  Edge burst is a phenomenon in non-Hermitian quantum dynamics discovered by a
recent numerical study [W.-T. Xue, et al, Phys. Rev. Lett 2, 128.120401(2022)].
It finds that a large proportion of particle loss occurs at the system boundary
in a class of non-Hermitian quantum walk. In this paper, we investigate the
evolution of real-space wave functions for this lattice system. We find the
wave function of the edge site is distinct from the bulk sites. Using
time-dependent perturbation theory, we derive the analytical expression of the
real-space wave functions and find that the different evolution behaviors
between the edge and bulk sites are due to their different nearest-neighbor
site configurations. We also find the edge wave function primarily results from
the transition of the two nearest-neighbor non-decay sites. Besides, the
numerical diagonalization shows the edge wave function is mainly propagated by
a group of eigen-modes with a relatively large imaginary part. Our work
provides an analytical method for studying non-Hermitian quantum dynamical
problems.
