= Graph estimate for the harmonic oscillator
{title2=$\|u''\|_2^2+\|(c+x^2)u\|_2^2\leq\|-u''+(c+x^2)u\|_2^2+2\|u\|_2^2$}
For $c>0$ and smooth compactly supported $u$, <integration by parts> gives
$$
\|-u''+(c+x^2)u\|_2^2
=\|u''\|_2^2+\|(c+x^2)u\|_2^2+2\int(c+x^2)|u'|^2\,dx-2\|u\|_2^2.
$$
Dropping the nonnegative weighted derivative term proves the displayed bound. For a weak resolvent solution in the <one-dimensional harmonic oscillator form domain>, apply the bound to $\chi_Ru$, where $\chi_R$ is a smooth cutoff. The cutoff error $-2\chi_R'u'-\chi_R''u$ is bounded in <L2 space> and tends to zero. The <Fatou lemma> then shows that $u''$ and $x^2u$ belong to <L2 space>, identifying the strong operator domain.
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