= No-log boundary condition at a linearly vanishing Sturm-Liouville endpoint
{title2=$\lim_{x\downarrow a}p(x)y'(x)=0$}
Suppose $p(x)=c(x-a)+O((x-a)^2)$ with $c>0$, and the other coefficients in $(py')'+(\lambda w-q)y=0$ extend continuously to $a$, with $w(a)>0$. A bounded <eigenfunction> has $(py')'=O(1)$, hence $py'=C+O(x-a)$. A nonzero $C$ would give $y'\sim C/[c(x-a)]$ and an unbounded <logarithmic singularity>. Thus $C=0$, and the regular branch has bounded first <derivative>. The second local branch carries a logarithm. The title condition excludes that branch.
For two regular <eigenfunctions>, $p(\overline u v'-\overline{u'}v)\to0$. This removes the singular-end contribution to the <Lagrange identity for a Sturm-Liouville operator>. Together with a separated self-adjoint condition at the other endpoint, it selects the regular <self-adjoint differential operator> realization. The precise operator domain uses the maximal differential-operator domain and the no-log endpoint condition; boundedness of classical eigenfunctions is its convenient local expression, not a replacement for specifying an operator domain.
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