MD Nahidul Hasan Sabit

arXiv2026

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 $H = \sum_{X \subseteq Λ} Φ(X)$ acting on a tensor product Hilbert space, we associate to each subset $S \subseteq Λ$ a subsystem Hamiltonian $H_S$ and its spectrum $\mathcal{S}(S) = σ(H_S)$. 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 $\|H_S - H_{S,r}\| \le |S| e^{-μr} \|Φ\|_μ$. As a consequence, subsystem spectra are stable under truncation in the sense that $d_H(\mathcal{S}(S), σ(H_{S,r})) \le |S| e^{-μr} \|Φ\|_μ.$ We then prove that for disjoint subsets $S_1, S_2 \subseteq Λ$, the subsystem spectrum is approximately additive: $d_H\big(\mathcal{S}(S_1 \cup S_2), \mathcal{S}(S_1) + \mathcal{S}(S_2)\big) \le (|S_1| + |S_2|) e^{-μD} \|Φ\|_μ,$ 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.

quant-ph↗

arXiv2026

Stability of Partitions Induced by Nearest-Center Assignment Under Perturbations

We study clustering through the partitions it induces on a finite labeled set $[n]=\{1,\dots,n\}$, and analyze how these partitions change under perturbations of a point configuration $X=(x_1,\dots,x_n)\in(\mathbb{R}^d)^n$. We equip the space of partitions $Π_n$ with a normalized pairwise disagreement metric $d(\cdot,\cdot)$, and define the stability radius $r(X,A)=\sup\{\varepsilon\ge0: A(X')=A(X)$ whenever $|X-X'\|\le\varepsilon\}$, where $\|X-X'\|=\max_i\|x_i-x'_i\|$. Our main results concern nearest-center assignment with fixed centers $\{c_1,\dots,c_k\}\subset\mathbb{R}^d$. For each point, we define the margin $γ_i=\min_{j\ne\ell_i}(\|x_i-c_j\|-\|x_i-c_{\ell_i}\|)$ and $γ_{\min}=\min_iγ_i$, where $\ell_i$ denotes the assigned center. We show that if $\varepsilon<γ_{\min}/2$, then no assignments change under perturbation and hence $A(X')=A(X)$. Conversely, any point that changes its assigned center must satisfy $γ_i\le2\varepsilon$, showing that instability is localized near decision boundaries. We construct configurations in which arbitrarily small perturbations $\|X-X'\|\le\varepsilon$ alter the induced partition, demonstrating that the margin condition is sufficient but not necessary for stability. We further extend the framework to a discrete-time setting, showing that if $\sum_t δ_t<r(X(0),A)$, then $A(X(t))=A(X(0))$, and we give a probabilistic bound on $\mathbb{E}[d(A(X),A(X'))]$ in terms of tail probabilities relative to $γ_i$. This framework identifies the margin as the key quantity governing both worst-case and average stability, and provides explicit conditions under which clustering-induced partitions remain invariant in a fixed-center Euclidean model.

math.CO↗