On The Symmetry Of Higher-Order Quasi-Differential Operator Products

The theory of ordinary differential operators is a comprehensive branch of mathematics that combines with the theories and methods of ordinary differential equations,functional analysis,space theory,and operator algebras.It originated in the early 19th century from the mathematical modeling of heat transfer in solids and the study of various classical boundary value problems of mathematical physics.The theory of differential operators provides a unified theoretical framework and important mathematical tools for solving many applied problems in quantum mechanics,mathematical physics equations,and other engineering and technical fields.Its research content mainly includes spectral analysis of differential operators,self-adjoint extensions,symmetric extensions,the theory of deficiency indices,completeness of characteristic functions,and inverse spectral problems,etc.The characterization of boundary conditions for ordinary differential operators is an important and fundamental research topic.Currently,there have been many research results on the characterization of self-adjoint boundary conditions,but the characterization of symmetric boundary conditions is extremely rare.This paper will study the symmetry of the product of two high-order quasi-differential operators,and give what boundary conditions can be chosen to determine the symmetric product operators.The paper is divided into four parts:(1)Briefly introduce the research significance and development of differential operators;present the problems that will be investigated in this essay;recall some basic knowledge,symbols,and related lemmas needed in the proofs.(2)Study the symmetry of the product of two regular higher-order quasi-differential operators.(3)Characterize the symmetry of the product of two singular higher-order quasi-differential operators.(4)Summarize and prospect the research work.