= Exponential ground state of a linearly weighted Sturm-Liouville problem
{title2=$-(xy')'+\omega^2xy=\lambda y,\quad\lambda_0=\omega,\quad y_0=e^{-\omega x}$}
For $\omega>0$ on $x>0$, use the regular finite branch at zero, decay at infinity, and a finite quadratic-form energy. For trial functions with $[xy^2]_0^\infty=0$, <integration by parts> gives
$$
\int_0^\infty x(y'^2+\omega^2y^2)dx=\omega\int_0^\infty y^2dx+\int_0^\infty x(y'+\omega y)^2dx.
$$
The <Rayleigh quotient> is at least $\omega$, with equality exactly for a nonzero multiple of $e^{-\omega x}$. Direct substitution verifies that this is an <eigenfunction> with <eigenvalue> $\omega$, so the lower bound is attained. The zero coefficient $p(0)=0$ makes this a singular-endpoint <Sturm-Liouville problem>; regularity replaces a prescribed zero endpoint value. Continuous piecewise differentiable trials of finite form energy are allowed in the <Rayleigh-Ritz method>, even if their derivative has a jump away from the endpoint.
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