Given a smooth radially symmetric potential $V = V(|x|)$ in the unit ball $B_1\subset\mathbb{R}^n$, consider the eigenvalue problem \begin{equation} \begin{cases} \Delta \varphi + V \varphi = \lambda \varphi & \text{ in } B_1\\ \varphi = 0 & \text{ on } \partial B_1, \end{cases} \end{equation} where $\Delta$ is the Laplacian in $\mathbb{R}^n$.
Assume that all eigenfunctions as above are radially symmetric for some $\lambda \in \mathbb{R}$.
Can we prove that these solutions are also unique?
The eigenfunctions won’t be radially symmetric in general. Using separation of variables, you can write $\phi=f(r)g(\omega)$ with $\omega$ the coordinates over the unit sphere. $g$ is therefore a spherical harmonic: $$ \Delta_{S^{n-1}}g=-l(l+n-2)g $$ with $l\in\mathbb N$. Your radial problem is now: $$ f’’+\frac{n-1}{r}f’-\frac{l(l+n-2)}{r^2}f+Vf=\lambda f $$ and using the using change of variable $h=r^{(n-1)/2}f$, you reduce the problem to a 1D Schrödinger equation: $$ h’’-\frac{\left(l+\frac{n-1}{2}\right) \left(l+\frac{n-3}{2}\right)}{r^2}h+Vh=\lambda h $$ with the norm given by the usual 1D $L^2$ norm: $$ ||h||^2=\int_0^\infty|h|^2dr $$ You can interpret the additional potential $-\frac{\left(l+\frac{n-1}{2}\right) \left(l+\frac{n-3}{2}\right)}{r^2}$ as the centrifugal potential.
The usual results for 1D system therefore apply like the usual Wronskian argument. Generally, it comes from the Sturm-Liouville theory on the half line, which guarantees the nondegeneracy of the eigenvectors characterized by their number of nodes.
Hope this helps.