2 {L}<^>{2} space.
Citation - WoS: 1
On Quantum Star Graphs With Eigenparameter Dependent Vertex Conditions
(Springer Basel Ag, 2023) Ugurlu, Ekin; Mutlu, Gokhan
We investigate the spectral properties of two different boundary value problems on a compact star graph in which the vertex conditions are dependent on the spectral parameter. We treat these boundary value problems as eigenvalue problems in some extended Hilbert spaces by associating them with vector-valued operators. We prove that the corresponding operators are self-adjoint. We construct the characteristic functions of these eigenvalue problems and prove that the corresponding operators have discrete spectrum. Moreover, we present some examples where we construct fundamental solutions and derive the resolvent operators.