The Asymptotic Form of Eigenvalues for a Class of Sturm-Liouville Problem with One Simple Turning Point (Journal of Sciences, Islamic Republic of Iran)
The purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of Sturm-Liouville problem with equation of the form w??=(?2f(x)?R(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in C2(a,b which has a single zero (so called turning point) at point 0x=x and R(x) is a continuously differentiable function. We prove that, as a classical case, the asymptotic form of eigenvalues of (1) with periodic boundary condition w(a)=w(b), as well as with Semi-periodic boundary condition w?(a)=w?(b)w(a)=?w(b), are the same as Dirichlet boundary condition w?(a)=?w?(b)w(a)=0=w(b). We also study the asymptotic formula for the eigenvalues of (1) with boundary condition w?(a)=0=w(b), as well as w(a)=0=w?(b) and w?(a)=0=w(b).