Manifestation of instability in the quasiclassical limit of the spectrum of the quartic double well

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.