Bound on Lyapunov exponent in $c=1$ matrix model

Classical particle motions in an inverse harmonic potential show the exponential sensitivity to initial conditions, where the Lyapunov exponent $\lambda_L$ is uniquely fixed by the shape of the potential. Hence, if we naively apply the bound on the Lyapunov exponent $\lambda_L \le 2\pi T/ \hbar$ to this system, it predicts the existence of the bound on temperature (the lowest temperature) $T \ge \hbar \lambda_L/ 2\pi$ and the system cannot be taken to be zero temperature when $\hbar \neq 0$. This seems a puzzle because particle motions in an inverse harmonic potential should be realized without introducing any temperature but this inequality does not allow it. In this article, we study this problem in $N$ non-relativistic free fermions in an inverse harmonic potential ($c=1$ matrix model). We find that thermal radiation is {\em induced} when we consider the system in a semi-classical regime even though the system is not thermal at the classical level. This is analogous to the thermal radiation of black holes, which are classically non-thermal but behave as thermal baths quantum mechanically. We also show that the temperature of the radiation in our model saturates the inequality, and thus, the system saturates the bound on the Lyapunov exponent, although the system is free and integrable. Besides, this radiation is related to acoustic Hawking radiation of the fermi fluid.