4.1 The Sturm–Liouville Problem
Definition 4.1.1 (Sturm–Liouville problem). A Sturm–Liouville problem on an interval \([a,b]\) is the differential equation \begin {equation} \frac {d}{dx}\left [p(x)\,\frac {dy}{dx}\right ] + \left [q(x) + \lambda \, w(x)\right ] y = 0, \label {eq:sl} \end {equation} together with boundary conditions at \(a\) and \(b\). Here \(p\), \(p'\), \(q\) and \(w\) are continuous on \([a,b]\), with \(p(x)>0\) and \(w(x)>0\) on the open interval. The values of \(\lambda \) for which a non-trivial solution exists are the eigenvalues, and the corresponding solutions the eigenfunctions.
The function \(w\) is called the weight, and the name is not a coincidence: it is exactly the weight against which the eigenfunctions will turn out to be orthogonal.
Definition 4.1.2 (Regular and singular problems). The problem is regular if \([a,b]\) is finite, \(p(x)>0\) and \(w(x)>0\) throughout the closed interval, and the boundary conditions are separated, \[\alpha _1 y(a) + \alpha _2 y'(a) = 0,\qquad \beta _1 y(b) + \beta _2 y'(b) = 0 .\] It is singular if the interval is infinite, or if \(p\) or \(w\) vanishes at an endpoint. In the singular case the boundary condition is usually replaced by the requirement that \(y\) remain bounded at the offending endpoint.
Note 4.1.3. Every classical family in Chapter 3 belongs to the singular case, and it is worth seeing why before the machinery starts. Legendre’s equation has \(p(x)=1-x^{2}\), which vanishes at both \(x=\pm 1\); Hermite’s interval is the whole line; Laguerre’s is a half-line and its \(p\) vanishes at the origin. In each case the boundary condition that selects the polynomial solutions is simply that the solution stay bounded — and it is that requirement, not any choice made by the analyst, which forces \(\lambda \) to take the discrete values \(n\) or \(n(n+1)\) and so produces a polynomial of degree \(n\) rather than an infinite series.
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