= Resolvent of the shifted harmonic oscillator
{title2=$\|(\lambda-A)^{-1}\|_{2\to2}\leq(\lambda+1)^{-1}$}
For $Au=u''-(1+x^2)u$ on $D(A)=\{u\in H^2(\mathbb R):x^2u\in L^2\}$, the form $a_\lambda(u,v)=\int u'\overline{v'}+(1+\lambda+x^2)u\bar v$ is coercive on the <one-dimensional harmonic oscillator form domain> when $\lambda>0$. The <Lax-Milgram theorem> gives a unique weak solution of $(\lambda-A)u=g$. The <graph estimate for the harmonic oscillator> puts it in $D(A)$, and $a_\lambda(u,u)\geq(1+\lambda)\|u\|_2^2$ gives the displayed <resolvent> bound. The domain is dense and the bounded resolvent proves closedness, so the <Hille-Yosida theorem> gives a <contraction semigroup>. An <exponentially shifted semigroup> with generator $A+I$ solves the heat equation with oscillator potential.
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