Suppose I have some matrix exponential $U(t)=\exp(-iAt)$ where $t$ is some real valued number, $A$ is a hermitian matrix (so $U(t)$ is unitary) where $A=B+B^{-1}$ and $B$ is unitary. Because $B$ and $B^{-1}$ commute we have $$ U(t)=\exp(-iBt).\exp(-iB^{-1}t)$$ Can this expression for $U(t)$ then be simplified any further without diagonalising $B$ and $B^{-1}$?
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1$\begingroup$ Even in 1-d, I see no good way of simplifying $e^{-ibt}e^{-it/b}$... $\endgroup$– Sangchul LeeOct 21, 2018 at 9:41
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1$\begingroup$ Since $B$ is unitary, $B^{-1} = B^\dagger$. This means that $\exp(-iB^{-1}t) = \exp(iBt)^\dagger$. Writing $U(t) = \exp(-i B t)\exp(iBt)^\dagger$ at least makes it clear that $U$ is unitary. $\endgroup$– eyeballfrogOct 21, 2018 at 9:46
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$\begingroup$ @eyeballfrog true. So lets say I diagonalised $B=V\Lambda V^\dagger$ then I should get \begin{align} U(t)&=V \exp(-i \Lambda t) V^\dagger (V \exp(i \Lambda t) V ^\dagger)^\dagger\\ &=V \exp(-i \Lambda t) V^\dagger V \exp(-i \Lambda^\dagger t) V^\dagger\\ &= V \exp(-i (\Lambda + \Lambda^\dagger)t)V^\dagger \end{align} Since $B$ is unitary its eigenvalues have modulus 1 and hence $\Lambda$ is unitary as well. $\endgroup$– asettOct 21, 2018 at 10:13
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1$\begingroup$ @amitsett Yes. And since $\Lambda$ is diagonal, $\Lambda + \Lambda^\dagger = 2\mathrm{Re}[\Lambda]$. Not sure how much this helps, but it's neat. $\endgroup$– eyeballfrogOct 21, 2018 at 19:09
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One obvious simplification is \begin{align} U(t) &=\exp(-itB)\exp(-itB^{-1})\\ &=\exp(-it(B+B^{-1}))\ \text{ (because $B$ commutes with $B^{-1}$)}\\ &=\exp(-it(B+B^\ast)) \end{align} but you probably already know this.
answered Oct 21, 2018 at 10:31
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$\begingroup$ Yep. I guess to be more clear I'm trying to break it up into exponentials of matrices that are easier to exponentiate. I could diagonalise $B$ such that $B=V\Lambda V^\dagger$ then $$\exp(-itB)=V\exp(-it \Lambda)V^\dagger$$ However I want to check if there's any further tricks I could use before finding eigenvectors and eigenvalues. $\endgroup$– asettOct 21, 2018 at 10:36