In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := - \Delta_g + |A|^2$ (usually called Jacobi operator) with Dirichlet boundary conditions, where $g$ is the metric on $\Sigma$. If we know that $\Sigma$ is a rotationally symmetric, with polar coordiantes $(r,\theta)$ and $g = dr^2 + h(r) \, d\theta^2$, the potential $|A|^2$ is a radial function.
Are there standard references for studying the Fourier mode decomposition of the eigenvectors of $J$ in this $N$-dimensional but symmetric setting? In particular I am interested in the following questions:
$\bullet$ Is the $0$th-mode eigenspace unidimensional? Are the eigenspaces associated to radial eigenfunctions unidimensional for all modes?
$\bullet$ Is the eigenvalue of the radial eigenfunctions in the $k$-th mode always less than the one in the $(k+1)$-th mode?
$\bullet$ In the same mode, is the eigenvalue of a radial eigenvector less than the eigenvalue of the eigenvectors which include angular variables?
$\bullet$ Let us call radial index of $J$ the sum of the dimensions of all the eigenspaces of $J$ on $[c,d]\times \mathbb{S}^{N-1}$ associated to the negative eigenvalues and only radial eigenvectors. Can one easily compare the radial index of $J$ and the Morse index of the 1-dimensional operator $J_1 := - \partial_r^2 + |A(r)|^2$ on $[c,d]$?
I am new in this field. Those questions seem natural (or naive) to me and I think they could be easily answered (under the technical assumptions and usual hypotheses in the topic), if specific results on the stability of rotationally symmetric minimal hypersurfaces are known. I do not know where to study the subject, theory and examples of computations.
Finally, do the answers of the above questions change totally if one consider instead weighted minimal hypersurfaces with weight $e^{f(r)}$? In that case the functional spaces are weighted and the operator takes the form $J + b(r) \, \partial_r$, with $b$ a radial function but not necessarily non-negative (the function $b$ I have in mind is smooth, bounded and has two horizontal distinct asymptotes).
The general idea is to ask if the study of the radial equation and radial eigenvectors give bounds or relations for the computation of the index of instability of rotationally symmetry (possibly weighted, with radial weigth) minimal hipersurfaces.
