Characterization Of Self-adjoint Domains For Two-Interval Odd Order Differential Operators

In this paper,first of all,we give a complete characterization of all self-adjoint do-mains of odd order differential operators on two intervals.These two intervals with all four endpoints are singular(one endpoint of each interval is singular or all four endpoints are regulars are the special cases).And these extensions yield 'new' self-adjoint operators,which involve interactions between the two intervals.These interactions are the inter-actions between singular endpoints involve jump discontinuities of the lagrange bracket of solutions when the four endpoints are singular in two intervals.Interactions maybe'though' regular or singular endpoints when the one endpoint of each interval is singular and other endpoints are regular.That is too say,there are regular self-adjoint interactions and singular interactions.Furthermore,we study all self-adjoint two-interval realizations of the two equations of odd order using Hilbert spaces but with the new inner products.We give a characterization of all self-adjoint extensions of two-interval minimal operator with all four endpoints are singular.And the self-adjoint realizations in direct sum Hilbert spaces can enlarged by using inner product multiples.