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Figure 12:
Stability surface for the relativistic Mathieu potential with a sharpening third harmonic.

Figure 13:
The relativistic Mathieu potential sharpened by its third harmonic.

Figure 14:
Stability surface for the relativistic Mathieu potential with a flattening third harmonic.

Figure 15:
The relativistic Mathieu potential flattened by its third harmonic.

Figure 16:
Stability surface for the relativistic Mathieu potential with a deltafunctionapproximating potential
.

Figure 17:
Terms in the coefficient matrix plus one solution for the potential
.
The potential is fairly flat on the bottom with a dent, while on top the peak approximating the delta function is quite conspicuous. The result is that negative energy solutions see a mild reflecting corrugation while positive energy solutions see an ever rising barrier through which they can tunnel.

Figure 18:
Contour map for the relativistic Mathieu potential with a delta function approximating potential
.
The squares outline the regions whose perspective view is shown in Figure 16.

Finding a relativistic version of the KronigPenney model has created some consternation in the physics literature. The contour map of Figure 18 shows the result of approximating the delta function by the Fourier series
.
A notable discrepancy develops between solutions with negative energy and those of positive energy.
Although the average potential in this approximation is zero, it reaches an appreciable maximum while retaining a minimum close to zero. After all, that is the desired result, but the environment which it presents to wave functions of opposite energies is quite different. Instead of a similar mass shell for the two alternatives, the shell has a negligible presence around the high peaks, while constituting a uniform barrier along its base.
The massdominated region in Figure 18 rises at a high angle on the left because of the relation of the factor 6/11 in the potential definition to the step size in the contour plotter. effects can be seen on the right but they are barely visible.
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