On power series representing solutions of the one-dimensional time-independent Schrödinger equation

Abstract

For the equation χ″(x) = u(x)χ(x) with infinitely smooth u(x), the general solution χ(x) is found in the form of a power series. The coefficients of the series are expressed via all derivatives u^(m)(y) of the function u(x) at a fixed point y. Examples of solutions for particular functions u(x) are considered.