Complete spectrum of the Robin eigenvalue problem on the ball

We investigate the following Robin eigenvalue problem \begin{equation*} \left{ \begin{array}{ll} -Δu=μu,, &$\text{in}$,, B,\ \partial_$\mathtt{\text{n}}$ u+αu=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 $k{J}_{\nu +l+1}(k)-(\alpha +l)J_{\nu +l}(k)$. 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 $i\in \{0,\dots ,l-1\}$ where ${\widehat{k}}_{\nu +l,1}$ denotes the unique zero of $\alpha I_{\nu +l}(k)+l{I}_{\nu +l}(k)+k{I}_{\nu +l+1}(k)$; while, for $\alpha =-l$, besides $l$ negative (increasing) eigenvalues, $0$ is also an eigenvalue.