9.1 Green’s formula and self-adjointness
Lemma 9.1.1 (Lagrange identity and Green’s formula). For twice continuously differentiable \(u\) and \(v\), \[ uLv - vLu = -\bigl (p(uv' - vu')\bigr )', \qquad \int _a^b(uLv - vLu)\,dt = -\Bigl [p\,(uv' - vu')\Bigr ]_a^b . \] If \(u\) and \(v\) both satisfy the boundary conditions, the boundary term vanishes, and \(\langle u, Lv\rangle = \langle Lu, v\rangle \), where \(\langle f, g\rangle = \int _a^bfg\,dt\).
Proof. \(uLv - vLu = -u(pv')' + v(pu')' = -\bigl (upv' - vpu'\bigr )'\), since the terms \(u'pv'\) cancel; the \(q\) terms cancel directly. Integrate. If \(\alpha _1u(a) + \alpha _2u'(a) = 0\) and \(\alpha _1v(a) + \alpha _2v'(a) = 0\) with \((\alpha _1, \alpha _2) \neq 0\), the vectors \((u(a), u'(a))\) and \((v(a), v'(a))\) are both orthogonal to \((\alpha _1, \alpha _2)\) in the plane, hence parallel, and \(u(a)v'(a) - v(a)u'(a) = 0\); the same holds at \(b\). □
The operator \(L\) with these boundary conditions is therefore symmetric, the analogue in infinite dimensions of a symmetric matrix, and the theory of this section is the analogue of the spectral theorem for symmetric matrices. Every second-order operator \(a_2y'' + a_1y' + a_0y\) with \(a_2 \neq 0\) can be put in this form by multiplying by \(\exp \int (a_1/a_2)/a_2\), so the restriction is only apparent.
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