4.3 Orthogonality: the Central Theorem
Everything now follows from a single identity.
Lemma 4.3.1 (Lagrange’s identity). For any twice-differentiable \(u\) and \(v\), writing \(\mathcal {L}y = \left (p\,y'\right )' + q\,y\), \[u\,\mathcal {L}v - v\,\mathcal {L}u = \frac {d}{dx}\left [p\left (u\,v' - v\,u'\right )\right ].\]
Proof. Expanding, \[u\left (pv'\right )' - v\left (pu'\right )' = u\left (pv'\right )' + u'pv' - v\left (pu'\right )' - v'pu' = \frac {d}{dx}\left [u\,p\,v'\right ]-\frac {d}{dx}\left [v\,p\,u'\right ],\] since the added and subtracted terms cancel. The \(q\) terms cancel identically. □
Theorem 4.3.2 (Orthogonality of eigenfunctions). Let \(y_m\) and \(y_n\) be eigenfunctions of the Sturm–Liouville problem (4.1) corresponding to distinct eigenvalues \(\lambda _m\neq \lambda _n\), and suppose the boundary conditions are such that \begin {equation} \Big [p(x)\left (y_m\,y_n' - y_n\,y_m'\right )\Big ]^{b}_{a} = 0 . \label {eq:bc} \end {equation} Then \[\int ^{b}_{a} y_m(x)\,y_n(x)\,w(x)\,dx = 0 .\]
Proof. By definition \(\mathcal {L}y_m = -\lambda _m w\,y_m\) and \(\mathcal {L}y_n = -\lambda _n w\,y_n\). Multiply the first by \(y_n\), the second by \(y_m\), and subtract: \[y_n\mathcal {L}y_m - y_m\mathcal {L}y_n = \left (\lambda _n-\lambda _m\right )w\,y_m\,y_n .\] Integrating over \([a,b]\) and applying Lemma 4.3.1 to the left-hand side gives \[\Big [p\left (y_n\,y_m'-y_m\,y_n'\right )\Big ]^{b}_{a} = \left (\lambda _n-\lambda _m\right )\int ^{b}_{a}y_m\,y_n\,w\,dx .\] The left-hand side vanishes by (4.2), and since \(\lambda _n\neq \lambda _m\) the integral must be zero. □
This one theorem contains every orthogonality proof of Chapter 3. Each of those proofs was a separate calculation using a Rodrigues formula and repeated integration by parts; each is an instance of the argument just given, with the boundary term (4.2) vanishing for its own reason.
Theorem 4.3.3 (Reality of the spectrum). Under the same conditions, every eigenvalue of a Sturm–Liouville problem is real, and eigenfunctions may be taken real-valued.
Proof. Suppose \(\mathcal {L}y = -\lambda w y\) with \(\lambda \) possibly complex. Taking complex conjugates and using that \(p,q,w\) are real gives \(\mathcal {L}\overline {y} = -\overline {\lambda }\,w\,\overline {y}\), so \(\overline {y}\) is an eigenfunction with eigenvalue \(\overline {\lambda }\). Applying the computation of Theorem 4.3.2 to the pair \(y,\overline {y}\), \[\left (\overline {\lambda }-\lambda \right )\int ^{b}_{a}\left |y\right |^{2}w\,dx = 0 .\] Since \(w>0\) and \(y\not \equiv 0\), the integral is strictly positive, forcing \(\overline {\lambda }=\lambda \). □
Note 4.3.4. Theorem 4.3.3 is why the eigenvalues in the table below are the familiar integers and integer combinations rather than anything more exotic. It is also the property that makes Sturm–Liouville problems the right setting for the physical problems they arise from: an eigenvalue is a squared frequency or an energy, and those are real quantities.
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