= Solution
The <Lagrange identity for a Sturm-Liouville operator> gives
$$
\int_\alpha^\beta\bigl(\overline u\,Mv-\overline{Mu}\,v\bigr)\,dx
=\bigl[a(\overline u\,v'-\overline{u'}\,v)\bigr]_\alpha^\beta.
$$
In the intended class of <functions> whose values and first <derivatives> stay finite at the endpoints, the endpoint factor $a$ makes this entire boundary term vanish. Thus \b[no separately prescribed endpoint values are needed to eliminate the boundary form]. The same <integration by parts> argument then proves that the <eigenvalues> are real.
The admissible endpoint behaviour is part of this conclusion: it is the <vanishing endpoint coefficient and the Sturm-Liouville boundary form>, together with endpoint regularity, that replaces explicit <boundary conditions>. If singular <functions> are allowed, a vanishing coefficient alone need not suffice: $a(x)=x(1-x)$ and $v(x)=\log x$ have $a(x)v'(x)\to1$ as $x\downarrow0$. A <self-adjoint differential operator> therefore still needs a specified domain of admissible <functions>.
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