The J-Selfadjoint Extensions Characterization Of Two-Interval Forth-Order J-Symmetric Operators

Over the years,J-symmetric differential operators has been payed attention by many scholars.In particular,the boundry conditions,deficiency index and spectral analysis of J-symmetric differential operators have been widely applied in a large number of scientific research.This article mainly centre on the characterization of all J-selfadjoint extensions for two-interval forth-order J-symmetric differential operators.In the direct sum of Hilbert spaces,we will promote the theory of J-symmetric differential operators from one-interval to two-interval,and characterize all J-selfadjoint extensions for two-interval forth-order J-symmetric differential operators by means of fourth order differential equation.First,when the four endpoints of two-interval are regular points,we will give the characterization and demonstration of J-selfadjoint extension domains of forth-order J-symmetric differential operators.Meanwhile,we will discuss boundary conditions for the separation and coupling,while we will give specific examples.Second,when the four endpoints of two-interval include limit points,according to the number of limit points and being in terms of the difference of the deficiency index,we will give the characterization of J-selfadjoint extension domains of forth-order J-symmetric differential operators.And again,under the assumption that the regularity field of the minimal operator is non-empty,we will give the characterization of J-selfadjoint extension domains of forth-order J-symmetric differential operators on two-interval with one endpoint is limit circle of each interval or with the endpoints all are limit circle by the I.Knowles’ s theory.Finally,when the four endpoints of two-interval have middle deficiency index,in the case of singular we we will give the characterization of J-selfadjoint extension domains of forth-order J-symmetric differential operators.