On a general solution of the one-dimensional stationary Schrödinger equation

The general solution of the one-dimensional stationary Schrödinger equation in the form of a formal power series is considered. Its efficiency for numerical analysis of initial value and boundary value problems is discussed. Consider the equation (pu) + qu = ωu (1) where we suppose that p, q and u are complex-valued functions of an independent real variable x ∈ [0, a], ω is an arbitrary complex number. p and q are supposed to be such that there exists a solution g0 of the equation (pg 0) ′ + qg0 = 0 on (0, a) such that g0 ∈ C(0, a) together with 1/g0 are bounded on [0, a] and p ∈ C(0, a) is a bounded nonvanishing function on [0, a]. Denote g = √ pg0. In [3] with the aid of pseudoanalytic function theory [2] the following result was obtained. Theorem 1 The general solution of (1) has the form u = c1u1 + c2u2 (2)