New method to solve certain differential equations

Abstract

A new method to solve stationary one-dimensional Schroedinger equation is investigated. Solutions are described by means of representation of circles with multiple winding number. The results are demonstrated using the well-known analytical solutions of the Schroedinger equation. 1. Introduction One of methods to obtain the equation of a unit circle with winding number equal to n [1] is raising the equation Q20 + P02 = 1 (1) to the n-th power, where Q0, P0 ∈R. For instance, in the case n = 2, by appropriate grouping of terms we get 2 2 Q1 + P 1 = 1, (2) where 2 Q1 = (2Q20 −1)2, (3) 2 P 1 = 4Q20(1 −Q20). (4) The results are illustrated in Figure 1. Generally, a circle with winding number equal to n is represented by the sum of polynomials of 2(n + 1) degree and proper number of zeros. Moreover, it can be seen that in this representation the domain of Q0 does not change regardless of the winding number.