michael_nielsen

Rather than framing this as something which should be better known, might it be better to say the spectral theorem and its implications should be deeper in peoples bones?

Ofc I love the spectral theorem - it's coincidentally the result I've proven most often in public writing - but this result of Thompson's is very different (and seems to be much harder).

Hmm, okay, I'm clearly taking too quick a glace then. Just to check myself, am I right in thinking U and V are the change of basis matrices which diagonalize H and K? So e^(iUHU*) will be diagonal with e^{i lambda} entries on the diagonal?

Hmm, I guess not, actually. If that were the case, UHU* + VKV* would be diagonal, with eigenvalues lambda_i + mu_i, if lambda_i are the eigenvalues of H and mu_i are the eigenvalues of K. So the rhs above should be diagonal with e^{i(lambda_i + mu_i)} entries on the diagonal.

That's right. When H and K don't share a common basis there's no obvious reason to expect there to be any relationship between the spectrum of exp(iH) exp(iK) and H and K. Thompson tells you there's a really strong relationship.

Interesting. Is there a short answer to what U and V are in this case?

Not as far as I know. In fact, I'm not even certain there's a constructive proof known - unless the situation has changed, Thompson's result depends on the proof of Horn's Conjecture by Allen Knutson, Terry Tao, and Alexander Klyachko, and I've never mastered that.