  The gap of the Liouvillian spectrum gives the asymptotic decay rate of a
quantum dissipative system, and therefore its inverse has been identified as
the slowest relaxation time. In contrary to this common belief, we show that
the relaxation time due to diffusive transports in a boundary dissipated
many-body quantum system is determined not by the gap or low-lying eigenvalues
of the Liouvillian but by superexponentially large expansion coefficients for
Liouvillian eigenvectors with non-small eigenvalues at an initial state. This
finding resolves an apparent discrepancy reported in the literature between the
inverse of the Liouvillian gap and the relaxation time in dissipative many-body
quantum systems.
