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.
, 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.
Let f(x) and g(x) be solutions of the Sturm-Liouville equation (1) corresponding to eigenvalues and respectively. Multiply the equation for g(x) by
f(x) (the complex conjugate of f(x)) to get:
may be complex; all other quantities are real.) Complex conjugate
this equation, exchange
g(x), and subtract the new equation from the original:
Integrate this between the limits
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 as at , or
- that independently at and at either:
- the condition cited in equation (2) or (3) holds or:
If we set
, 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
have distinct eigenvalues, then they are orthogonal. QED.