2.3. SOLUTION OF THE PROBLEM

All that remains in solving the problem is to solve the finite set of Harper equations using numerical solution (computational) techniques. In other words determining the energy values in each of the equations in this finite set. It was solved byDougles R. Hofstadter in 1976.

Now that we have the solutions, let’s look at their dependence on external magnetic field (alpha values) . Let’s consider alpha values a rational number. In other words, the magnetic field that we apply to the system should only be in rational values. In this case, if the energy solutions corresponding to each alpha values