Subsystem-Resolved Spectral Theory for Quantum Many-Body Hamiltonians

We study spectral properties of quantum many-body Hamiltonians through a subsystem-based framework. Given a Hamiltonian of the form $H = \sum_{X \subseteq \Lambda} \Phi(X)$ acting on a tensor product Hilbert space, we associate to each subset $S \subseteq \Lambda$ a subsystem Hamiltonian $H_S$ and its spectrum $\mathcal{S}(S) = \sigma(H_S)$. This produces a family of spectra indexed by subsystems, allowing spectral data to be organized according to interaction structure. We show that subsystem Hamiltonians admit local approximations: $H_S$ can be approximated by operators supported on finite neighborhoods with an error bounded by $\|H_S - H_{S,r}\| \le |S| e^{-\mu r} \|\Phi\|_\mu$. As a consequence, subsystem spectra are stable under truncation in the sense that $d_H(\mathcal{S}(S), \sigma(H_{S,r})) \le |S| e^{-\mu r} \|\Phi\|_\mu.$ We then prove that for disjoint subsets $S_1, S_2 \subseteq \Lambda$, the subsystem spectrum is approximately additive: $d_H\big(\mathcal{S}(S_1 \cup S_2), \mathcal{S}(S_1) + \mathcal{S}(S_2)\big) \le (|S_1| + |S_2|) e^{-\mu D} \|\Phi\|_\mu,$ where $D = d(S_1, S_2)$. In the finite-range case, this relation becomes exact. The results show that spectral properties reflect the locality of interactions not only at the level of operators, but also at the level of spectra. The framework provides a way to study many-body systems in which interaction geometry directly shapes spectral behavior.