Spectral filtering of Hamiltonian terms ($g_Z=\int w(t)e^{itH}{h}_Z{e}^{-itH}dt$)

A real nonnegative normalized filter $w$ defines $g_Z=\int w(t)e^{itH}{h}_Z{e}^{-itH}dt$. In an energy eigenbasis, it multiplies each matrix element by $\hat{w}(E_i-E_j)$. 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.

In a telescoping local-shell decomposition, split the spectral filter integral at a time proportional to the shell distance. A Lieb-Robinson bound controls short times, while the two-sided almost-exponential spectral filter tail controls long times. Polynomial shell-weight growth then gives the displayed decay. The estimate is asymptotic at large distance; small shells retain the elementary operator-norm bound. The evenization of a nonnegative bandlimited filter can supply the two-sided tail from a one-sided existence statement.

If each local term is already minimized by the global ground state, subtract its minimum $a_Z$ to obtain a positive operator. Nonnegative normalized spectral filtering of Hamiltonian terms preserves positivity and leaves the ground eigenvalue $a_Z$ unchanged. Thus the same state minimizes every filtered term. This preserves frustration freeness but does not create it for an initially frustrated decomposition.

The condition $[g_Z,P_0]=0$ makes a unique global ground state an eigenvector of every filtered term, but its termwise eigenvalues can exceed the corresponding minima. For example $h_{12}=|1⟩⟨1{|}_1+|0⟩⟨0{|}_2$ and $h_{23}=|1⟩⟨1{|}_3+2|1⟩⟨1{|}_2$ commute with their sum. Their sum is uniquely minimized by $|000⟩$, yet $h_{12}$ is lower on $|010⟩$. Filtering leaves these commuting terms unchanged, so it cannot force frustration freeness.

Filter a fixed local term using a nested sequence of neighbourhood Hamiltonians $H_s$, and call the results $G_s$. The differences $G_s-G_{s-1}$ give a shell decomposition. When $H_0=h_Z$ and $H_D=H$, normalization gives $G_0=h_Z$ and telescoping gives $g_Z=h_Z+{\sum }_{s=1}^D(G_s-G_{s-1})$. Individual shell terms need not commute with the full ground projector.