Asymptotics of the eigenvalues for the operators with transition points and Neumann conditions

In this paper we consider the following differential equation LW?(d^2 W)/?d??^2 - (?(?)-u^2 (1-?^2 ) ) W=0, ??[a,b] (1) Where ??[a,b],a=-b > 1, u is a large parameter and ?(?) is a continuous function on [a,b]. For equation (1), ?=±1 are transition points and r(?)=1-?^2 be a weight function. Using the asymptotic solution constructed by Olver in [3] and [5], we study the asymptotic behavior of the eigenvalues of the operatorL with Neumann boundary conditions W^ʹ (a)=W^ʹ (b)=0.