  We consider physical properties of a superconductor with a recently proposed
type of odd-frequency pairing that exhibits diamagnetic Meissner response
("odd-dia state"). Such a state was suggested in order to address stability
issues arising in an odd-frequency superconducting state with paramagnetic
Meissner response ("odd-para state"). Assuming the existence of an odd-dia
state (due to a proper retarded interaction), we study its coexistence with an
odd-para state. The latter is known to be generated as an induced
superconducting component in, e.g., singlet superconductor/ferromagnet
proximity structures or triplet superconductor/normal metal systems.
Calculating the superfluid density of the mixed odd-para/odd-dia state and the
Josephson current between the odd-para and odd-dia states, we find that the
expressions for the currents in both cases have non-vanishing imaginary
contributions and are therefore unphysical. We show that a realization of the
odd-dia state implies the absence of a Hamiltonian description of the system,
and suggest that there exists no physically realizable perturbation that could
give rise to the spontaneous symmetry breaking necessary for an actual
realization of the odd-dia superconducting state.
