  Identifying dynamical signatures of excited state quantum phase transitions
(ESQPTs) in experimentally realizable quantum many-body systems is helpful for
understanding the dynamical effects of ESQPTs. In such systems, the highly
controllable spinor Bose-Einstein condensates (BECs) offer an exceptional
platform to study ESQPTs. In this work, we investigate the dynamical
characteristics of the ESQPT in spin-$1$ BEC by means of the Loschmidt echo
spectrum. The Loschmidt echo spectrum is an extension of the well-known
Loschmidt echo and definded as the overlaps between the evolved state and the
excited states of the initial Hamiltonian. We show that both the time evolved
and long time averaged Loschmidt echo spectrum undergo a remarkable change as
the system passes through the critical point of the ESQPT. Moreover, the
particular behavior exhibited by the Loschmidt echo spectrum at the critical
point stand as a dynamical detector for probing the ESQPT. We further
demonstrate how to capture the features of the ESQPT by using the energy
distribution associated with the Loschmidt echo spectrum for time evolved and
long time averaged cases, respectrively. Our findings contribute to a further
verification of the usefulness of the Loschmidt echo spectrum for witnessing
various quantum phase transitions in many-body systems and provide a new way to
experimentally examine the dynamical consequences of ESQPTs.
