Question originally posted here, but I'm more likely to get an answer on MO. I'm looking for a reference on the following topic, should one exist. (Or a firm "stop looking and do it yourself".)

Let $H(s)=(1−s)H(0)+sH(1)$ where $s\in[0,1]$ and $H(0)$,$H(1)$ are Hermitian. Let $\lambda_0(s) < \lambda_1(s)$ be the two lowest eigenvalues of $H(s)$ and let $\gamma(s) = \lambda_1(s) - \lambda_0(s)$.

Define $I_G=\{s\in[0,1]:γ(s)≤G\}$. I'm looking for bounds of the form $\int d \mu(I_G) \leq f(G,H(0),H(1))$. Has this question previously been studied and, if so, does anyone have a reference?