Cayley Transform, Exponential Mapping and more…

Question

Assume a self-adjoint operator, represented as hermitian matrix $H=H^\dagger$. To my knowledge there are at least 2 mappings of $H$ onto unitary matrices:

1. Cayley's Transformation with $U_1=\frac{H+i1}{H-i1}$ ($1$ is a unit matrix of appropriate dimension)
2. Exponential Mapping with $U_2=e^{iH}$.

Are there other mappings with a different structure and if not can one prove that?

Answer 1

Assume that a mapping $H\mapsto U$ is of such a form that $U$ can be expressed as a power series in $H$ (I don't think this is a huge loss of generality to assume this, perhaps none.) Then $$ U=c_{0}I+c_{1}H+c_{2}H^{2} + c_{3}H^{3} + \cdots $$ Since $U$ unitary implies $e^{i\theta}U$ is unitary, then WLOG we may take $c_{0}$ to be real. Moreover, if $U$ is unitary, then $U^{-1}= U^{*}$, so (assuming $U^{-1}$ can be represented as a power series in $H$) $$ U^{-1}=c_{0}I+c_{1}^{*}H+c_{2}^{*}H^{2} + c_{3}^{*}H^{3}+ \cdots $$ The condition $UU^{-1} = I$ gives $$ I=UU^{-1}=c_{0}^{2}I+c_{0}(c_{1}+c_{1}^{*})H+\cdots $$ whereupon we immediately determine that $c_{0}=1$ and that $c_{1}+c_{1}^{*}=0$. Hence $c_{1}$ is purely imaginary, and we write $c_{1}=ib_{1}$. With this in hand, we expand the products of these power series again: $$ I=UU^{-1}=I+(c_{2}+c_{2}^{*}+b_{1}^{2})H^{2}+\cdots $$ so that $2\mathfrak{R}(c_{2})=-b_{1}^{2}$ but the imaginary part of $c_{2}$ is a free parameter. So we've already found 2 free parameters, each of which generate infinite families of unitary maps $H\mapsto U$. So there are a fuck-ton of these maps.