Complete spectrum of the Robin eigenvalue problem on the ball

We investigate the following Robin eigenvalue problem \begin{equation*} \left\{ \begin{array}{ll} -\Delta u=\mu u\,\, &\text{in}\,\, B,\\ \partial_\texttt{n} u+\alpha u=0 &\text{on}\,\, \partial B \end{array} \right. \end{equation*} on the unit ball of $\mathbb{R}^N$. We obtain the complete spectral structure of this problem. In particular, for $\alpha>0$, the first eigenvalue is $k_{\nu,1}^2$ and the second eigenvalue is $k_{\nu+1,1}^2$, where $k_{\nu+l,m}$ is the $m$th positive zero of $kJ_{\nu+l+1}(k)-(\alpha+l) J_{\nu+l}(k)$. Moreover, when $\alpha\in(-l,1-l)$ with any $l\in \mathbb{N}$, one has $l$ negative (strictly increasing) eigenvalues $-\widehat{k}_{\nu+i,1}^2$ with $i\in\{0,\ldots,l-1\}$ where $\widehat{k}_{\nu+l,1}$ denotes the unique zero of $\alpha I_{\nu+l}(k)+lI_{\nu+l}(k)+kI_{\nu+l+1}(k)$; while, for $\alpha=-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue.