The width of the minimum gap of an interpolated matrix

Question

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?

Answer 1

A bound with a function $f(G)$ that does not depend on $H(0)$ and $H(1)$ seems unlikely. In a typical situation the function $\gamma(s)$ can be extended to the complex plane and $\gamma(z)$ vanishes at a point $z=s_0+i\delta$ for some $s_0\in(0,1)$ close to the real axis. The inverse gap $1/\gamma(s)$ will show a peak at $s_0$ with height $1/\gamma_{\rm min}=1/\gamma(s_0)\gg 1$ and width $\delta\ll 1$. The corresponding estimate for $f(G)$ would be $f(G)=0$ for $G<\gamma_{\rm min}$ and $f(G)=\delta$ for $\gamma_{\rm min}<G\ll 1$.

The problem of the estimation of $f(G)$ has been studied extensively in the context of adiabatic quantum computation (see for example General error estimate for adiabatic quantum computing).

A parameterization of $\gamma(s)$ that is used in those studies is $$\gamma(s)=[(s-s_0)^{2a}+\gamma_{\rm min}^b]^{1/b},$$ with integer $a,b$, which has $\delta=\gamma_{\rm min}^{b/2a}$. This gives a useful functional form for $f(G)$ in terms of a few parameters.