Chapter 12.22, Problem 26P

Solve the following eigenvalue problem (see end of Section 2 and Problem 11): Given the differential equation $y^{\prime \prime }+\left(\frac{\lambda }{x}-\frac{1}{4}-\frac{l(l+1)}{x^2}\right)y=0$ where $l$ is an integer $\ge 0,$ find values of $\lambda$ such that $y\to 0$ as $x\to \infty ,$ and find the corresponding eigenfunctions. Hint: let $y=x^{l+1}{e}^{-x/2}v(x),$ and whow that $v(x)$ satisfies the differential equation $x{v}^{\prime \prime }+(2l+2-x)v^{\prime }+(\lambda -l-1)v=0$.

Compare (22.26) to show that if $\lambda$ is an integer $>l$, there is a polynomial solution $v(x)=L_{\lambda -t-1}^{2t+1}(x)$.