Abstract
We study the vertex conditions of local Sturm-Liouville operators on metric graphs. Our aim is to give a new description of vertex conditions defining the self-adjoint Sturm-Liouville operators and to clarify the natural geometric structure on the space of complex vertex conditions. Based on this description, we give the self-adjointness results for local Sturm-Liouville operators on finite graphs and the Povzner-Wienholtz-type self-adjointness results for local Sturm-Liouville operators on infinite graphs.
Highlights
A graph we consider in this paper is an ordered pair of disjoint sets (V, E), where V is a countable vertex set and E is a countable edge set
A Sturm-Liouville operator on the graph is a system of Sturm-Liouville operators on intervals complemented by appropriate matching conditions at inner vertices and some boundary conditions at the boundary vertices
If V and E are finite sets, the description of the self-adjoint vertex conditions can be treated as the description of the boundary conditions in self-adjoint multi-interval SturmLiouville problems. (The results about multi-interval Sturm-Liouville problems can be found in [ ].) For example, in [ ], Harmer described the self-adjoint boundary conditions for the Schrödinger operators on the finite graphs in terms of a unitary matrix
Summary
A graph we consider in this paper is an ordered pair of disjoint sets (V , E), where V is a countable vertex set and E is a countable edge set. (The results about multi-interval Sturm-Liouville problems can be found in [ ].) For example, in [ ], Harmer described the self-adjoint boundary conditions for the Schrödinger operators on the finite graphs in terms of a unitary matrix. Based on the methods in [ ] and [ ], we give a new description of all vertex conditions defining the domains of local (essentially) self-adjoint Sturm-Liouville operators on weighted directed graphs. In Section we give some properties of self-adjoint vertex conditions for local Sturm-Liouville operators on the graph. Based on these properties we get the necessary conditions for local Sturm-Liouville operators to be self-adjoint. In the fourth section we give the sufficient conditions for local Sturm-Liouville operators to be self-adjoint, which are PovznerWienholtz-type self-adjointness results for Sturm-Liouville operators on graph.
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