Graph estimate for the harmonic oscillator ($\Vert u^{\prime \prime }{\Vert }_2^2+\Vert (c+x^2)u{\Vert }_2^2\le \Vert -u^{\prime \prime }+(c+x^2)u{\Vert }_2^2+2\Vert u{\Vert }_2^2$)

For $c>0$ and smooth compactly supported $u$, integration by parts givesDropping 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 ${\chi }_{R}u$, where ${\chi }_R$ is a smooth cutoff. The cutoff error $-2{\chi }_R^{\prime }{u}^{\prime }-{\chi }_R^{\prime \prime }u$ is bounded in L2 space and tends to zero. The Fatou lemma then shows that $u^{\prime \prime }$ and $x^{2}u$ belong to L2 space, identifying the strong operator domain.

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.