Hi--
Yeah, I'm sure it's possible to "convert" your JAA type to an FAA type but I have no experience in this matter.
When I was a flight instructor I did have a Swiss student that converted his private pilot certificate. He simply presented his credentials to the FAA and they awarded him an FAA Private certificate with certain limitations.
My advice to you is this:
1) Do a search on Pprune for something like, "Convert JAA type to FAA type."
2) Start another thread with a more specific title to attract someone who has had the same experience.
3) Go to
www.propilot.com In the menu, click on DOC's FAR Bulletin Board and post your question there. Doc is an authority on FAA regulations and probably has the resources to help you.
Good luck.