Theorem: Orthogonality of Solutions to the Sturm-Liouville Equation with Distinct Eigenvalues

Let $\map f x$ and $\map g x$ be solutions of the Sturm-Liouville equation:
:$(1): \quad -\map {\dfrac \d {\d x} } {\map p x \dfrac {\d y} {\d x} } + \map q x y = \lambda \map w x y$
where $y$ is a function of the free variable $x$.
The functions $\map p x$, $\map q x$ and $\map w x$ are specified.
In the simplest cases they are continuous on the closed interval $\closedint a b$.
In addition:
:$(1a): \quad \map p x > 0$ has a continuous derivative
:$(1b): \quad \map w x > 0$
:$(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$.
Assume that the Sturm-Liouville problem is regular, that is, $\map p x^{-1} > 0$, $\map q x$, and $\map w x > 0$ are real-valued integrable functions over the closed interval $\closedint a b$, with ''separated boundary conditions'' of the form: 
:$(2): \quad \map y a \cos \alpha - \map p a \map {y'} a \sin \alpha = 0$
:$(3): \quad \map y b \cos \beta - \map p b \map {y'} b \sin \beta = 0$
where $\alpha, \beta \in \hointr 0 \pi$.
Then:
:$\ds \innerprod f g = \int_a^b \overline {\map f x} \map q x \map w x \rd x = 0$
where $\map f x$ and $\map g x$ are solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues and $\map w x$ is the "weight" or "density" function.

Proof:
Multiply the equation for $\map g x$ by $\overline {\map f x}$ (the complex conjugate of $\map f x$) to get:
:$-\overline {\map f x} \dfrac {\map \d {\map p x \map {\dfrac {\d g} {\d x} } x} } {\d x} + \overline {\map f x} \map q x \map g x = \mu \overline {\map f x} \map w x \map g x$
Only $\map f x$, $\map g x$, $\lambda$ and $\mu $ may be complex.
All other quantities are real.
Complex conjugate this equation, exchange $\map f x$ and $\map g x$, and subtract the new equation from the original:
{{begin-eqn}}
{{eqn | l = -\overline {\map f x} \frac {\map \d {\map p x \map {\frac {\d g} {\d x} } x} } {\d x} + \map g x \frac {\map \d {\map p x \map {\frac {\d \bar f} {\d x} } x} } {\d x}
      | r = \frac {\map \d {\map p x \paren {\map g x \map {\frac {\d \bar f} {\d x} } x - \overline {\map f x} \map {\frac {\d g} {\d x} } x} } } {\d x}
}}
{{eqn | r = \paren {\mu - \bar \lambda} \overline {\map f x} \map g x \map w x
}}
{{end-eqn}}
Integrate this between the limits $x = a$ and $x = b$:
:$\ds \paren {\mu - \bar \lambda} \int_a^b \overline {\map f x} \map g x \map w x \rd x = \map p b \paren {\map g b \map {\frac {\d \bar f} {\d x} } b - \overline {\map f b} \map {\frac {\d g} {\d x} } b} - \map p a \paren {\map g a \map {\frac {\d \bar f} {\d x} } a - \overline {\map f a} \map {\frac {\d g} {\d x} } a}$
The right side of this equation vanishes because of the boundary conditions, which are either:
:periodic boundary conditions, that is, that $\map f x$, $\map g x$, and their first derivatives (as well as $\map p x$) have the same values at $x = b$ as at $x = a$
or:
:that independently at $x = a$ and at $x = b$ either:
::the condition cited in equation $(2)$ or $(3)$ holds
:or:
::$\map p x = 0$.
So:
:$\ds \paren {\mu - \bar \lambda} \int_a^b \overline {\map f x} \map g x \map w x \rd x = 0$
If we set $f = g$, so that the integral surely is non-zero, then it follows that $\bar \lambda = \lambda$.
That is, the eigenvalues are real, making the differential operator in the Sturm-Liouville equation self-adjoint (hermitian).
So:
:$\ds \paren {\mu - \lambda} \int_a^b \overline {\map f x} \map g x \map w x \rd x = 0$
It follows that, if $f$ and $g$ have distinct eigenvalues, then they are orthogonal.
{{qed}}
Category:Sturm-Liouville Theory
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