The number of energy levels of a particle in an MQW structure

A method is proposed for calculating the number of energy levels of a quantum particle moving in a one-dimensional piecewise constant potential field of a certain kind, which is called a multiple quantum well (MQW) structure. It consists of several layers, namely, of potential wells with a zero potential separated by walls with a potential $U>0$. The external walls have an infinite width. The method is based on a recently obtained multilayer equation that makes it possible to calculate the eigenvalues of the energy $E$ of a quantum particle in an arbitrary MQW structure. The equation has the form $F^*_j(E)=0$, where $F^*_j(E)$ is a rather complex function that is constructed from the given MQW structure and depends on the index j of an arbitrarily chosen bounded layer. The key property is that the functions $F^*_j(E)$ corresponding to the external bounded layers are strictly monotone on the intervals of their continuity. For internal bounded layers, these functions may not be monotone. The formula for the number of energy levels holds in the case of general position. This means that it is not valid for specially chosen well and wall widths (these cases are rather rare and are called resonant). An example is presented in which the number of energy levels does not increase when the potential well is doubled.