Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each entry $G_{i,j} \sim {\cal N}(0,1)$. Let $\epsilon>0$ be some parameter. Does there exist a function $f(\epsilon)$ such that the following holds with high probability:

The eigenvalues $\lambda_1 \geq \lambda_2 \geq ... \geq \lambda_n$ of $A+ \epsilon G$ have pairwise spacing at least $f(\epsilon)$, i.e. $\lambda_i - \lambda_j \geq f(\epsilon)$, for all $i<j$.

Note that since some eigenvalues may appear with multiplicity, variational techniques may not work here, due to non-existence of the second order derivative.

In addition, some eigenvalues can be at arbitrarily close distance to each other so that while variational techniques collapse, degenerate perturbation analysis is not completely accurate either.