Avoided crossings of the quartic oscillator

The phenomenon of avoided crossings of energy levels in the spectrum of quantum systems is well known. However, being of an exponentially small order it is hard to calculate. In particular, this is the case when the potential is generating a Schrodinger equation of a type which is beyond the hypergeometric one. Recently, there have been attempts to understand this phenomenon in connection with Heun-type differential equations. The most famous example of this class is the quantum quartic oscillator which is governed by the triconfluent case of Heun's differential equation. In the following we consider situations where the fourth-order potential has two minima and we calculate the avoided crossings of its eigenvalue curves in dependence on the asymmetry and the barrier height between the two wells. The results are compared with those obtained from an asymptotic approach of the problem for large values of the control parameter that governs the barrier height.