= Compact parabolic trial bound for the quantum harmonic oscillator
{title2=$\lambda_0\leq\sqrt{10/7}$}
For the dimensionless <quantum harmonic oscillator> $H=-d^2/dx^2+x^2$, the normalized compact trial function is $\psi_a=\sqrt{15/(16a^5)}(a^2-x^2)$ on $|x|\leq a$ and zero elsewhere. Its kinetic and potential contributions to the energy <quadratic form> are $5/(2a^2)$ and $a^2/7$. Minimizing their sum gives $a^4=35/2$ and the displayed upper bound, approximately $1.19523$, compared with exact <ground state> eigenvalue $1$. The <weak derivative> is square-integrable because the trial is continuous at its endpoints. Jumps in the ordinary first derivative do not invalidate the <Rayleigh-Ritz variational principle>: it requires the energy form domain, rather than the smaller <operator domain>.
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