Characterization Of Symplectic Geometry And The Friedrichs Extension Of Some Differential Operators

Ordinary differential operators theory can supply differential equations,classical physics,modern physics and other technique fields the theory basis,which is a compositive and edging mathematics banch of ordinary differential equations,functional analysis,space theory,operators theory etc..It contains a great impoetant problems,such as deficiency index theory,adjoint extension,spectral analysis,press eigenfunction to spread,numerical method,inverse questions,and so on.In this paper, our attention is paid to the Friedrichs extension of some classes ordinary differential operators. First,making use of W.T.Reid's characterization of principal solutions for high order equation,we disscuss the Friedrichs extenion of powers and products of singular quasi-differential expression of order 2n ,and obtain the sufficienl and necessary condtions that boundary conditions satisfy and some other result ;Second,characterize the Friedrichs extension of the minimal operator of order 2n reqular differential operators in direct sum space by the basic theory of first-order systems of differential equation and their relationship to general 2n-th order quasi-differential expressions, and meanwhile,also give the new characterization for above result . Last, the Friedrichs extension for singular Sturm-Liouville operators by the basic theory of complex symplectic geometry.This paper contains four chapters.The first chpter:we give the background and advance of the Friedrichs extension and basic knowledge of symplectic geometry and so on.The second chapter: the Friedrichs extenion of powers and products of singular quasi-differential expression of order 2n. The third chapter: the Friedrichs extension of the minimal operator of order 2n reqular differential operators in direct sum space and it's new characterization by the basic theory of complex symplectic geometry. The fourth chapter:the new characterization of the Friedrichs extension for singular Sturm-Liouville operators in terms of symplectic geometry.