  We study the non-Markovian dynamics of a two-level giant atom interacting
with a one-dimensional semi-infinite waveguide through multiple coupling
points, where a perfect mirror is located at the endpoint of the waveguide. The
system enters a non-Markovian process when the travel time of the photon
between adjacent coupling points is sufficiently large compared to the inverse
of the bare relaxation rate of the giant atom. The photon released by the
spontaneous emission of the atom transfers between multiple coupling points
through the waveguide or is reabsorbed by the atom with the photon emitted via
the atom having completed the round trip after reflection of the mirror, which
leads to the photon being trapped and forming bound states. We find that three
different types of bound states can be formed in the system, containing the
static bound states with no inversion of population, the periodic equal
amplitude oscillation with two bound states, and the periodic non-equal
amplitude oscillation with three bound states. The physical origins of three
bound states formation are revealed. Moreover, we consider the influences of
the dissipation of unwanted modes and dephasing on the bound states. Finally,
we extend the system to a more general case involving many giant atoms coupled
into a one-dimensional semi-infinite waveguide. The obtained set of delay
differential equations for the giant atoms might open a way to better
understand the non-Markovian dynamics of many giant atoms coupled to a
semi-infinite waveguide.
