The past and future wave operators on the singular spectrum

TL;DR

This paper investigates the existence and properties of averaged wave operators for singular unitary operators, revealing conditions under which past and future wave operators coincide and exploring implications for boundary behavior of certain integrals.

Contribution

It establishes the simultaneous existence or non-existence of averaged wave operators for singular unitary operators with rank-two commutator and links this to boundary behavior of Cauchy-type integrals.

Findings

Past and future averaged wave operators either both exist or both do not.

When they exist, the past and future wave operators coincide.

Results provide insights into boundary behavior of Cauchy-type integrals.

Abstract

We consider averaged wave operators constructed for singular unitary operators $U_1$, $U_2$ and a bounded identification operator $A$. In the case of rank-two commutator $AU_1 - U_2A$, we show that averaged wave operators of past and future exist or do not exist simultaneously, and if they exist, they must coincide. As a consequence, we obtain some results concerning boundary behaviour of Cauchy-type integrals.