Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite/Lemma 1

Theorem
If the function $ h \left ( { x } \right ) $ satisfies the equation


 * $ \displaystyle -\frac{ \mathrm d }{ \mathrm d x} \left ( { Ph' } \right ) + Qh = 0$

and the boundary conditions


 * $ h \left ( { a } \right ) = h \left ( { b } \right ) = 0 $

then


 * $ \displaystyle \int_a^b \left ( { P h'^2 + Q h^2 } \right ) \mathrm d x = 0$