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Frequency-dependent Inter-pseudospin Solutions to Superconducting Strontium Ruthenate

The lasting puzzle of the superconducting order parameter of Sr$_2$RuO$_4$ calls for theoretical studies that include seldom-considered effects. Here we include spin-orbit coupling effects on the electronic structure and then solve the linearized Eliashberg equation in a pseudospin basis, including the possibility that spin and charge fluctuations induce frequency-dependent superconducting order parameters. We find that spin-orbit coupling mixes even and odd contributions in orbital, spin and frequency spaces and that leading inter-pseudospin symmetries, B$_{1g}^+$ and A$_{2g}^-$, have intra-orbital components respectively even and odd in Matsubara frequency. An accidental degeneracy between these could resolve apparent experimental contradictions.