The Symplectic Geometric Characteri- Zation Of The Self-adjoimt Domains Of Two Interval Secomd Oder Differential Operator

In this paper, we describe the self-adjoint domains of the two interval of 2-nd order lifferential operator in terms of symplectic geometry.The differential operators are essentially the unbounded closable operators in the inear operator. Its research areas include the deficiency index of the differential opera-or, self adjoint extension, spectrum analysis and many important branch. The choices of the domains of the differential operators are always the very important. Given lifferential expression, the specific demands for the differential operators eventually eflect on the restrictions on the domains.In 1986, Everitt W. N. and Zettl A. proposed Two-Interval Theory, and given two rder differential operator self adjoint domain description. In 2012, Suo Jian qing given in complete characterization of all self-adjoint domains of differential operators on two ntervals in the direct sum of Hilbert spaces in terms of real-parameter solutions. In 1999, Everitt W. N. and Zettl A. considered the self adjoint of the the differential >perator in term to symplectic geometry and given the conclusion which there is a one to one correspondence between the set of self-adjoint extensions of the minimal perator Mmin and the set of complex symplectic Lagrangian subspaces.We use the complete subspaces of the symplectic spaces to describe two order ;elf adjoint differential operators domain. Constructing different symplectic space on he maximal operator domain, given the algebraic structure of self adjoint boundary conditions. Two order symmetric differential operators according to the deficiency ndex is divided into the limit point case and limit circle case, limit point condition of he differential operators is equal to deficiency index (1,1), limit circle condition of the lifferential operators is equal to deficiency index (2,2). And the deficiency of the two nterval differential is from one to four, according to discussing the self-adjoint domains the two interval of 2-nd order differential operator in terms of symplectic geometry.Firstly, u={u1,u2},υ={υ1,υ2} be real-valued maximal domain functions such hat uj,υj(j= 1,2) are linearly independent modulo Djmin. According to the defi-ciency index of choice, giving the symplectic geometry on the two interval of two order ifferential operator.Secondly, the real-parameter characterization of two interval of two order differen-tial operator the self-adjoint extensions in terms of symplectic geometry. In the limit point case and the limit circle case were given symplectic geometry description and classification of self adjoint boundary conditions.