Energy levels of the magnetic monopole in the Prasad-Sommerfield limit

By analyzing the asymptotic behavior of the 't Hooft magnetic monopole in the Prasad-Sommerfield limit we see that there is no uniqueness of the finite-energy solution due to one degree of freedom recovered. Moreover, this degree of freedom should be quantized in order to get a finite-energy solution. Having at our disposal the asymptotic form of solutions at $r\to 0$ and $r\to \infty$, we match them up at some finite value $r=a$ in a $C^0$-class way, then insert the result into the energy integral and find the minimum with respect to the parameter $a$. Thus we find the approximate energy levels of the monopole in the Prasad-Sommerfield limit.