GLYPH<230> GLYPH<230> GLYPH<230> - 1.0 - 0.5 0.5 1.0 E - 2 - 1 1 2 Re @ F H E LD , Im @ F H E LD GLYPH<230> GLYPH<230> GLYPH<230> GLYPH<230> GLYPH<230> - 1.0 - 0.5 0.5 1.0 E - 2 - 1 1 2 Re @ F H E LD , Im @ F H E LD E$_{1}$ E$_{2}$ E$_{3}$ E$_{4}$ E$_{5}$ m = const. − 1 − 0 . 759003 − 0 . 273555 +0 . 271144 +0 . 788942 m ( x ) − 0 . 633251 − 0 . 00806737 +0 . 605869 - - FIG. 4: The energy spectrum is derived by solving numerically with respect to the real variable E , the transcendental equations Re[ F ( E )] = 0, Im[ F ( E )] = 0. In the figure we plot Re[ F ( E )] ( solid line ) and Im[ F ( E )] ( dotted line ) as functions of the energy E for the position dependent mass case (left panel) and for the constant mass case (right panel). The eigenvalues (full disks) are given by the points on the E -axis where the two curves cross. The corresponding numerical values are given in table I. TABLE I: Numerical values of the energy eigenvalues (discrete spectrum) for the bound states. The model parameters are: m$_{0}$ = 1, W = 2, a = 10 and L = 2. In the upper row we have the spectrum of the constant mass case, while in lower row we have the position dependent mass case. 19 constant mass we have used the results of [24]. We observe that the in the position dependent mass case the number of bound states decreases relative to the constant mass. In Figures 5 and 6 we provide some example of the (normalized) wave-functions and probability densities both for constant mass case and position dependent mass. Also, comparing Figure 5 with Figure 6 one can infer that in the position dependent mass case the probability density is almost flat in the region inside the potential well as opposed to the constant mass case where, for the highest excited states it oscillates strongly. We note, in the constant mass case (right panel of Fig.4), an eigenvalue corresponding to E = − m = − 1 which merges with the negative continuum. This situation has previously been considered in the literature [24, 29, 30] and has been referred to as super-criticality. Such super-critical states are also called half-bound states and are characterized by the fact that one of the spinor components (the upper, u$_{1}$ , or the lower, u$_{2}$ ) are not strictly normalizable. We show in table I the eigenvalue E = − 1 only because it is a solution of Eq. 52.