= Compact cosine trial bound for the quantum harmonic oscillator
{title2=$\lambda_0\leq\sqrt{(\pi^2-6)/3}$}
For the dimensionless <quantum harmonic oscillator> $H=-d^2/dx^2+x^2$, use the compactly supported trial $\psi_\gamma=\cos(\gamma x)$ on $|x|\leq\pi/(2\gamma)$ and zero elsewhere, with $\gamma>0$. It is continuous and belongs to the oscillator energy <quadratic form> domain: its <weak derivative> is square-integrable even though its ordinary derivative jumps at the endpoints. Exact <integration> gives
$$
R(\gamma)=\frac{\int(|\psi_\gamma'|^2+x^2|\psi_\gamma|^2)dx}{\int|\psi_\gamma|^2dx}=\gamma^2+\frac{\pi^2-6}{12\gamma^2}.
$$
The minimum occurs at $\gamma^4=(\pi^2-6)/12$, giving the displayed upper bound, approximately $1.13572$, compared with exact <ground state> value $1$. This illustrates why <Rayleigh-Ritz variational principle> trials need the energy form domain rather than the smaller <operator domain>.
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