Sturm-Liouville theory/Proofs

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.

Orthogonality Theorem

$\langle f,g\rangle =\int _{a}^{b}{\overline {f(x)}}g(x)w(x)\,dx\ =\ 0$ , 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

Let $f\left(x\right)$ and $g\left(x\right)$ be solutions of the Sturm-Liouville equation (1) corresponding to eigenvalues $\lambda$ and $\mu$ respectively. Multiply the equation for $g\left(x\right)$ by ${\bar {f}}\left(x\right)$ (the complex conjugate of $f\left(x\right)$) to get:

$-{\bar {f}}\left(x\right){\frac {d\left(p\left(x\right){\frac {dg}{dx}}\left(x\right)\right)}{dx}}+{\bar {f}}\left(x\right)q\left(x\right)g\left(x\right)=\mu {\bar {f}}\left(x\right)w\left(x\right)g\left(x\right)$ .

(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:

$-{\bar {f}}\left(x\right){\frac {d\left(p\left(x\right){\frac {dg}{dx}}\left(x\right)\right)}{dx}}+g\left(x\right){\frac {d\left(p\left(x\right){\frac {d{\bar {f}}}{dx}}\left(x\right)\right)}{dx}}={\frac {d\left(p\left(x\right)\left[g\left(x\right){\frac {d{\bar {f}}}{dx}}\left(x\right)-{\bar {f}}\left(x\right){\frac {dg}{dx}}\left(x\right)\right]\right)}{dx}}=\left(\mu -{\bar {\lambda }}\right){\bar {f}}\left(x\right)g\left(x\right)w\left(x\right).$ Integrate this between the limits $x=a$ and $x=b$

$\left(\mu -{\bar {\lambda }}\right)\int \nolimits _{a}^{b}{\bar {f}}\left(x\right)g\left(x\right)w\left(x\right)dx=p\left(b\right)\left[g\left(b\right){\frac {d{\bar {f}}}{dx}}\left(b\right)-{\bar {f}}\left(b\right){\frac {dg}{dx}}\left(b\right)\right]-p\left(a\right)\left[g\left(a\right){\frac {d{\bar {f}}}{dx}}\left(a\right)-{\bar {f}}\left(a\right){\frac {dg}{dx}}\left(a\right)\right]$ .

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

$\bullet$ 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

$\bullet$ that independently at $x=a$ and at $x=b$ either:

$\bullet$ the condition cited in equation (2) or (3) holds or:

$\bullet$ $p\left(x\right)=0$.

So: $\left(\mu -{\bar {\lambda }}\right)\int \nolimits _{a}^{b}{\bar {f}}\left(x\right)g\left(x\right)w\left(x\right)dx=0$.

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:

$\left(\mu -\lambda \right)\int \nolimits _{a}^{b}{\bar {f}}\left(x\right)g\left(x\right)w\left(x\right)dx=0$ .

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