= Polynomial Rayleigh-Ritz trial space for a one-dimensional box
{title2=$E_{\min}^{\rm trial}=(7-\sqrt{133}/2)\hbar^2/(ma^2)$}
For the <Dirichlet boundary conditions> on $(-a,a)$, take the two-dimensional even trial space spanned by $(a^2-x^2)$ and $x^2(a^2-x^2)$. Using the kinetic-energy <quadratic form> gives, for $b=\lambda a^2$,
$$
\frac{ma^2}{\hbar^2}R[(a^2-x^2)(1+\lambda x^2)]=\frac{3(11b^2+14b+35)}{4(b^2+6b+21)}.
$$
The minimum occurs at $b=(-49+4\sqrt{133})/13$ and equals $7-\sqrt{133}/2\simeq1.2337187027$. It is slightly above the exact <infinite square well> coefficient $\pi^2/8\simeq1.2337005501$. The zero extension of a trial function has derivative jumps at the walls, so the appropriate admissibility statement uses the Dirichlet kinetic-energy form; it need not belong to the unrestricted full-line second-derivative <operator domain>.
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