# Generalization of a problem on canonical transforms of operators

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 $F$?

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Welcome to math.SE! a) Please choose a more informative title. b) Questions should be as self-contained as possible. (Please see meta.math.stackexchange.com/questions/2674.) –  joriki Oct 4 '12 at 8:23
@joriki: Thanks for the welcome! I have edited the question accordingly. –  Yankar Oct 4 '12 at 11:44