  We provide a quasi-one-dimensional model which can support a
pair-density-wave (PDW) state, in which the superconducting (SC) order
parameter modulates periodically in space, with gapless Bogoliubov
quasiparticle excitations. The model consists of an array of
strongly-interacting one-dimensional systems, where the one-dimensional systems
are coupled to each other by local interactions and tunneling of the electrons
and Cooper pairs between them. Within the interchain mean-field theory, we find
several SC states from the model, including a conventional uniform SC state,
PDW SC state, and a coexisting phase of the uniform SC and PDW states. In this
quasi-1D regime we can treat the strong correlation physics essentially exactly
using bosonization methods and the crossover to the 2D system by means of
interchain mean field theory. The resulting critical temperatures of the SC
phases generically exhibit a power-law scaling with the coupling constants of
the array, instead of the essential singularity found in weak-coupling BCS-type
theories. Electronic excitations with an open Fermi surface, which emerge from
the electronic Luttinger liquid systems below their crossover temperature to
the Fermi liquid, are then coupled to the SC order parameters via the proximity
effect. From the Fermi surface thus coupled to the SC order parameters, we
calculate the quasiparticle spectrum in detail. We show that the quasiparticle
spectrum can be fully gapped or nodal in the uniform SC phase and also in the
coexisting phase of the uniform SC and PDW parameters. In the pure PDW state,
the excitation spectrum has a reconstructed Fermi surface in the form of Fermi
pockets of Bogoliubov quasiparticles.
