Consider the operator $A=D_p^2+ip$, where $Dp=-i ∂_p$, and the domain of A is $$D(A)=\{u \in L^2(R,dp) : Au \in L^2(R,dp)\}.$$ Using the fact that $$ \|Au\|^2=\|D_p^2u\|^2 +\|pu\|^2 +2\langle u,D_pu\rangle $$ and $\|u\|\cdot\|Au\|\ge\|D_pu\|^2$ for every $u \in D(A)$, I would like to prove that there exists $c>1$ such that: $$ c\left[\| D_p^2u\|^2 +\|pu\|^2 +\|u\|^2 \right] \le \|Au\|^2≤c^{-1} \left[ \|D_p^2u\|^2+\|pu\|^2+\|u\|^2 \right] $$
here is what i tried to do: $||Au||^2≤||D^2_pu||^2+ ||pu||^2 +2||u||.||D_pu||$ then $||Au||^2-||u||^2-||u||.||Au||≤||D^2_pu||^2 +||pu||^2 +2||u||.||D_pu||-||u||^2 -||D_pu||^2≤||D^2_pu||^2 +||pu||^2 $
or $-||u||.||Au||≥-1_{/2}(||u||^2 +||Au||^2)$ then $-3_{/2}||u||^2 +1_{/2}||Au||^2 ≤||D^2_pu||^2 +||pu||^2$ then $1_{/2}||Au||^2≤||D^2_pu||^2 +||pu||^2 +3_{/2}||u||^2≤3_{/2}(||D^2_pu||^2 +||pu||^2 +||u||^2)$
hense $||Au||^2≤3(||D^2_pu||^2 +||pu||^2 +||u||^2) $
For the other inegality i wrote:
$||Au||^2≥||D^2_pu||^2+ ||pu||^2 -2||u||.||D_pu|| $ hense $||Au||^2+||u||^2+||u||.||Au||>=||D^2_pu||^2 +||pu||^2 -2||u||.||D_pu||+||u||^2 +||D_pu||^2≥||D^2_pu||^2 +||pu||^2 $
or $||u||.||Au||≥1_{/2}(||u||^2 +||Au||^2) $ hence $3_{/2} (||Au||^2+||u||^2)>=||D^2_pu||^2 +||pu||^2 $
Hence $||Au||^2≥2_{/3}(||D^2_pu||^2 +||pu||^2)-||u||^2$
but this doesn't answer the question .Can someone please help me to correct my work.
