= Normalized Sturm-Liouville spectral Green kernel
{title2=$G_\mu(x,\xi)=w(\xi)\sum_n\frac{y_n(x)y_n(\xi)^*}{\lambda_n-\mu}$}
Let $L=w^{-1}[-(d/dx)(p\,d/dx)+q]$ be the regular self-adjoint operator of a <Sturm-Liouville problem>, with <eigenfunctions> orthonormal for the <weighted inner product> $\int w\overline uv$. For $\mu$ outside its <spectrum>, the displayed kernel satisfies $(L_x-\mu)G_\mu=\delta(x-\xi)$ with an ordinary, unweighted <Dirac delta>. Indeed weighted completeness gives $w(\xi)\sum_ny_n(x)y_n(\xi)^*=\delta(x-\xi)$. Thus $u(x)=\int G_\mu(x,\xi)f(\xi)d\xi$ solves $(L-\mu)u=f$. A kernel integrated against $w(\xi)d\xi$ instead omits the prefactor. For real coefficients, the weighted reciprocity relation is $G_\mu(x,\xi)/w(\xi)=G_\mu(\xi,x)/w(x)$ when the spectral parameter is real and outside the spectrum. A zero eigenvalue prevents inversion at $\mu=0$ without a forcing compatibility condition or a projected inverse.
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