  We study the spectra for a class of differential operators with
asymptotically constant coefficients.These operators widely arise as the
linearizations of nonlinear partial differential equations about patterns or
nonlinear waves. We present a unified framework to prove the perturbation
results on the related spectra. The proof is based on exponential dichotomies
and the Brouwer degree theory. As applications, we employ the developed theory
to study the stability of quasi-periodic solutions of the Ginzburg-Landau
equation,fold-Hopf bifurcating periodic solutions of reaction-diffusion systems
coupled with ordinary differential equations, and periodic annulus of the
hyperbolic Burgers-Fisher model.
