Sturm-Liouville theory/Proofs

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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

, 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:

(Only f(x), g(x), , and 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 and


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 and have distinct eigenvalues, then they are orthogonal. QED.