  Recently, the out-of-time-ordered correlator(OTOC) and Krylov complexity have
been studied actively as a measure of operator growth. OTOC is known to exhibit
exponential growth in chaotic systems, which was confirmed in many previous
works. However, in some non-chaotic systems, it was observed that OTOC shows
chaotic behavior and cannot distinguish saddle-dominated scrambling from
chaotic systems. For K-complexity, in the universal operator growth hypothesis,
it was stated that Lanczos coefficients show linear growth in chaotic systems,
which is the fastest. But recently, it appeared that Lanczos coefficients and
K-complexity show chaotic behavior in the LMG model and cannot distinguish
saddle-dominated scrambling from chaos. In this paper, we compute Lanczos
coefficients and K-complexity in an inverted harmonic oscillator. We find that
they exhibit chaotic behavior, which agrees with the case of the LMG model. We
also analyze bounds on the quantum Lyapunov coefficient and the growth rate of
Lanczos coefficients and find that there is a difference with the chaotic
system. Microcanonical K-complexity is also analyzed and compared with the OTOC
case.
