AVOIDED CROSSINGS OF THE QUARTIC OSCILLATOR

The phenomenon of avoided crossings of energy levels in the spectrum o f quantum systems is well known. However, being of an exponentially sm all order it is hard to calculate. In particular, this is the case whe n the potential is generating a Schrodinger equation of a type which i s 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 quarti c oscillator which is governed by the triconfluent case of Heun's diff erential equation. In the following we consider situations where the f ourth-order potential has two minima and we calculate the avoided cros sings of its eigenvalue curves in dependence on the asymmetry and the barrier height between the two wells. The results are compared with th ose obtained from an asymptotic approach of the problem for large valu es of the control parameter that governs the barrier height.