  Self-adjoint Toeplitz operators have purely absolutely continuous spectrum.
For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a
further spectral classification determined by propagation properties of the
operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$.
It turns out that the spectrum is naturally partitioned into three disjoint
subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation
properties are modeled by the continuous part of the symbol, whereas on the
thin spectrum, the model operator is determined by the jumps of the symbol. On
the mixed spectrum, these two types of the asymptotic evolution of $\exp(-iTt)
f$ coexist. This classification is justified in the framework of scattering
theory. We prove the existence of wave operators that relate the model
operators with the Toeplitz operator $T$. The ranges of these wave operators
are pairwise orthogonal, and their orthogonal sum exhausts the whole space,
i.e., the set of these wave operators is asymptotically complete.
