Chapter 12.22, Problem 11P

Show that we have solved the following eigenvalue problem (see Problem 5 and end of Section 2): Given the differential equation $y^{\prime \prime }+\left(ϵ-x^2\right)y=0$ [compare equation (22.1)]. find the possible values of $ϵ$ (eigenvalues) such that the solutions y(x)of the given differential equation tend to zero as $x\to \pm \infty ;$ for these values of $ϵ$, find the eigenfunctions $y(x).$ What is $ϵ,$ and what are the eigenfunctions?