Fractional Sturm-Liouville operators on compact star graphs

Abstract

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 αi{\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 αi→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 L2{L}^{2} space.

Keywords

• fractional sturm-liouville operator
• metric graph
• transmission condition
• fractional-order derivative
• star graph
• 26a33
• 81q10
• 34b37
• 34b45
• 34b24