Numerical Computation Of Eigenvalues Of Differential Operators And The Boundary Value Space Theory For The Self-Adjoint Extensions Of Symmetric Operators

In this paper,we study three important problems in the field of(differential) operator theory:the computation of eigenvalues,the self-adjoint extensions of symmetric operators and the description of self-adjoint domains of differential operators.For the numerical computation of eigenvalues of differential operators,it is very important for pure theory and real applications.In fact,for differential operators,only a few classes have closed form solutions,but it is more and more powerful,especially with the appearance and update of the high speed computer,for the numerical method to deal with various problems from engineering and technology.Meantime,we need to notice that the numerical results can enlighten people to obtain more profound qualitative results conversely.The numerical analysis will be more and more important in the research of theory.High order(non)-self-adjoint problems arose in hydrodynamic and magnetohydro-dynamic stability theory,the problems of heat-conduction and diffusion in multilayered medium,and the problems with spectral parameter in boundary conditions are several new important problems in the spectral theory of differential operators.But,the numerical methods for computation of eigenvalue appeared are designed mainly for the second order self-adjoint problems.In recent years,Greenberg and Marletta developed a kind of shooting method,which can deal with high order(non)-self-adjoint problems, based on the Atkinson-Pr(u|¨)fer oscillation theory.As the method they used to solve initial value problems of ordinary differential equations is to approximate the coefficient function with piecewise-constant function,it have a shortcoming for highly oscillatory coefficient function problems.For the problems above,we develop a new method,which offer a uniform frame for these problems.For a representation for the general solution of the eigen-equation ly(x,λ)=λy(x,λ)(where l denote the ordinary differential expression with order n andλis the eigen-parameter) as a eigen-parameter power series,we find and prove that the coefficients of the power series satisfy a recursive relation constructed by differential equations.From the properties of Volterra integral operator,we give and prove a method to obtain the power series solution,also,we prove that the method is numerical stable and give the truncated error estimation of this power series solution.According to the properties of characteristic determinant△(λ),we transform the eigenvalue problem to zero problem of characteristic determinant in the given area or interval.As the characteristic determinant we computed is the polynomial ofλ,applying the rootfinder, we can find the numerical solution of eigenvalues in a given domain or interval.Further, we give a method to compute the corresponding eigenfunctions.At last,we verify and analyze our method with concrete numerical examples.The method developed from us is simple,powerful and universal.Not only can it to deal with the second order self-adjoint problems,but also can it to deal with the high order(non)-self-adjoint problems and other new problems mentioned above.Meantime,the method of us overcome some of shortcoming of the method of Creenberg and Marletta.In the classical theory of boundary value space,the linear relation theory is the model of the description of the extensions of symmetric operators,but the theory of boundary value space using the method of linear relations is rather cumbersome.Whereas this fact,in this thesis,we study the theory of boundary value space constructively and discuss the problems related the description of self-adjoint extension.First,we study the relation between boundary maps and the self-adjoint domains only using the Neumann formula and the definition of self-adjointness,and prove that all self-adjoint domains can be unitary parameterized by arbitrary boundary value space constructively.The method above is more simple and directive compared to the linear relation method.In addition, we discover the structure of the general boundary maps,and redefine the boundary value space constructively.Further,we prove that there is a biholomorphic mapping between the unitary transformations for unitary parameterizing the self-adjoint extensions and the unitary transformations for Cayley extensions.This result will be the basis for the research of the dependence of the spectra and the boundary conditions.At last,we obtain and prove the necessary and sufficient condition for the self-adjointness of the general boundary conditions B(ψ):=MΓ_1ψ+NΓ_2ψ=0,where M,N are the square matrix with the order equal to the deficiency index,the corresponding unitary transformation and the boundary mappings.These results is true for all closed symmetric operators with equal deficiency indices,and provide a uniform tools for various ordinary differential operators.In this thesis,we give the description of self-adjoint domains for differential operators newly based on the theory of boundary value space for the general symmetric operators. Concretely,for the regular and singular differential operators and discontinuous Sturm-Liouville operators,we find a group of simple boundary maps and give the self-adjoint description by unitary transformations on C^m(where m is the deficiency index of the corresponding operator) parametrically,respectively.This thesis consists of eight chapters.In chapterⅠ,we introduce the background about the problems what we study and the main results of this thesis;ChapterⅡpresent some real backgrounds from which the spectral problems of differential operators arise;ChapterⅢintroduce the general method to obtain the general solution as eigenparameter power series of initial value of ordinary differential equation with the idea of separating the eigen-parameter,and the numerical method to compute the corresponding eigenfunction;In chapterⅣandⅤ,we discuss the numerical method of eigenvalue of self-adjoint and non-self-adjoint problems;ChapterⅥdiscuss the theory of boundary value space of symmetric operators;ChapterⅦandⅧdiscuss the analytical description of self-adjoint domain of ordinary differential operators and discontinuous Sturm-Liouville operators with boundary value space method respectively.