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

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This article proves that solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues are orthogonal.

Note that when the Sturm-Liouville problem is regular, distinct eigenvalues are guaranteed. For background see Sturm-Liouville Theory.


Theorem

Let $f \left({x}\right)$ and $g \left({x}\right)$ be solutions of the Sturm-Liouville equation:

$(1): \quad -\dfrac {\mathrm d} {\mathrm d x} \left({p \left({x}\right) \dfrac {\mathrm d y} {\mathrm d x}}\right) + q \left({x}\right) y = \lambda w \left({x}\right) y$

where $y$ is a function of the free variable $x$.


The functions $p \left({x}\right)$, $q \left({x}\right)$ and $w \left({x}\right)$ are specified.

In the simplest cases they are continuous on the closed interval $\left[{a \,.\,.\, b}\right]$.


In addition:

$(1a): \quad p \left({x}\right) > 0$ has a continuous derivative
$(1b): \quad w \left({x}\right) > 0$
$(1c): \quad y$ is typically required to satisfy some boundary conditions at $a$ and $b$.


Assume that the Sturm-Liouville problem is regular, that is, $p \left({x}\right)^{-1} > 0$, $q \left({x}\right)$, and $w \left({x}\right) > 0$ are real-valued integrable functions over the closed interval $\left[{a \,.\,.\, b}\right]$, with separated boundary conditions of the form:

$(2): \quad y \left({a}\right) \cos \alpha - p \left({a}\right) y' \left({a}\right)\sin \alpha = 0$
$(3): \quad y \left({b}\right) \cos \beta - p \left({b}\right) y' \left({b}\right)\sin \beta = 0$

where $\alpha, \beta \in \left[{0 \,.\,.\, \pi}\right)$.


Then:

$\displaystyle \left\langle{f, g}\right\rangle = \int_a^b \overline {f \left({x}\right)} q \left({x}\right) w \left({x}\right) \, \mathrm d x = 0$

where $f \left({x}\right)$ and $g \left({x}\right)$ are solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues and $w \left({x}\right)$ is the "weight" or "density" function.


Proof

Multiply the equation for $g \left({x}\right)$ by $\overline {f \left({x}\right)}$ (the complex conjugate of $f \left({x}\right)$) to get:

$-\overline {f \left({x}\right)} \dfrac {\mathrm d \left({p\left({x}\right) \dfrac {\mathrm d g} {\mathrm d x} \left({x}\right)}\right) } {\mathrm d x} + \overline {f \left({x}\right)} q \left({x}\right) g \left({x}\right) = \mu \overline {f \left({x}\right)} w \left({x}\right) g \left({x}\right)$

Only $f \left({x}\right)$, $g \left({x}\right)$, $\lambda$ and $\mu $ may be complex.

All other quantities are real.

Complex conjugate this equation, exchange $f \left({x}\right)$ and $g \left({x}\right)$, and subtract the new equation from the original:

\(\displaystyle -\overline {f \left({x}\right)} \frac {\mathrm d \left({p \left({x}\right) \frac {\mathrm d g} {\mathrm d x} \left({x}\right) }\right) } {\mathrm d x} + g \left({x}\right) \frac {\mathrm d \left({p \left({x}\right) \frac {\mathrm d \bar f} {\mathrm d x} \left({x}\right) }\right) } {\mathrm d x}\) \(=\) \(\displaystyle \frac {\mathrm d \left({p \left({x}\right) \left({g \left({x}\right) \frac {\mathrm d \bar f} {\mathrm d x} \left({x}\right) - \overline {f \left({x}\right)} \frac {\mathrm d g} {\mathrm d x} \left({x}\right) }\right) }\right) }{\mathrm d x}\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({\mu - \bar \lambda}\right) \overline {f \left({x}\right)} g \left({x}\right) w \left({x}\right)\) $\quad$ $\quad$

Integrate this between the limits $x = a$ and $x = b$:


$\displaystyle \left({\mu - \bar \lambda}\right) \int_a^b \overline {f \left({x}\right)} g \left({x}\right) w \left({x}\right) \, \mathrm d x = p \left({b}\right) \left({g \left({b}\right) \frac {\mathrm d \bar f} {\mathrm d x} \left({b}\right) - \overline {f \left({b}\right)} \frac {\mathrm d g} {\mathrm d x} \left({b}\right)} \right) - p \left({a}\right) \left({g \left({a}\right) \frac {\mathrm d \bar f} {\mathrm d x} \left({a}\right) - \overline {f \left({a}\right)} \frac {\mathrm d g} {\mathrm d x} \left({a}\right)}\right)$


The right side of this equation vanishes because of the boundary conditions, which are either:

periodic boundary conditions, i.e., that $f \left({x}\right)$, $g \left({x}\right)$, and their first derivatives (as well as $p \left({x}\right)$) 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:
$p \left({x}\right) = 0$.


So:

$\displaystyle \left({\mu - \bar \lambda}\right) \int_a^b \overline {f \left({x}\right)} g \left({x}\right) w \left({x}\right) \, \mathrm d x = 0$

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:

$\displaystyle \left({\mu - \lambda}\right) \int_a^b \overline {f \left({x}\right)} g \left({x}\right) w \left({x}\right) \, \mathrm d x = 0$

It follows that, if $f$ and $g$ have distinct eigenvalues, then they are orthogonal.

$\blacksquare$


Sources