The Schr (?) Dinger Operator Eigenvalue Spacing Of The First Two Estimated Transfer Boundary Conditions Differential Operator Of Sex-linked

This paper is composed of two mutual independent part.In this paper , the first part gives the optimal lower bound: G(C) =inf G(v) = 1 for the first two eigenvalues of one-dimensional Schrodinger operatorH|(v∈SW) = -D~2 + V(x) acting on L~2[0,π] with Neumann boundary conditions where SW issingle-well potential , and C is constant function in SW . This result generalizedthe result of Lavine , removed the convex condition for the potential of the later.Another result is , when the potential is a constant function , we give the estimate of uper and lower bound for the first two eigenvalues of one-dimensional Schrodinger operator H = —D~2 + V(x) acting on L~2[0,π] with general separate-type of boundary conditions .In the second part of this paper , we proved the self-adjointness of a class of differential operators with transfering boundary conditions in a new space , which is a generalization of the result of Ai-ping zhang and Jiong Sun .