Sturm-Liouville theory/Proofs

This article proves that solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues are orthogonal. For background see Sturm–Liouville theory.

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 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 or  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.