This question is inspired by the solution to Question 1 here.
$$H_0={p^2\over 2m}+{1\over 2}mw^2x^2\\$$ Perturbation $$H_1=g{w\over 2}(xp+px),g\in \mathbb R, |g|<1$$ We get rid of the perturbation by a unitary transformation $$p\mapsto P+\gamma x, x\mapsto x$$ This does not change the canonical commutation relation $$[x,p]=i\hbar$$
It says that we can get rid of the perturbation by doing the given transform and that the transform does not alter the commutation relations. I would like to know what is the most general form of the transform (hence perturbation) that could do the same.
I understand that this has a physics context, but I think it is a valid mathematical problem to ask on this site?
Thank you.
Added: Anyone? I have been researching "canonical transformations" but my search has not been able to answer my question.
Added II: Let's first consider a simpler problem. Suppose $p\mapsto P+F(x)$ what sort of $F$ is permissible? I believe all $F$ expandable as a power series in $x$ fits the mould. However, is this the most general $F$?