Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+a)^2+a$ with parameter $a>0$. It has a regular singularity $x=0$. We basically hope for something like homogeneous Dirichlet b.c.

It is solved by making the substitution $y(x)=e^{x/2}x^{-\frac{1}{2}+\sqrt{(a+\frac{1}{2})^2-\lambda^2}}u(x)$, leading to Kummer's equation with two independent solutions (1st & 2nd kind) $$xu''(x)+(\gamma-x)u'(x)-\alpha u(x)=0,$$ in which $\alpha=\sqrt{(a+\frac{1}{2})^2-\lambda^2}-a+\frac{1}{2}$ and $\gamma=1+2\sqrt{(a+\frac{1}{2})^2-\lambda^2}$.

Let's then follow the ubiquitous argument.

Requiring nondivergence at $x=0$, the 2nd kind solution is dropped. Requiring nondivergence at infinity, the 1st kind is reduced to a polynomial when $-\alpha$ is a non-negative integer and eigenvalue $\lambda^2$ is attained.
Seen from this condition for $\alpha$, *we only have a bounded and finite sequence of eigenvalues*. Therefore, one probably also expects an additional continuous spectrum. But I'm not sure whether it starts from the largest eigenvalue.

## Question

Is it possible to know the limit circle/point classification of this ODE near $0,\pm\infty$? Is the above solution useful for this purpose? How should one proceed? I also found page 13 of this paper has a limit point/circle classification of the Kummer equation I used. Not sure if relevant.