A Turning Point Problem Arising in Connection with a Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann—Wigner Potentials

Abstract

In connection with our study of “A Limiting Absorption Principle for Schrödinger Operators with Generalized Von Neumann—Wigner Potentials” [7,8] we needed to solve a turning point problem for a family of ordinary differential equations. This family arose by separation of variables from the partial differential equation (−Δ+p0 − λ)f = 0, where Δ is the Laplacian, p0 is a spherically symmetric potential function given by definition (2.1) and λ > 0 is the spectral parameter. Following the notation of [7, 8] we parametrize this family by j= 0,1,2,.... It is a key feature of this family of differential equations that for each j the corresponding equation has at least one turning point and that as j tends to infinity so do the turning points. Hence our family of operators is similar to the one of Langer [6,9]. It is different from the one of Langer inasmuch as we do not need a uniform expansion for the solution which is normalized near zero. All that we need is a lower estimate for the absolute values of the solutions at a family of points σ^(j) which satisfy the inequality of conclusion (2.15) of Theorem Theorem 2.1. Hence these points are to the left of the turning points. On account of these weaker conclusions, we can allow to add an oscillatory potential to the one of Langer.