Necessary and Sufficient Condition for Quadratic Functional to be Positive Definite

Theorem
The quadratic functional


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

where


 * $ \displaystyle P \left ( { x } \right ) > 0 \quad \forall x \in \left [ { a \,. \,. \, b } \right ]$

is positive definite for all $ h \left ( { x } \right )$:


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

iff the interval $ \left [ { a \,. \,. \, b } \right ] $ contains no points conjugate to $ a $.

Necessary condition
Let there be $ \omega \left ( { x } \right ) $ :


 * $ \displaystyle \omega \left ( { x } \right ) \in C^1 \left [ { a \,. \,. \,b } \right] $.

Then


 * $ \displaystyle 0 = \int _a^b \frac{ \mathrm d}{ \mathrm d x} \left ( { \omega h^2 } \right ) \mathrm d x$

Let $ \omega $ be a solution to the equation $ \displaystyle P \left ( { Q + \omega' } \right ) = \omega^2$

Then

Suppose


 * $ \displaystyle h' +\frac{ \omega h }{ P }=0 $

Then


 * $ h \left ( { a } \right ) = 0 \implies h \left ( { x } \right ) = 0 \quad \forall x \in  \left [ { a \,. \,. \,b } \right] $

due to Existence-Uniqueness Theorem for First-Order Differential Equation.

This implies an infinite number of conjugate points.

Conditions of the theorem do not allow any conjugate points.

Hence


 * $ h \left ( { x } \right ) \ne 0 \quad \forall x \in \left ( { a \,. \,. \,b } \right ) $

and


 * $ \displaystyle P \left ( {h' +\frac{ \omega h }{ P } } \right )^2 > 0$

Sufficient condition
Consider the functional


 * $ \displaystyle \int_a^b \left [ { t \left ( { Ph^2 + Q h'^2 } \right ) + \left ( { 1 - t  } \right ) h'^2 } \right ] \mathrm d x \quad \forall t \in \left [ { 0 \,. \,. \, 1 } \right ] $