  Pair density wave superconductivity constitutes a novel electronic condensate
proposed to be realized in certain unconventional superconductors. Establishing
its potential existence is important for our fundamental understanding of
superconductivity in correlated materials. Here we compute the dynamical
magnetic susceptibility in the presence of a pair density wave ordered state,
and study its fingerprints on the spin-wave spectrum including the neutron
resonance. In contrast to the standard case of d-wave superconductivity, we
show that the pair density wave phase exhibits neither a spin-gap nor a
magnetic resonance peak, in agreement with a recent neutron scattering
experiment on underdoped La$_{1.905}$Ba$_{0.095}$CuO$_4$ [Z. Xu et al., Phys.
Rev. Lett. 113, 177002 (2014)].
