On energy levels of a particle in a comblike structure

Abstract

Previously, two similar one-dimensional problems have been considered [1‐4] concerning the eigenvalues of the effective refractive index of a multilayer waveguide and the energy eigenvalues of a quantum particle in a piecewise constant potential field. In these problems, the main attention is focused on an equation that determines the eigenvalues of the indicated parameters for an arbitrary number of layers (in the first problem) and of the segments of constant potential (which are also considered as layers) in the second case. The form of this equation was guessed by the author during the calculation of multilayer waveguides, which was performed under the guidance of A.A. Ma œ er. This equation is referred to as the “multilayer” equation, since it is obtained by equating to zero the so-called multilayer function, which depends on the energy E of a particle. The outer layers of the layered structures under consideration are assumed to be infinite. If the total number of layers (including the unbounded outer layers) is n , the multilayer function F j ( E ) and, consequently, the multilayer equation for the same layered structure can be written in n ‐ 2 ways. Each of these ways corresponds to the choice of a distinguished j th layer of a finite width. For certainty, let us concentrate below on the second problem, that is, the problem of energy eigenvalues of a particle in a multilayer structure. In the case of three layers, the multilayer equation reduces to the well-known equation [5] for energy eigenvalues of a particle in a square potential well. In the case of a greater number of layers, the results obtained with the use of the multilayer equation are in complete agreement with the data obtained using numerical methods. Note also that the multilayer equation can be used not only in the problem under consideration, but also in other problems and situations as was done, for example, in [6]. Our method of determining the number of eigenvalues can also be extended to these