  We provide a detailed examination of a thermal out-of-time-order correlator
(OTOC) growing exponentially in time in systems without chaos. The system is a
one-dimensional quantum mechanics with a potential whose part is an inverted
harmonic oscillator. We numerically observe the exponential growth of the OTOC
when the temperature is higher than a certain threshold. The Lyapunov exponent
is found to be of the order of the classical Lyapunov exponent generated at the
hilltop, and it remains non-vanishing even at high temperature. We adopt
various shape of the potential and find these features universal. The study
confirms that the exponential growth of the thermal OTOC does not necessarily
mean chaos when the potential includes a local maximum. We also provide a bound
for the Lyapunov exponent of the thermal OTOC in generic quantum mechanics in
one dimension, which is of the same form as the chaos bound obtained by
Maldacena, Shenker and Stanford.
