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Is there a way to solve the equation: $T^2 = -\kappa\, \mathrm{Tr}\, (\log(e^{i T \hat{H}_0} \hat{O}) )^2$ for $T$?

Here $\kappa$ is an arbitrary positive constant, $\hat{H}_0 \in \mathfrak{su}(N)$ and $\hat{O} \in SU(N)$. $\log$ is the principle matrix log. I'm only interested in positive $T$ solutions.

I have solved it in the case that $e^{iT\hat{H}_0}$ and $\hat{O}$ commute, the answer is obtained from the quadratic formula and is not an especially pleasing form.

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    $\begingroup$ What are $r$ and $T$? $\endgroup$
    – Vít Tuček
    Commented Jul 5, 2014 at 22:14
  • $\begingroup$ The $Tr$ together means the matrix trace. $\endgroup$
    – Benjamin
    Commented Jul 5, 2014 at 22:17
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    $\begingroup$ You seem to be missing an open parenthesis. I'm guessing it's before the log? $\endgroup$
    – user44191
    Commented Jul 5, 2014 at 23:14
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    $\begingroup$ @CarloBeenakker: Perhaps there could be a closed form solution in terms of spectra of $H_0$ and $\widehat{O}$. Since the OP claims that he obtained some formula in the case when these two operators commute, it is probably time to investigate spectral mapping theorem in the context of functional calculus in several variables for non-commuting operators. ;) $\endgroup$
    – Vít Tuček
    Commented Jul 6, 2014 at 0:00
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    $\begingroup$ @VítTuček --- if $H_0$ and $O$ commute, the trace depends only on the $N$ eigenvalues of each of these two matrices; once you abandon that, the eigenvectors enter as well so knowing the spectra of $H_0$ and $O$ will not suffice. $\endgroup$
    – Carlo Beenakker
    Commented Jul 6, 2014 at 7:21

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