gauge has been given recently in Ref. \[\], and this problem was first highlighted in Ref. \[\], where the author presented a more general gauge condition to replace the simple wave function relation in Eq. (\[psi\_transform\]) when degeneracy is present. We will return to this issue later.

For the physical samples considered in this paper, the system is finite, and the electron is confined to some extent, so there must be some finite number of $\ell$’s that constitute the degeneracy number for a given Landau level (since the wave functions grow outward with increasing $|\ell |$). The standard argument (already provided above) is that the expected degeneracy $G$ of the $LLL$, given a sample of area $A$, is the ratio of the total magnetic flux $\Phi$ to the flux quantum, $$G=\frac{\Phi}{\Phi_0}=\frac{BA}{h/e}.
    \label{degeneracy}$$ As demonstrated later, this is not quite true, as sample edges confine the electron and result in higher energies for states near the edges.

Also note that the difference between centres of two successive $\ell$ eigenstates $\Psi_{n_r,\ell}$ and $\Psi_{n_r,\ell+1}$ close to the edge will be much less than the spread $l_B$ \[see Eq. (\[lll\_wavefunction\])\] and will decrease like $\sim\sqrt{|\ell|}-\sqrt{|\ell + 1 |}$, implying that the corresponding eigenstates will 