  We calculate the spectrum of transfer matrix eigenvalues associated with
Polyakov loops in finite-density lattice QCD with static quarks. These
eigenvalues determine the spatial behavior of Polyakov loop correlations
functions. Our results are valid for all values of the gauge coupling in $1+1$
dimensions, and valid in the strong-coupling region for any number of
dimensions. When the quark chemical potential $\mu$ is nonzero, the spatial
transfer matrix $T_s$ is non-Hermitian. The appearance of complex eigenvalues
in $T_s$ is a manifestation of the sign problem in finite-density QCD. The
invariance of finite-density QCD under the combined action of charge
conjugation $\mathcal{C}$ and complex conjugation $\mathcal{K}$ implies that
the eigenvalues of $T_s$ are either real or part of a complex pair. Calculation
of the spectrum confirms the existence of complex pairs in much of the
temperature-chemical potential plane. Many features of the spectrum for static
quarks are determined by a particle-hole symmetry. For $\mu$ small compared to
the quark mass $M$, we typically find real eigenvalues for the lowest lying
states. At somewhat larger values of $\mu,$ pairs of eigenvalues may form
complex-conjugate pairs, leading to damped oscillatory behavior in Polyakov
loop correlation functions. However, near $\mu=M$, the low-lying spectrum
becomes real again. This is a direct consequence of the approximate
particle-hole symmetry at $\mu=M$ for heavy quarks. This behavior of the
eigenvalues should be observable in lattice simulations and can be used as a
test of lattice algorithms. Our results provide independent confirmation of
results we have previously obtained in PNJL models using complex saddle points.
