Resolvent of the shifted harmonic oscillator ($\Vert (\lambda -A{)}^{-1}{\Vert }_{2\to 2}\le (\lambda +1{)}^{-1}$)

For $Au=u^{\prime \prime }-(1+x^2)u$ on $D(A)=\{u\in H^2(\mathbb{R}):x^{2}u\in L^2\}$, the form $a_{\lambda }(u,v)=\int u^{\prime }\overline{v^{\prime }}+(1+\lambda +x^2)u\bar{v}$ 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 $a_{\lambda }(u,u)\ge (1+\lambda )\Vert u{\Vert }_2^2$ 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.