  According to its Lax pair formulation, the nonlinear Schr\"odinger (NLS)
equation can be expressed as the compatibility condition of two linear ordinary
differential equations with an analytic dependence on a complex parameter. The
first of these equations---often referred to as the \emph{$x$-part} of the Lax
pair---can be rewritten as an eigenvalue problem for a Zakharov-Shabat
operator. The spectral analysis of this operator is crucial for the solution of
the initial value problem for the NLS equation via inverse scattering
techniques. For space-periodic solutions, this leads to the existence of a
Birkhoff normal form, which beautifully exhibits the structure of NLS as an
infinite-dimensional completely integrable system. In this paper, we take the
crucial steps towards developing an analogous picture for time-periodic
solutions by performing a spectral analysis of the \emph{$t$-part} of the Lax
pair with a periodic potential.
