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.
[edit] Orthogonality Theorem
, where f(x) and g(x) are solutions to the Sturm-Liouville equation corresponding to distinct eigenvalues and w(x) is the "weight" or "density" function.
[edit] Proof
Let f(x) and g(x) be solutions of the Sturm-Liouville equation (1) corresponding to eigenvalues λ and μ respectively. Multiply the equation for g(x) by f(x) (the complex conjugate of f(x)) to get:
(Only f(x), g(x), λ, and μ may be complex; all other quantities are real.) Complex conjugate this equation, exchange f(x) and g(x), and subtract the new equation from the original:
Integrate this between the limits
x = a
and
x = b
.
The right side of this equation vanishes because of the boundary conditions, which are either:
- periodic boundary conditions, i.e., that f(x), g(x), and their first derivatives (as well as p(x)) have the same values at x = b as at x = a, or
- that independently at x = a and at x = b either:
So:
If we set f = g , so that the integral surely is non-zero, then it follows that λ =λ that is, the eigenvalues are real, making the differential operator in the Sturm-Liouville equation self-adjoint (hermitian); so:
It follows that, if f and g have distinct eigenvalues, then they are orthogonal. QED.
Some content on this page may previously have appeared on Citizendium. |