Try looking up some of the more renowned analysis departments, this is all subjective of course. But off the top of my head, I think UCLA has a well regarded analysis department. I know Texas has a very good analysis department (for PDEs anyway).
I don't have the link with me, but try looking for the US News grad school rankings. They rank math grad schools by specialty, so check out the analysis rankings. Of course, this should only serve as a rough guide, not a definitive source. Some schools have excellent advisers but don't have the overall strength in their departments. With those analysis rankings, check out the departments, see how many specialize in functional analysis, how often they publish, do they advise students on a regular basis, etc.
Also, look for a functional analysis math journal (there's gotta be one right?) and lookat who publishes and what they are publishing. This is a good way to spot potential advisers.
Once you have found some prospective departments, then you need to consider location, postdoc placement rate, percentage of students who complete their thesis, like the typical questions you ask about any grad school. But to me, the most important thing should be that they have some potential advisers who are working in subjects you like. Then you can worry about non-math stuff.
Finally, just to add, have you read Barry Simon's 4 volume text on functional analysis and mathematical physics? Simon is a top notch functional analyst over at Caltech. Check him out, he does functional analysis, Schrodinger Operators, etc.
Best of luck!