  Within the Bohr's correspondence principle, the quantum theory should
reproduce the classical world when $\hbar\to 0$. In practice, the discrete
energies come close to each other before passing to continuum, causing that
highly excited states become inaccessible. However it is possible to identify
signatures that the spectrum is approaching the continuum limit. In this work,
for $\hbar \to 0$, we focus our attention on the spectrum of the 1D quartic
double well and find signatures of the classical instability in the quantum
spectrum near the critical energy, and explore how the tunneling phenomenon
remains only at energies close to this critical energy.
