Inverse Spectral Problems Of Differential Operators With Bessel Potentials

In quantum mechanics,Schr(?)dinger equations and Dirac equations with spherically symmetric potentials can be used to describe many physical motions in the central force field.For example,the motions of electrons in the Coulomb field of the nucleus;the motions of particles in infinite spherical square potential well,etc.These two kinds of equations play an important role in quantum mechanics,which have enjoyed widespread popularity.The differential operators with Bessel potentials studied in this thesis are derived from the Schr(?)dinger equations and the Dirac equations with spherically symmetric po-tentials.Due to the singularity of the potentials,their research can not be described by classical differential operator theory.Therefore,it is necessary to study differential oper-ators with Bessel potentials,which can not only provide theoretical basis for the quantum behavior of particles,but also enrich the theoretical research of differential operators.We mainly study the inverse spectral problems for Bessel operators and Dirac-Bessel opera-tors.This thesis involves six chapters,which are as follows:In the first chapter,we introduce the physical background and research advances of Bessel operators and Dirac-Bessel operators,the development of inverse spectral prob-lems,and the main results of this paper.In the second chapter,we present some basic concepts and relative results.In the third chapter,we study the inverse spectral problems of Dirac-Bessel opera-tors with the potential known on a subinterval.We show that the potential on the whole interval can be uniquely determined in terms of appropriate partial information on the spectral data including eigenvalues and norming constants.By virtue of the asymptotics of eigenvalues and a classical estimate of Levinson,we reconstruct a series of inequalities.Based on this,we associate the known information of potential and spectral data with the Weyl function.And then,we prove the potential can be uniquely determined by the Weyl function.In the forth chapter,the inverse spectral problems of Dirac-Bessel operators on a star-shaped graph are considered.We put forward the definition of Weyl vector and Weyl solution of Dirac-Bessel operator on a star-shaped graph for the first time.And we study the asymptotics of the Weyl solutions by means of the Green matrix.Finally,we prove that the Dirac-Bessel operators can be uniquely determined by the Weyl vector.In particular,our results also work for regular Dirac operators.The fifth chapter is devoted to investigate the Bessel operators with distributional potentials.The continuous dependence of the nth eigenvalue on the boundary conditions will be investigated.First of all,using the generalized Prufer transformation,we describe the location of eigenvalues;in particular,we also obtain the oscillation properties of the eigenfunctions.We next construct a sequence of Sturm-Liouville operators with distri-butional potentials.By virtue of the properties of Prufer angle we obtain a relationship between the eigenvalues of Sturm-Liouville operators with distributional potentials and the eigenvalues of the operators to be discussed.Based on this new relationship,we final-ly establish our results in light of the deficiency index theory and the properties of Prufer angle.Finally,in the sixth chapter,we first of all investigate the inverse nodal problems for the classical Bessel operators.We prove that any twin dense nodal subset on interval(a,1),a?(0,1/2],can uniquely determine the potential on the whole interval.However,if a?(1/2,1),then the twin dense nodal subset on interval(a,1)fails to establish the unique-ness results.In this case,we provide additional informations to prove the uniqueness results.Secondly,as the application of the results in chapter three,the inverse spectral problems for Bessel operators with distributional potentials are considered.We show that the potential known on a subinterval can be determined completely by partial information on the eigenvalues and norming constants.