The Boundary Value Problem of Scattering Theory

Abstract

This chapter is devoted to a detailed study of the boundary value problem generated on the half line 0 ≤ x < ∞ by the differential equation

$$-y+q\left(x\right)y={\lambda^2}y$$

((3.1.1))

and the boundary condition

$$y\left(0\right)=0$$

((3.1.2))

for the important special case where the function q(x) is real and satisfies the condition

$$\int\limits_0^\infty{x\left|{q\left(x\right)}\right|dx}<\infty$$

((3.1.3))

which is assumed to hold throughout the chapter. From condition (3.1.3) it is clear that (3.1.1) reduces to the simpler equation -y″ = λ² y when x → ∞. This permits us a complete investigation of the properties of the solution to equation (3.1.1), and this is our goal in the present section.