A-Equation and Its Connections to Nonlinear Integrable System

Yilian Zhang

doi:10.4236/jamp.2017.56110

Abstract

A novel approach to inverse spectral theory for Schr&oumldinger Equation operators on a half-line was first introduced by Barry Simon and actively studied in recent literatures. The remarkable discovery is a new object A-function and intergo-differential Equation (called A-Equation) it satisfies. Inverse problem of reconstructing potential is then directly connected to finding solutions of A-Equation. In this work, we present a large class of exact solutions to A-Equation and reveal the connection to a class of arbitrarily large systems of nonlinear ordinary differential Equations. This non-linear system turns out to be C-integrable in the sense of F. Calogero. Integration scheme is proposed and the approach is illustrated in several examples.

Highlights

IntroductionBarry Simon investigated a new approach to inverse spectral theory for the half-line Schrödinger operator,
Several years ago, Barry Simon investigated a new approach to inverse spectral theory for the half-line Schrödinger operator, − d2 dx2 + q(x) in L2 (0, ∞) in [3].A new A-function introduced in [1] [2] [3], is related to Weyl-Titchmarsh function by the following relation: ( ) m x, −κ 2 =−κ ∞
A novel approach to inverse spectral theory for Schrödinger Equation operators on a half-line was first introduced by Barry Simon and actively studied in recent literatures

Summary

Introduction

Barry Simon investigated a new approach to inverse spectral theory for the half-line Schrödinger operator,. The inverse problem to determine q from m, becomes a problem to solve the integro-differential Equation (2). To construct numerical solvers to this integro-differential equation, one needs to study sets of exact analytic solutions. ∑ A(α , x) = n ( ) f j x e−2αγ j(x) This ansatz is motivated by the explicit example in [1], where A(α , 0) is calculated for Bargmann potentials using inverse scattering theory (which is valid only under restrictive assumptions). ∑ c j = γ γ i1≤i2≤ ≤ij i1 i2 γ ij Via this “change of variable”, (5) yields a new nonlinear system: 2= c′j 2c1′c j−1 + c′j′−1 1 ≤ j ≤ n + 1. The latter is solved by finding a nonlinear analogue of the method of integrating factors (Theorem 13).

The γ Equation

Second Order Nonlinear System to First Order Nonlinear System

Method of Integrating Factor

Exact Analytic Examples

Conclusion