We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.
We solve the static isoperimetric problem underlying the Mandelstam-Tamm bound. Among one-dimensional confining potentials with a fixed spectral gap, we prove that the harmonic trap is the unique maximizer of the ground-state position variance. As a consequence, we obtain a sharp geometric quantum speed-limit bound on the position-position component of the quantum metric, and we give a necessary-and-sufficient condition for when the bound is saturated. Beyond the exact extremum, we establish quantitative rigidity. We control the Thomas-Reiche-Kuhn spectral tail and provide square-integrable structural stability for potentials that nearly saturate the bound. We further extend the analysis to magnetic settings, deriving a longitudinal necessary-and-sufficient characterization and transverse bounds expressed in terms of guiding-center structure. Finally, we outline applications to bounds on static polarizability, limits on the quantum metric, and benchmarking of trapping potentials.
Abstract We resolve the static isoperimetric problem underlying the Mandelstam–Tamm limit: among one-dimensional confining potentials at fixed gap Δ, the harmonic trap uniquely maximizes Var 0 ( x ) , yielding the exact geometric quantum speed limit g x x ⩽ ( 2 m Δ ) − 1 with an iff criterion. Beyond the extremum we prove quantitative rigidity (Thomas–Reiche–Kuhn-tail and structural L 2 control), extend to magnetic settings (longitudinal iff; transverse guiding-center bounds), and note applications to static polarizability, quantum-metric limits, and trap benchmarking.
We study the center-of-mass observable in one-dimensional many-body systems with translation-invariant interactions and extend the harmonic-rigidity mechanism from the one-body setting to an interacting many-body problem. We prove a sharp upper bound on the ground-state center-of-mass fluctuation in terms of the active spectral gap associated with the center-of-mass probe, and show that this bound does not require any positivity assumption on the ground state. In the positivity class, we characterize the equality case completely. Exact saturation occurs if and only if the external one-body traps are harmonic with a common frequency, while the interaction may remain arbitrary within the translation-invariant class. We also identify a natural rigidity defect measuring deviation from the harmonic extremal situation and prove quantitative near-saturation estimates controlling both the variance deficit and the spectral weight outside the first active shell. In this way, the paper establishes harmonic confinement as the unique static extremizer for the rigid interacting center-of-mass mode at fixed active gap.
This paper examines the averaged quantum metric for multiparametric random-matrix Hamiltonians in the vicinity of an integrable point, revealing a local conical geometry that arises from specific spectral conditions. The central object is the eigenstate-averaged quantum geometric tensor, which defines a Riemannian metric on the parameter space of the Hamiltonian family. For the two-parameter model studied by Sharipov and colleagues, the averaged metric components admit the asymptotic expansion Grr∼a/rGrr∼a/r and Gϕϕ∼brGϕϕ∼br as the integrable point is approached, with aa and bb positive. By embedding this surface of revolution into Euclidean three-space, the author demonstrates that the metric is locally isometric to a right circular cone whose opening angle satisfies tanα=b/atanα=b/a. Thus, the ratio of the two leading coefficients determines the cone's aperture, linking geometry directly to the relative weight of radial and angular perturbation matrix elements at low frequencies. The paper then turns to the spectral origin of this conical singularity. Using the spectral representation of the metric, the radial component GrrGrr is expressed as an integral over the spectral density of squared matrix elements divided by ω2ω2. The divergence 1/r1/r requires that this density retains finite weight at frequencies on the order of the mean level spacing, a condition characteristic of Poissonian spectra where level repulsion is absent. However, if a hard spectral gap ΔΔ is introduced in the matrix-element spectrum, the integral acquires a lower cutoff, yielding a bound Grr≤C/Δ2Grr≤C/Δ2 that removes the singularity entirely. This regularization mirrors classical mechanisms in which spectral gaps suppress low-order resonances and confine instability. Between the conical near-integrable regime and the hemispherical geometry of the fully ergodic phase, an intermediate scaling window is identified with the non-ergodic extended phase, where multifractal estimates suggest a smooth interpolation. The discussion draws structural parallels to classical theories of thin versus thick chaos, where low-frequency spectral weight determines whether local instability remains confined or becomes transport-effective. However, the author emphasizes that the quantum calculation only establishes kinematic consistency, not a quantitative dictionary with classical measures of confinement. Two propositions are rigorously derived: the local conical embedding follows directly from the metric expansion, and a hard spectral gap bounds the radial component. All conclusions are strictly limited to the specific ensembles studied, and the paper makes no claim of universality beyond those models.
