Spectral Properties Of Sturm-liouville Operators On Infinite Metric Graphs

Differential operators on metric graphs are abstract mathematical models used to study the problems in mesoscopic physics and the problems on chemical structures,and they have been applied extensively in chemistry,particle physics and nanotechnology.Since the fifties of the last century,the theory of differential operators on metric graphs has been a hot research topic in mathematics,physics,biology,chemistry and their interdisciplinaries.After decades of development,it has become an important part of the differential equation theory and the spectral theory of differential operators.The main object of this paper is to study the self-adjointness and the spectral properties of Sturm-Liouville operators on infinite metric graphs.This paper involves six chapters.In the first chapter,we introduce the background of quantum graphs and the main results in this paper.In the second chapter,basic concepts and relative results are presented.The third chapter is devoted to study the properties of three classes of local vertex conditions: the vertex conditions whose coefficient matrices are of the rank ?(v),the self-adjoint vertex conditions and the J-self-adjoint vertex conditions.Then we study the structures of the sets consisted of vertex conditions.When the differential form is symmetric,the self-adjoint conditions of Sturm-Liouville operators on compact metric graphs are given.Moreover,the Glazman-Povzner-Wienholtz-type self-adjointness results of Sturm-Liouville operators on noncompact metric graphs are obtained.When the differential form is J-symmetric,the J-adjoint operators and the J-self-adjoint extensions of the local Sturm-Liouville operators are studied.In the fourth chapter,we consider the self-adjointness and the spectral properties of Schr?dinger operators on the infinite regular metric trees with ?-type vertex conditions.An operator of this type is unitary equivalent to the direct sum of infinitely many auxiliary operators with transmission conditions.By constructing the quadratic forms of the auxiliary operators and proving the compactness of the embedding operators from auxiliary operators' energy spaces to the function spaces containing their domains,the Molchanov discrete criteria of the auxiliary operators are obtained.Based on the relationship between the spectrum of a Schr?dinger operator with ?-type conditions and the spectra of the auxiliary operators,we obtain a necessary and sufficient condition for the Schr?dingeroperators to have pure discrete spectrum.Finally,the compactness of the perturbation of quadratic forms implies the stability of the essential spectrum and the discreteness of the negative spectrum of the Schr?dinger operators on trees.The self-adjointness and the spectral properties of Schr?dinger operators on the infinite regular metric trees with??-type vertex conditions are investigated in the fifth chapter.The Molchanov discrete criteria are extended to operators of this type in this chapter.Since the energy space of the Schr?dinger operators with ?'-type vertex conditions is contained in the direct sum spaces (?) instead of the Sobolev space (?),some new inequalities should be proven.Moreover,we obtain the stability of the essential spectrum of this type of operators when potential function and the coupling constants of ?'-type vertex conditions change.The last chapter is focus on the infinite regular metric trees with infe (?).And we consider the Sturm-Liouville operators on these trees.We prove that the selfadjointness of the auxiliary operators implies the self-adjointness of Sturm-Liouville operators on trees.On this basis,we get two types of conditions to ensure the self-adjointness of the Sturm-Liouville operators.Based on a series of embedding inequalities,we extend the Molchanov discrete criteria to the auxiliary operators and the Sturm-Liouville operators on trees.