The Self-Adjointness Of Product Of Differential Operators And The Dependence Of Eigenvalues On The Boundary

In this paper, we study the self-adjointness of product of differential operators and the dependence of eigenvalues on the boundary. The dif-ferential operators are essentially the unbounded closable operators, and the domains of the unbounded closable operators must not be the whole space. Hence the choice of the domains of the differential operators are always important and difficult. Given a differential expression, the specific demands for the differential operators eventually reflect on the restrictions on the domains. Differential operators with different domains will have different spectral distributions, especially the discrete spectrum. Among the choice of the domains, the choice of self-adjoint domains is one of the important. Self-adjoint differential operator, because of its important ap-plication background, not only makes its spectrum with inverse spectrum problem become a hot topic of mathematicians, at the same time the prob-lem of identification and description of self-adjointness was also mentioned the important position.Firstly we consider the problem of description of self-adjoint domains of product of differential expression in terms of real-parameter solutions. Under appropriate assumptions, using solutions corresponding to value of a pair of opposite each other, we present a characterization of self-adjoint boundary conditions for product of differential expression, making a matrix of self-adjoint boundary conditions be determined only associated with the initial value of the solutions in the regular point.For fourth order singular symmetric differential expression, there will be a middle deficiency indices case. Next we study the problem of self-adjointness of product of two fourth-order and higher-order differential operators in middle deficiency indices case. By using the theorem of char-acterization of real parameter solutions on half line for self-adjoint domains, and analytical skills, we give a sufficient and necessary condition in the form of matrix, and get some results related to the self-adjointness of product operators.Again, in the engineering practice people note:a elastic rod, its di-ameter are negligible compared with its length and both ends are fixed in some meaningful way, then we go to play it and find that the sound stem from the rod will gradually strengthen with its length shortening, that is, the natural frequency of the rod increases gradually, this phenomenon is highly known by dynamicists. Using mathematical language, we translate this problem to the problem of dependence of eigenvalues of fourth-order boundary value problems on the boundary. With the help of spectrum theory of differential operator, combining the work of Dauge, Q., Kong and others ([38],[51],[87]), we study the dependence of eigenvalues of two kinds of fourth order boundary value problems and a kind of higher order boundary value problems on the boundary. We give specific form of the equations for derivative of the nth eigenvalue as a function of an endpoint, and proved that as the length of the interval shrinks to zero all eigenvalues march off to plus infinity for all boundary conditions considered in this paper. In addition, we give some examples.Finally, we study the matrix representations of fourth order boundary value problems with periodic boundary conditions, and consider its inverse process, i.e. the representations of fourth-order boundary value problems of matrix eigenvalue problem.This paper contains six parts.1. The background and main results in this paper;2. The associated fundamental definitions and important lemmas;3. Characterization of real-parameter solutions of self-adjoint do-mains for the product of differential expressions;4. The self-adjointness of product of two singular differential operators;5. The dependence of eigen-values of differential equation boundary value problems on the boundary;6. Matrix representations of fourth order boundary value problems with periodic boundary conditions.