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We consider physical properties of a superconductor with a recently proposedtype of odd-frequency pairing that exhibits diamagnetic Meissner response("odd-dia state"). Such a state was suggested in order to address stabilityissues arising in an odd-frequency superconducting state with paramagneticMeissner response ("odd-para state"). Assuming the existence of an odd-diastate (due to a proper retarded interaction), we study its coexistence with anodd-para state. The latter is known to be generated as an inducedsuperconducting component in, e.g., singlet superconductor/ferromagnetproximity structures or triplet superconductor/normal metal systems.Calculating the superfluid density of the mixed odd-para/odd-dia state and theJosephson current between the odd-para and odd-dia states, we find that theexpressions for the currents in both cases have non-vanishing imaginarycontributions and are therefore unphysical. We show that a realization of theodd-dia state implies the absence of a Hamiltonian description of the system,and suggest that there exists no physically realizable perturbation that couldgive rise to the spontaneous symmetry breaking necessary for an actualrealization of the odd-dia superconducting state.

Odd-frequency superconducting states with different types of Meissner response: Problem of coexistence