  Quantum many-body scars (QMBS) appear in a flat-band model with interactions
on the saw-tooth lattice. The flat-band model includes a compact support
localized eigenstates, called compact localized state (CLS). Some
characteristic many-body states can be constructed from the CLSs at a
low-filling on the flat-band. These many-body states are degenerate. Starting
with such degenerate states we concretely show how to construct multiple QMBSs
with different eigenenergies embedded in the entire spectrum. If the degeneracy
is lifted by introducing hopping modulation or weak perturbations, these states
lifted by these ways can be viewed as multiple QMBSs. In this work, we focus on
the study of the perturbation-induced QMBS. Perturbed states, which are
connected to the exact QMBSs in the unperturbed limit, indicate common
properties of conventional QMBS systems, that is, a subspace with sub-volume or
area law scaling entanglement entropy, which indicates the violation of the
strong eigenstate thermalization hypothesis (ETH). Also for a specific initial
state, slow-thermalization dynamics appears. We numerically demonstrate these
subjects. The flat-band model with interactions is a characteristic example in
non-integrable systems with the violation of the strong ETH and the QMBS.
