Limit-point and limit-circle

Remarks

The definition does not depend on $c$. Weyl's alternative says that if every solution is square integrable near $b$ for one $\lambda \in \C$, then this holds for every $\lambda \in \C$. For $\lambda \in \C \setminus \R$ there is always at least one nonzero solution that is square integrable near $b$. So $b$ is limit-point exactly when, for such $\lambda$, this solution is unique up to a constant factor. Example: for $p = w = 1$, $q = 0$ on $(0, \infty)$ and $\lambda = 0$, the solutions are $y = c_1 + c_2 x$, and $y = 1$ is not square integrable near $\infty$. By Weyl's alternative, $\infty$ is in the limit-point case. Non-example of limit-point: a regular endpoint, such as $b < \infty$ with $1/p$, $q$, $w$ integrable near $b$, is always in the limit-circle case, since every solution then extends continuously to $b$. For $p = w = 1$ and $q(x) = \kappa / x^2$ on $(0, 1]$, the endpoint $0$ is in the limit-point case exactly when $\kappa \ge 3/4$. If $p = w = 1$ and $q$ is bounded below near $b = \infty$, then $\infty$ is in the limit-point case.