  Winding number and density of states are two fundamental physical quantities
for non-self-adjoint quasi-periodic Schr\"odinger operators, which reflect the
asymptotic distribution of zeros of the characteristic determinants of the
truncated operators under Dirichlet boundary condition, with respect to
complexified phase and the energy respectively. We will prove that the winding
number is in fact Avila's acceleration and it is also closely related to the
density of states by a generalized Thouless formula for non-self-adjoint
Schr\"odinger operators and Avila's global theory.
