Manifestation of classical instability in the quantum density of states of a double well potential

Abstract

Within Bohr's correspondence principle, quantum theory should reproduce the classical world when ħ→0. In practice, obtaining the complete spectrum is challenging since the discrete energies come close to each other before passing to continuum. However it is possible to identify signatures that the spectrum is approaching the continuum limit. In this work, for ħ→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.