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We investigate the possible superconducting instabilities of stronglycorrelated electron materials using a generalization of linear response theoryto external pairing fields depending on frequency. We compute a pairingsusceptibility depending on two times, allowing us to capture dynamical pairingand in particular odd-frequency solutions. We first benchmark this method onthe attractive one-band Hubbard model and then consider the superconductivityof strontium ruthenate Sr$_2$RuO$_4$ within single-site dynamical mean-fieldtheory, hence restricting ourselves to pairing states which are momentumindependent in the orbital basis. The symmetry of the superconducting orderparameter of this material is still debated, and local odd-frequency stateshave been proposed to explain some experimental discrepancies. In thetemperature range studied, we find that the leading eigenvectors areodd-frequency intra-orbital spin-triplet states, while the eigenvectors withthe highest predicted transition temperature correspond to even-frequencyintra-orbital spin-singlet states. The latter include a state with $d$-wavesymmetry when expressed in the band basis.

Frequency-Dependent Superconducting States From the Two-Time Linear Response Theory: Application To Sr$_2$RuO$_4$