This article proves that solutions to the Sturm–Liouville equation corresponding to distinct eigenvalues are orthogonal. For background see Sturm–Liouville theory.
Theorem
, where
and
are solutions to the Sturm–Liouville equation corresponding to distinct eigenvalues and
is the "weight" or "density" function.
Proof
Let
and
be solutions of the Sturm-Liouville equation [1] corresponding to eigenvalues
and
respectively. Multiply the equation for
by
(the complex conjugate of
) to get:
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67bfbb7bffdcfe6e080e543e6c8086c741e5eb18)
(Only
,
,
, and
may be complex; all other quantities are real.) Complex conjugate
this equation, exchange
and
, and subtract the new equation from the original:
![{\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de362722465e914a457a1df7065729d824136d78)
Integrate this between the limits
and
![{\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].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/572c1ec0058b8a1e2182c346b25abc3275381a77)
The right side of this equation vanishes because of the boundary conditions, which are either:
periodic boundary conditions, i.e., that
,
, and their first derivatives (as well as
) have the same values at
as at
, or
that independently at
and at
either:
the condition cited in equation [2] or [3] holds or:
![{\displaystyle p\left(x\right)=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/150ebd9b5dc473ed277f31658033a398ba10a094)
So:
.
If we set
, so that the integral surely is non-zero, then it follows that
; that is, the eigenvalues are real, making the differential operator in the Sturm-Liouville equation self-adjoint (hermitian); so:
![{\displaystyle \left(\mu -\lambda \right)\int \nolimits _{a}^{b}{\bar {f}}\left(x\right)g\left(x\right)w\left(x\right)\,dx=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/220269913cd8e98d843b6417c097ef8469fc13c4)
It follows that, if
and
have distinct eigenvalues, then they are orthogonal. QED.
See also
References
1. Ruel V. Churchill, "Fourier Series and Boundary Value Problems", pp. 70–72, (1963) McGraw–Hill, ISBN 0-07-010841-2.