= Ground-space perturbation bound
{title2=$\delta\nu-\delta^2/(g-\delta)\leq\lambda_{\min}(H+\delta V)\leq\delta\nu$}
Suppose $H\geq0$ has ground energy zero and positive <spectral gap> at least $g$, and $0\leq V\leq I$. Let $\nu$ be the minimum of $V$ restricted to the <ground space>. Splitting a normalized vector into ground and excited components bounds its energy below by $\delta\nu+(g-\delta)y^2-2\delta y$, with $y$ the excited norm. Completing the square yields the displayed bound for $0<\delta<g$. The error is controlled by the gap and cannot be treated as a uniform $O(\delta^2)$ when $g$ closes.
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