Spectrum Of Several Differential Operators With Indefinite Weight

In this paper, we investigate spectrum properties of several ordinary differential op-erators with indefinite weight function.Firstly, we study a discontinuous Sturm-Liouville operator with eigenparameter de-pendent boundary conditions and indefinite weight function. The eigenparameter exists in two boundary conditions, so we establish a new operator A associated with the problem. Thus the eigenvalues and eigenfunctions of the problem are defined as the eigenvalues and the first components of the corresponding eigenelements of the operator A respectively. We prove that the operator A is self-adjoint in an appropriate Krein space K and its real eigenvalues are unbounded from below and above, show that the eigenvalues of the prob-lem are analytically simple, discuss asymptotic approximation formula of its eigenvalues and eigenfunctions, obtain completeness of its eigenfunctions, construct Green's function and the resolvent of this operator in terms of Green's function. At the end of the chapter we extend these conclusions to the operator with finite turning points. Especially, we notice that the signs of ρ and θ influence spectrum properties of the operator A.Next we consider a discontinuous fourth-order differential operator with indefinite weight function. The eigenparameter exists in four boundary conditions, so we establish a new operator A associated with the problem in Krein space K:=(Lrθ2(I)(?) Cθρ1(?) Cθρ2(?) Cρ3(?) Cρ4,[·,·]. Compare to Sturm-Liouville operator, the spectrum of fourth-order differential operator is more complicate. Here by means of the Green formulae, we prove that the new operator is symmetric in Krein space K. And by constructing fundamental solutions of this problem, we know the truth that the eigenvalues of the problem consist of the zeros of function ω(λ). Thus we can obtain some properties of its eigenvalues. Especially, we notice that the sign of θ influence space K and point spectrum properties of the operator A when we fix the weight function.Further, we consider a singular Sturm-Liouville expression and a high-order singular differential expression with the indefinite weight function. Moreover, we characterize the local definitizability of the corresponding self-adjoint operator in a Krein space in a neighborhood of∞. Here, the weight function r(x) is more general, its turning points are not uniqueness. We introduce the operators A_, Aab and A+. A_is self-adjoint extension of Amin-in the anti-Hilbert space Lr2((-∞a)) and A+is self-adjoint extension of Amin+in the Hilbert space Lr2+((b,+∞)). Moreover, Aab is self-adjoint extension of Sab in the Krein space LT2ab((a,b)). If the operator A+is semi-bounded from below and A_is semi-bounded from above, then the resolvent set p(A) of the self-adjoint operator A in (5.1.6) in the Krein space Lr2(R) is nonempty, the essential spectrum σess(A) of the self-adjoint Sturm-Liouville operator A in the Krein space Lr2(R) s.t. σess(A)＝σaess(A-) Uσess(A+) and there exists R>0such that the operator A is definitizable over the domain{λ∈C:|λ|> R}.Finally, we consider an indefinite Sturm-Liouville expression with two singular end-points. Its turning points are also not uniqueness. We construct a boundary triplets for Sturm-Liouville operator and obtain corresponding abstract Weyl function. We present a sufficient condition and a necessary condition for similarity of indefinite Sturm-Liouville operators to self-adjoint operators respectively. Using these results, we construct a ex-ample of operator and prove that this operator is similar to a self-adjoint one and we construct two examples and prove that none of them is similar to a self-adjoint operator.This thesis consists of six chapters. In chapter Ⅰ, we introduce the background about the problems what we study and the main results of this thesis; Chapter Ⅱ simply present Krein space approach of ordinary differential operators; Chapter Ⅲ study a discontin-uous Sturm-Liouville operator with eigenparameter-dependent boundary conditions and indefinite weight function; In Chapter Ⅳ, we study a discontinuous four order differential operator with eigenparameter-dependent boundary conditions; In chapter Ⅴ, we discuss the local deflnitizability a singular Sturm-Liouville operator and a high-order singular dif-ferential operator with an indefinite weight function; In chapter Ⅵ, we consider similarity of an indefinite Sturm-Liouville operator with two singular endpoints.