Okay to state very specific research interest in statement of purpose?
First, how about asking the opinion of the professor who has the common interest with yours?  S/he will hopefully be able to assess how much influence s/he might have with the admissions committee.

For the rest of my answer I will assume the professor advises you not to make your statement too narrow.

I suggest you start off rather broadly, for approximately one paragraph, talking about an area that includes at least one of your specific research questions.  At this point you are filming the city from a helicopter or drone.  In the next paragraph, you can bring your camera down to street level, and start describing one or more of your specific questions, as an example.

If you are really creative, you could write the broad paragraph in such a way that at least one other professor in the department could see some spark of commonality.