  We study the scattering theory of delicate topological insulators (TIs),
which are novel topological phases beyond the paradigm of the tenfold way,
topological quantum chemistry, and the symmetry indicator method. We
demonstrate that the phase of the reflection amplitude can probe the delicate
topology by capturing a characteristic feature of a delicate TI. This feature
is the returning Thouless pump, where an integer number of charges are pumped
forward and backward in the first and second half of the adiabatic cycle
respectively. As a byproduct of our analysis we show that requiring additional
symmetries can stable the boundary states of a delicate TI beyond the
conventional requirement of a sharply defined surface. Furthermore, we propose
a photonic crystal experiment to implement a delicate TI and measure its
reflection phase which reveals the delicate topology.
