The bookUnit 3 · Ordinary Differential Equations56 / 83

Sturm–Liouville problems and Green's functions

Why this is asked: Eigenvalues of a regular Sturm–Liouville problem are real, simple and unbounded above; eigenfunctions for distinct eigenvalues are orthogonal with respect to the weight. Green's function is built from the two boundary solutions.

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Sturm–Liouville: what self-adjointness buysinteractive

Real eigenvalues, orthogonal eigenfunctions and simplicity — from one integration by parts.

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The trap here

“Every boundary value problem has a Green's function” — false

$y'' + \pi^{2}y = f$ on [0,1] with y(0) = y(1) = 0

$\lambda = \pi^{2}$ is an eigenvalue of the homogeneous problem, so the operator is not invertible and no Green's function exists.