Deficiency Indices Of Higher Order Singular Differential Operator With A Class Of Equations Appropriate Qualitative Schr (?) Dinger Operator Of The First Eigenvalue Estimates

This paper is composed of two mutual independent part.The first part of this paper gives a sufficient and necessary condition of the relationship between the deficiency indices d(M) of even order(2n) singular real symmetric regular ordinary differential operator M and the well-posedness of a kind of ordinary differential equations P_m with original value conditions ,that. is d(M) ≤ m is equivalent to the fact that the solution of the problem P_m is existent and unique,where n ≤ m ≤ 2n and the equality holds if and only if M is lim-n.As the second part of this paper,we give the optimal estimate of lower bound λ₁ > π²/d² for the first eigenvalue of the Schrodinger operator H = -△ + W(x) acting on L²(Ω) with Drichlet boundary conditions where Ω is a bounded smooth convex subset of Rⁿ and W is a nonnegative potential. Meanwhile,we give the approximate values of the λ₁ of the conresponding physical potential functions with Rayleigh method.