Could someone please give me a reference for the eigenvalues and eigenstates of operators related to the angular momentum of a spinless, non-relativistic 2-D quantum particle?


In particular, I'm interested to know the eigenvalues of the operator that takes the angular momentum eigenstates


$\varphi\_{q,k}(\mathbf{x}) := \sqrt{\frac{q!}{(k+q)!}} \left[ \sqrt{\frac{b}{2}}(x+iy)\right]^{k} L\_q^{(k)}\left(\frac{b \lvert \mathbf{x} \rvert^2}{2}\right)\sqrt{\frac{b}{2\pi}}e^{-\frac{b \lvert \mathbf{x} \rvert^2}{4}} ,\quad q\in \mathbb{Z}\_+,\ k \in \mathbb{Z}\_+ -q$


to $\varphi\_{q,k+1}$ or $\varphi\_{q,k+2}$, with the angular momentum operator defined as $-i(x\,\partial/\partial y - y\, \partial/\partial x)$ in $L^2(\mathbb{R}^2)$.



Spectrum of an angular-momentum related operator
Maybe these references will be helpful: 


<http://scitation.aip.org/content/aip/journal/jmp/28/2/10.1063/1.527672> (Coherent angular momentum states for the two‐dimensional oscillator, by A.J. Bracken, 
D.S. McAnally and O.A. Odundun)


<http://arxiv.org/abs/1401.4985> (Lost and found: the radial quantum number of Laguerre-Gauss modes, by E. Karimi et al.)


<http://www.redalyc.org/articulo.oa?id=45326937008> (An approach to ladder operators for the two-dimensional harmonic oscillator, by J. Lopez-Bonilla and
L. Cruz-Beltran).


