  The commutator $[x(t),p]$ in an inverted harmonic oscillator (IHO) in
one-dimensional quantum mechanics exhibits remarkable properties. It reduces to
a c-number and does not show any quantum fluctuations for arbitrary states.
Related to this nature, the quantum Lyapunov exponent computed through the
out-of-time-order correlator (OTOC) $\langle [x(t),p]^2 \rangle $ precisely
agrees with the classical one. Hence, the OTOC may be regarded as an ideal
indicator of the butterfly effect in the IHO. Since IHOs are ubiquitous in
physics, these properties of the commutator $[x(t),p]$ and the OTOCs might be
seen in various situations, too. In order to clarify this point, as a first
step, we investigate OTOCs in one-dimensional quantum mechanics with polynomial
potentials, which exhibit butterfly effects around the peak of the potential in
classical mechanics. We find two situations in which the OTOCs show exponential
growth reproducing the classical Lyapunov exponent of the peak. The first one,
which is obvious, is using a suitably localized wave packet near the peak, and
the second one is taking a limit akin to the large-$N$ limit in the noncritical
string theories.
