  We investigate Krylov complexity of the fermion chain operator which consists
of multiple Majorana fermions in the double-scaled SYK (DSSYK) model with
finite temperature. Using the fact that Krylov complexity is computable from
two-point functions, the analysis is performed in the limit where the two-point
function becomes simple and we compare the results with those of other previous
studies. We confirm the exponential growth of Krylov complexity in the very low
temperature regime. In general, Krylov complexity grows at most linearly at
very late times in any system with a bounded energy spectrum. Therefore, we
have to focus on the initial growth to see differences in the behaviors of
systems or operators. Since the DSSYK model is such a bounded system, its
chaotic nature can be expected to appear as the initial exponential growth of
the Krylov complexity. In particular, the time at which the initial exponential
growth of Krylov complexity terminates is independent of the number of degrees
of freedom. Based on the above, we systematically and specifically study the
Lanczos coefficients and Krylov complexity using a toy power spectrum and
deepen our understanding of those initial behaviors. In particular, we confirm
that the overall sech-like behavior of the power spectrum shows the initial
linear growth of the Lanczos coefficient, even when the energy spectrum is
bounded.
