Optimal Recovery Of Potentials Or Weights For Sturm-Liouville Problems With One Given Eigenvalue

Sturm-Liouville theory has been an important mathematical tool in engi-neering technology,mathematical physics,life science and other fields since it generates.In addition to the classical Sturm-Liouville problems(SLP),some phenomena in physics and engineering technology,reaction-diffusion processes and nonlocal quantum mechanics etc.give rise to SLP with spectral param-eters in boundary conditions and nonlocal SLP.Driven by the application in many fields,the extremal problems in S-L theory,especially the extremal prob-lem of eigenvalues,and inverse spectral problems have aroused great interest and attention of many scholars.In the study of classical inverse spectral prob-lems,it is generally necessary to know two sets of full spectral information to uniquely determine the potential functions,however,there can only detect a finite number of eigenvalues in practice.This dissertation focuses on the quan-titative expression of infimum of integral modulus for potentials or weights of SLP and analytic expression of attainable function when one eigenvalue is giv-en,so as to realize the optimal recovery of potentials or weights,and obtain the extremal values of eigenvalues when potentials or weights are on the L1 sphere.There are two parts in this dissertation:1.This paper proves that the m-th eigenvalue of a linear Hamiltonian sys-tem with distribution coefficients is continuous on the coefficients with respect to a weak*topology.The continuity of eigenvalues on the coefficients of equa-tions or systems is the basis of extremal value problems.In order to highlight the role of Dirac functions in extremal problems we first introduce the linear space D[0.1]formed by the finite lincar combinationis of L1 integrable func-tions and Dirac functions,and weak*topological space(D[0.1],w*).Then we extend the Dieudorine's theorem in metric space to(D[0.1].u*).Final-ly.we show the strong continuity of eigenvalues by means of the generalized Dieudonne's theorem,the properties of eigenvalue characteristic function and the uniform lower bound of eigenvalues.As the general form of S-L equations.the linear Hamiltonian system includes the even order scalar equations,and its coefficients are allowed to be Dirac functions in this paper.So the result in this part covers the existing strong continuity of eigenvalues for scalar equa-tions[36,56,97,100]and second-order linear systems[75]in weak topological space(Lp.wp).p?1.It also covers the strong continuity of the eigenvalues on nonsingular continuous measure potentials or weights with respect to weak*topology for second order measure differential equations[76,102].2.Given one eigenvalue,this paper explicitly formulates the infimum of the L1-norm of potentials or weights for SLP with separated boundary condi-tions,SLP with spectral parameters in boundary conditions and nonlocal SLP with nonlocal potentials,and specifies the attainable functions.It involves:(1)For SLP with separated boundary conditions,we extend the classic Lyapunov inequality by Mercer's theorem.Then using generalized Lyapunov-type inequality we set up the infimum of the norms for potentials in L1[0.1]explicitly in terms of a given nth eigenvalue and parameters in boundary value conditions.and specify where the infimum can be attained.As an application,we obtain extremum of the nt.h eigenvalue of the problem for potentials on a sphere in L1[0.1].(2)For SLP with spectral parameters in boundary conditions which gen-erates by a kind of string vibration system.we first transforms it into the SLP with a distribution weight and boundary conditions without spectral pa-rameters.Then using the Min-Max principle.Lyapunov-type inequality and the spectral theory of SLP with Dirac distribution weight,we study in detail the minimization of the total mass and string mass of the system when the minimum vibration frequency is known.(3)For the nonlocal SLP with a nonlocal potential,this paper is concerned with its eigenvalue problems and the corresponding extremal problems.We give the operator description of this problem,the characteristic function,the lower bound and localization of eigenvalues.Furthermore,we investigate the norm infimum of the nonlocal potentials in L1[0,1]if the first eigenvalue ? is known.The explicit quantity of this value is given in terms of ?.As appli-cations,the extremal values of the first eigenvalue for nonlocal potentials on a sphere in L1[0,1]are obtained,and a Lyapunov-type inequality for nonlocal potentials is given.This dissertation is divided into five chapters:the first chapter is the introduction,which introduces the signifi cance,status of the studied problem,the main contents and methods of the work in this paper;the second chapter is the first part of this paper,which proves that the mth eigenvalue of a linear Hamiltonian system is strongly continuous on coefficients with respect to the weak*topology of space D[0,1];the third,fourth and fifth chapters are the second part of this paper,which mainly discuss the extremal values of the potentials of SLP with separated boundary conditions,the weights of SLP with spectral parameters in boundary conditions and the nonlocal potentials of nonlocal SLP with nonlocal potentials,respectively.