For completeness, I will sketch the solution of a particle in an infinite circular well first and then get to my question. I apologize in advance since the introduction is standard undergraduate quantum mechanics.
Let us consider the problem of a particle in an infinite circular well. This is described by the potential
$$V(r) = \begin{cases}0&r \leq R \\ \infty&r> R \end{cases}$$
The Schrödinger equation then splits into a radial and an angular part and we can write the eigen-function as
$$\psi(r,\theta) = u(r) e^{i l \theta}$$
where $l = 0,\pm 1,\pm2,..$ due to the single-valuedness of the wave-function. The radial part of the equation is
$$\left(\frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} - \frac{l^2}{r^2} + \frac{2mE}{\hbar^2}\right)u(r) = 0$$
with the boundary condition: $u(R) = 0$. The solutions to this are given by the regular Bessel functions (we discard $Y_l$'s since they blow up at r = 0)
$$u_{n,l}(r) = J_{l}\left(\alpha_{n,l}\frac{r}{R}\right)$$
where $\alpha_{n,l}$ is the $n^{th}$ zero of $J_{l}(r)$.
In this way, we can construct normalizable wavefunctions
$$\psi_{n,l}(r,\theta) = \mathcal{N}_{n,l}J_{l}\left(\alpha_{n,l}\frac{r}{R}\right) e^{il\theta}$$
where $\mathcal{N}_{n,l} = \frac{1}{\sqrt{\pi}R |J_{l+1}(\alpha_{n,l})|}$. These are simultaneous eigenfunctions of the Hamiltonian and of angular momentum, with energy $E_{n,l} = \frac{\hbar^2 \alpha_{n,l}^2}{2mR^2}$ and angular momentum = $l \hbar$.
Now, my question is the following: Suppose I want to evaluate
$$(\partial_x \pm i \partial_y) \psi_{n,l} = e^{\pm i \theta}\left(\frac{\partial}{\partial r} \pm \frac{i}{r}\frac{\partial}{\partial \theta} \right)\psi_{n,l}(r,\theta)$$
Using the properties of Bessel functions, I find that this evaluates to
$$\mp\mathcal{N}_{n,l}\frac{\alpha_{n,l}}{R}e^{i(l\pm 1)\theta}J_{l\pm 1}\left(\alpha_{n,l}\frac{r}{R} \right)$$
This is no longer an eigenfunction since $J_{l\pm 1}\left(\alpha_{n,l}\frac{r}{R} \right) \neq 0$ when $r = R$.
So, is there any way of constructing eigenfunctions with fixed angular momentum, $l$, such that when I act on them by the operator(s) $\partial_x \pm i \partial_y$, I get another eigenfunction?