= Spectral filtering of Hamiltonian terms
{title2=$g_Z=\int w(t)e^{itH}h_Ze^{-itH}\,dt$}
A real nonnegative normalized filter $w$ defines $g_Z=\int w(t)e^{itH}h_Ze^{-itH}\,dt$. In an energy <eigenbasis>, it multiplies each matrix element by $\widehat w(E_i-E_j)$. A Fourier cutoff at the <spectral gap> removes couplings between a unique <ground state> and excitations. The filtered terms still sum to $H$, because the total Hamiltonian is unchanged by its own <unitary time evolution>. Each term commutes with the ground projector, but need not be minimized there.
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