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

## TL;DR

This paper investigates how quantum spectra of a 1D quartic double well reflect classical instability as Planck's constant approaches zero, revealing signatures of the continuum limit and the persistence of tunneling near critical energies.

## Contribution

It identifies quantum spectral signatures of classical instability in the quartic double well as 5 approaches zero, linking quantum behavior to classical chaos.

## Key findings

- Signatures of classical instability appear in the quantum spectrum near critical energy.
- Tunneling persists only at energies close to the critical energy.
- Quantum spectrum approaches the continuum limit as 5 0.

## Abstract

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.