This problem is from $Spectral \ Theory \ and \ its \ Applications,\ Bernard\ Helffer, \ Page \ 37$. It says that if $u \in H^1(\mathbf{R^m}), x_j u \in L^2(\mathbf{R^m}),(-\Delta+\vert x \vert^2+1 )u \in L^2(\mathbf{R^m}) $ then \begin{equation} u\in B^2(\mathbf{R^m}) :=\lbrace u\in H^2(\mathbf{R^m}): x^{\alpha} u \in L^2(\mathbf{R^m}),\forall \alpha,s.t.\vert \alpha \vert \leq 2 \rbrace \end{equation} The book ssys it needs a regularity theorem based on the method of differential quotients, but there are no details. And I really want to know how it works. I've tried to prove this when $m=1$, but it seems not easy. Now I have an idea:
Show that if \begin{equation} (\pm\frac{d}{dx}+x)f \in L^2, f \in L^2,\ then \ xf \in L^2 \end{equation} Is this helpful?
Okay, now a $2010$ version of this book says that this differential method can be find in $Non-Homogeneous \ Boundary \ Value \ Problems \ and \ Applications,\ Lions\ and\ Magenes$. But where can I find this book?