# Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

## TL;DR

This paper develops a spectral classification for Toeplitz operators with piecewise continuous symbols based on propagation properties, and establishes a scattering theory framework with asymptotic completeness of wave operators.

## Contribution

It introduces a new spectral classification into thick, thin, and mixed spectra for such Toeplitz operators and proves the existence and completeness of related wave operators.

## Key findings

- Spectral classification into three disjoint subsets: thick, thin, and mixed.
- Existence of wave operators relating model operators to Toeplitz operators.
- Orthogonal decomposition of the space via wave operators, ensuring asymptotic completeness.

## Abstract

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.