Use integration by parts with a complex conjugate, so that the argument also covers complex-valued eigenfunctions. The coefficient functions are understood to be real. For a nonzero eigenfunction

u

λ \int_{α}^{β} ∣ u ∣^{2} d x = \int_{α}^{β} \overline{u} M u d x = [a \overline{u} u^{'}]_{α}^{β} - \int_{α}^{β} a ∣ u^{'} ∣^{2} d x + \int_{α}^{β} b ∣ u ∣^{2} d x = - \int_{α}^{β} a ∣ u^{'} ∣^{2} d x + \int_{α}^{β} b ∣ u ∣^{2} d x .

(1)

The denominator is strictly positive and the numerator is real. Consequently the eigenvalue satisfies

λ = \frac{\int _{α}^{β} ( b ∣ u ∣ ^{2} - a ∣ u ^{'} ∣ ^{2} ) d x}{\int _{α}^{β} ∣ u ∣ ^{2} d x} \in R .

(2)

This is the Rayleigh quotient for the sign convention used here. Positivity of

a

is unnecessary for this particular reality argument; real coefficients and disappearance of the endpoint term are what matter.

Solution

 0  0

The Lagrange identity for a Sturm-Liouville operator gives

\int_{α}^{β} (\overline{u} M v - \overline{M u} v) d x = [a (\overline{u} v^{'} - \overline{u^{'}} v)]_{α}^{β} .

(1)

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 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: