Landau levels in rotationally invariant gauge

Question

I try to find wave-function of electron in external constant magnetic field in gauge $$A=\frac{B}{2}(-y,x,0).$$ I substitute anzats, $\psi=e^{-i\omega t}e^{ip_zz}F(x,y)$. Then, I rewrite equation in polar coordinates and obtain (I write only differential operator): $$\partial_r^2+\frac{1}{r}\partial_r+\frac{1}{r^2}\partial_{\theta}^2-ieB\partial_{\theta}-\frac{e^2B^2}{4}r^2+\Omega,$$ where $\Omega=2m\omega-p_z^2+eBs$ and $s=\pm 1$. Then, I use $F(r,\theta)=f(r)e^{i\theta n}$, $$\partial_r^2+\frac{1}{r}\partial_r-\frac{n^2}{r^2}+eBn-\frac{e^2B^2}{4}r^2+\Omega.$$ To solve this equation, I changle variables, $\xi=r^2$ and find $$\partial_{\xi}^2+\frac{1}{\xi}\partial_{\xi}+\frac{eBn+\Omega}{4\xi}-\frac{n^2}{4\xi^2}-\frac{(eB)^2}{16}.$$ Using asymptotes, I know that $$f(r)=\rho(r)e^{-\xi/2}\xi^{n/2}.$$ Finally, equation for $\rho(r)$ is $$\xi\rho''+(n+1-\xi)\rho'+\frac{\rho}{2}\left(\frac{\Omega+eBn}{2}-\frac{(eB)^2\xi}{8}+\frac{\xi}{2}-n-1\right)=0.$$ I know that solution of this equation should be Laguerre polynomial up to factor with exp function. Using Wolfram Mathematica, I see that solution should be $$\exp\left(\frac{\xi}{2}+\frac{eB\xi}{4}\right)L_{n}^{(\Omega-eB)/(2eB)}\left(\frac{eB\xi}{2}\right).$$ Moreover, Mathematica says me that confluent hypergeometric function is also the solution.

I do not understand several facts:

1. How to rewrite equation for $\rho$ in the "canonical" form and explicitly see that solutions are Laguerre polynomials with exp prefactor?
2. How can I choose the correct solution? It seems that both functions, Laguerre polynomial and confluent hypergeometric function are related to Hermite polynomials. I compare with Hermtie because I know that the solution of electron in external magnetic field in gauge $A=B(-y,0,0)$ is Hermite polynomial.
3. What should I do to find spectrum? It seems that all the information of spectrum should be encoded in upper index of Laguerre polynomial. So, my guess is that for specific values
4. Where I can find normalization factor? To be honest, I do not want to perform calculation for it

Answer 1

The answer is given in the recent paper by Orion Ciftja, "Detailed solution of the problem of Landau states in a symmetric gauge".

Starting from the equation after variables separation, I have obtained $$\partial_r^2+\frac{1}{r}\partial_r-\frac{n^2}{r^2}-\frac{e^2B^2}{4}r^2+\Omega(n),$$ where I now absorb term $eBn$ into $\Omega$. Introducing magnetic length, $l^2=1/(eB)$ and new variable $\xi=r^2/(2l_0^2)$, one can obtain $$\xi\partial_{\xi}^2+(n+1-\xi)\partial_{\xi}+\left(\lambda-\frac{n+1}{2}\right),$$ where $\lambda=\Omega l_0^2$. This equation is nothing more than confluent hypergeometric function equation and with arguments of finiteness of w-f at infinity, it is easy to find the spectrum and normalization factor (both are given in the paper).