We establish a necessary condition for quantum heuristics to achieve nontrivial computational advantage relative to a classical problem generator A. We prove that any unitary procedure whose dynamics nearly commute with A is spectrally confined: its ability to transfer amplitude across separated spectral sectors is bounded strictly by the commutator defect divided by the spectral gap. This yields a quantitative operator-theoretic obstruction to global state-space exploration. Consequently, repeated application of such procedures (e.g., in deep variational circuits) produces apparent algorithmic activity without generating meaningful information gain. We classify this regime as a "degenerate quantum heuristic". We apply this framework to Variational Quantum Eigensolvers (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), demonstrating that circuit depth cannot overcome transversality failure. The theoretical results are fully closed within trace-ideal (S₂) operator theory. We extend the spectral confinement bound to the entire resolvent geometry, proving that degenerate heuristics preserve the Birman-Schwinger kernel and regularized Fredholm determinants (det₂), thereby mathematically obstructing any computational phase transition or new eigenvalue emergence. Finally, we demonstrate that genuine quantum speedup requires structural noncommutativity, which can be deterministically enforced via an Ω-Σ dissipative control layer. The manuscript includes complete finite-dimensional numerical validation, exhibiting the exact quantitative rigidity between commutator defect, spectral transport, and resolvent deformation.
We establish a sharp quantitative stability theorem for the von Neumann entropy under a natural class of convex constraints induced by orthogonal block decompositions of finite-dimensional Hilbert spaces. Given a block–convex set of block-diagonal quantum states, we prove that any state whose entropy is within ε of the minimal value must lie within O(ε)O(\sqrt{\varepsilon})O(ε) in trace norm of the manifold of entropy minimizers. The stability exponent is shown to be optimal, and purely quadratic stability is characterized precisely in the confining (fixed-support) regime. The proof relies on a canonical decomposition of entropy into classical (Shannon) and conditional components, combined with sharp relative entropy estimates and geometric properties of the marginal polytope. As an application, we obtain quantitative non-concentration bounds for spectral measures associated with finite spectral decompositions of self-adjoint operators. The framework is entirely finite-dimensional, independent of arithmetic or dynamical assumptions, and provides a general geometric stability principle for entropy minimization under structured convex constraints.
Abstract We formulate a minimal operator-theoretic criterion—the Spectral Confinement Class (SCC)—that any quantum theory exhibiting confinement-like behavior in four space-time dimensions should satisfy. This criterion consists of three necessary spectral properties: (i) a strict spectral gap above the vacuum, (ii) exponential clustering of connected correlation functions, and (iii) the absence of massless non-singlet excitations with respect to an internal compact symmetry (e.g., color). Within a general multichannel lattice framework, we prove that gate-induced spectral localization provides an explicit mechanism realizing SCC. Under minimal structural assumptions—local spectral rigidity with a gap m₀>0 uniform in the channel dimension N, bounded finite-range interactions, and a structural gate condition preventing spectral percolation—we derive a uniform lower bound on the spectral gap Δ ≥ m₀ - 2·z(d)·g·Vₘₐₓ where z(d)=2d is the lattice coordination number. The bound is uniform in system volume and channel dimension and exhibits a dimensional penalty: for fixed coupling g>0, the admissible gap window shrinks with increasing spatial dimension. We emphasize that SCC is necessary but not sufficient for confinement in the Yang–Mills sense. Its role is that of a spectral admissibility filter: candidate confining theories should satisfy SCC at the operator level; failure to do so presents obstacles to confinement regardless of gauge structure. In this sense, SCC functions as a spectral admissibility filter: candidate confining theories must satisfy it at the operator level, independently of gauge-theoretic or geometric mechanisms. Keywords Primary Keywords: spectral confinement class • spectral gap • gate condition • dimensional penalty • Yang–Mills theory • confinement • lattice Hamiltonians • multichannel systems • spectral localization Secondary Keywords: exponential clustering • local spectral rigidity • excitation projectors • coordination number • four-dimensional quantum field theory • gauge theories • Wilson loops • operator theory • variational methods • mass gap problem Mathematical Subject Classification (MSC 2020): Primary: 81T13 (Yang-Mills and other gauge theories), 81Q10 (Selfadjoint operator theory in quantum mechanics) Secondary: 82B20 (Lattice systems), 47A10 (Spectrum and spectral theory)