Inverse Scattering for the Schrödinger Equation

Abstract

In Chapter 8 we studied the properties of the Schriidinger equation

$$\psi \prime \prime (x) + [\lambda - q(x)]\psi (x) = 0$$

(10.1.1)

on a finite interval [0, b] and on the half-line [0, ∞). For the finite interval we assumed that q(x) was continuous; for the half line we made additional assumptions regarding integrability, e.g.

$$\begin{array}{*{20}{c}} {\int_{0}^{\infty } {|q(x)|dx = {{C}_{0}},} } & {\int_{0}^{\infty } {x|q(x)|dx = {{C}_{1}}.} } \\ \end{array}$$

(10.1.2)

We interpreted these integrals in the ordinary (Rieman) sense.

Article Note

: my endeavours

Have ever come short of my desires

Yet fil’d with my abilities

Henry VIII. Act III, Scene 2.