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 [1] corresponding to eigenvalues $\lambda$ and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } respectively. Multiply the equation for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\left( x\right) } by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{f} \left( x\right) } (the complex conjugate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left( x\right) } ) to get:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left( x\right) } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\left( x\right) } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lambda } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu } may be complex; all other quantities are real.) Complex conjugate this equation, exchange Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left( x\right) } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\left( x\right) } , and subtract the new equation from the original:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=b}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bullet } periodic boundary conditions, i.e., that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f\left( x\right) } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g\left( x\right) } , and their first derivatives (as well as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\left( x\right) } ) have the same values at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=b} as at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a} , or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bullet } that independently at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=a} and at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=b} either:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bullet } the condition cited in equation [2] or [3] holds or:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bullet } Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p\left( x\right) =0} .

So: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=g} , so that the integral surely is non-zero, then it follows that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{\lambda} =\lambda } ; that is, the eigenvalues are real, making the differential operator in the Sturm-Liouville equation self-adjoint (hermitian); so:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g} have distinct eigenvalues, then they are orthogonal. QED.

See also

• Sturm–Liouville theory
• Normal mode
• Self-adjoint

References

1. Ruel V. Churchill, "Fourier Series and Boundary Value Problems", pp. 70-72, (1963) McGraw-Hill, ISBN 0-07-010841-2.