Study On Self-adjointness Of Some Differential Operators Product

Ordinary differential operators theory can supply the theory basis for differential equations, classical physics, modern physics and other technique fields, which is a comprehensive and marginal mathematics branch of ordinary differential equations, functional analysis, space theory, operators theory etc.. It investigates a great deal of important problems, such as spectral analysis, adjoint extension, deficiency index theory, completeness of eigenfunctions, inverse questions and so on.The study of ordinary differential operator originated from the solid heat transfer problem and variety of classical mathematics physics definite solutions in early 19th century. The self-adjointness of differential operator is a important part of differential operator theory and is interested in more mathematical researchers international. On self-adjointness of differential operator's product have obtained some results which focus on the same differential expression. In this paper, we get the self-adjointness of the product operators generated by different two differential expressions by operator theory and matrix calculation. First,we discuss the self-adjointness of product of operators generated by two different 4th-order differential expressions. Second, we investigate the self-adjointness of product of operators generated by one 2nd-order differential expression and another 4th-order differential expression.This paper contains four chapters.Chapter one is divided into two parts: in the first part, we give the simple summarize of adjointness of differential operator product; in the second part, we give the basic knowledge of symmetric differential operators.Chapter two contains two parts: in the first part, we discuss the self-adjointness of the product of operators generated by two different 4th-order differential expressions D⁴+ D²+qi (t)(i=1,2), ( D = d/dt,t∈I=[a,b]), obtain the sufficient and necessary conditions when product operator is self-adjoint. In the second part, we discuss the self-adjointness of the product of operators when coefficients are same.In Chapter three, we discuss the self-adjointness of the product of operator generated by symmetric differential expression D⁴+ D²+q₁(t) and D⁴+ q₂(t).In Chapter four, we investigates the self-adjointness of the product of operator generated by symmetric differential expression D ( 2)+ q₁(t) and D⁴+ q₂(t).