  We present an analytic Bogoliubov description of a BEC of polar molecules
trapped in a quasi-2D geometry and interacting via internal state-dependent
dipole-dipole interactions. We derive the mean-field ground-state energy
functional, and we derive analytic expressions for the dispersion relations,
Bogoliubov amplitudes, and dynamic structure factors. This method can be
applied to any homogeneous, two-component system with linear coupling, and
direct, momentum-dependent interactions. The properties of the mean-field
ground state, including polarization and stability, are investigated, and we
identify three distinct instabilities: a density-wave rotonization that occurs
when the gas is fully polarized, a spin-wave rotonization that occurs near zero
polarization, and a mixed instability at intermediate fields. These
instabilities are clarified by means of the real-space density-density
correlation functions, which characterize the spontaneous fluctuations of the
ground state, and the momentum-space structure factors, which characterize the
response of the system to external perturbations. We find that the gas is
susceptible to both density-wave and spin-wave response in the polarized limit
but only a spin-wave response in the zero-polarization limit. These results are
relevant for experiments with rigid rotor molecules such as RbCs,
$\Lambda$-doublet molecules such as ThO that have an anomalously small
zero-field splitting, and doublet-$\Sigma$ molecules such as SrF where two
low-lying opposite-parity states can be tuned to zero splitting by an external
magnetic field.
