Problem: Let $M = D + \sigma uu^\top$, where $D$ is a positive semidefinite diagonal matrix, in particular, $0 = d_1 < d_2 \leq d_3 \leq \ldots \leq d_n$ and $D = \operatorname{diag}(d_i)$. Also, let's consider $\sigma > 0$. Finally, let $\lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_n$ be the eigenvalues of $M$.
I am interested in a non-trivial lower bound for $\lambda_2$, but if there are more general results then those are welcome too.
Research: I was doing some research on this problem, and found Weyl's inequality, and some other related questions like these:
[1] eigenvalues of "diagonal matrix $+$ rank one matrix"
[2] maximum eigenvalue of a diagonal plus rank-one matrix
I could find that in general $d_i \leq \lambda_i$. However, I noticed the following by running some experiments on MATLAB: when $d_i = d_{i+1}$, then $\lambda_i = d_i$, i.e., the eigenvalue is not changed; but when $d_i < d_{i+1}$ then $d_i < \lambda_i$, i.e., the eigenvalue increases. I was wondering if one could get a lower bound like this $d_i + \epsilon_i \leq \lambda_i$ whenever $d_i<d_{i+1}$. My intuition was that maybe $\epsilon_i$ is related to the gap $(d_{i+1} - d_i)$ and the $\ell_2$-norm of $u$ but I was not able to get something.