Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.20353

Computing Spectral Size: Rigorous Algorithms and the Limits of Computation

Matthew J. Colbrook·Mark Embree·Jake Fillman
View original source ↗Read source document ↗Export metadata ↓

Abstract

Many structures in mathematical physics and dynamics exhibit intricate fractal geometry. Such behavior appears prominently in quantum mechanics and materials science through spectra of aperiodic and quasicrystalline operators, where questions of ``size'' (Lebesgue measure, fractal dimension, spectral gaps, etc.) are central. Yet the lack of rigorous computational tools for analyzing these quantities limits both theory and application. Naïve truncation often fails, and there is no overarching framework to explain what can, and cannot, be computed. We develop a unified program for the rigorous computation of spectral size for bounded self-adjoint operators, based on local spectral exclusions and adaptive covers. This constructive framework yields algorithmically optimal methods (under natural computational assumptions) that bridge spectral theory with computation to address problems previously deemed intractable. Their complexity is classified within the Solvability Complexity Index (SCI) hierarchy, extending Smale's program on the limits of computation. Sharp computational lower bounds are established through impossibility results for limit-periodic Schrödinger operators constructed from adversarial potentials. The methods enable state-of-the-art rigorous computations for one- and two-dimensional aperiodic systems, and pinpoint problems where numerics can feed directly into computer-assisted proofs. Beyond spectral analysis, they apply broadly to computing measures of size for general closed sets, opening new directions in the computational study of complex geometric structures.

Explore related subjects

math.SPcs.NAmath-phmath.MPmath.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Matthew J. Colbrook, Mark Embree, Jake Fillman. 2026-06-23. Computing Spectral Size: Rigorous Algorithms and the Limits of Computation. https://arxiv.org/abs/2407.20353

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Schrödinger operators with accretive potentials in weighted spaces

We analyse Schrödinger operators with accretive potentials in weighted spaces. We find conditions on potentials and weights for which the Dirichlet realisation, introduced by generalised form methods, has non-empty resolvent set. We establish a domain and graph norm separation property, as well as sufficient conditions for the compactness and Schatten class of the resolvent. Moreover, we investigate the relation between discrete spectra and eigenfunctions of operators in standard and weighted spaces. As applications we extend results on the completeness of eigensystems of operators with accretive potentials from standard to weighted spaces and analyse operator matrices exhibiting a Schur dominance property, in particular, related to a wave equation with strong accretive damping.

math.SP↗
arXiv2026

Eisenstein scattering and Plancherel decomposition on cuspidal Bruhat-Tits quotients

For arithmetic quotients of Bruhat--Tits trees with finitely many cusps, we establish an explicit unitary correspondence between the spherical Eisenstein transform, with the Eisenstein series normalized by their constant terms, and the scattering transform of an associated Jacobi operator with finite core. Tracking the Haar measure, stabilizer weights, height coordinates, and cusp widths yields the Plancherel measure and shows that the absolutely continuous spectrum has multiplicity equal to the number of cusps. From a discrete Green identity we derive a matrix-valued Maass--Selberg formula for the Hermitian matrix $iS(θ)^*\partial_θS(θ)$, where $S(θ)$ is the scattering matrix. Its trace is determined by $\det S(θ)$, while the full matrix retains additional cusp-to-cusp information. After the corresponding change of normalization, the finite Schur complement obtained by eliminating the cusp rays agrees with the resonance matrix of Arends-Peterson-Weich. Using their resonance computations as input, we distinguish eigenvalues supported entirely in the finite core from poles of the scattering matrix. The Nagao and $Γ_0(T)$ quotients, together with a four-cusp quotient arising from an elliptic curve over $\mathbb F_3$, make the normalizations and matrix-valued conclusions explicit.

math.SP↗
arXiv2026

Dirichlet--Neumann bracketing for nonlocal operators

We establish Dirichlet--Neumann bracketing for the Dirichlet eigenvalues of $ψ(-Δ)$ on bounded Lipschitz domains, where $ψ$ is an arbitrary complete Bernstein function. The eigenvalues lie between $ψ$ applied to the corresponding Neumann and Dirichlet eigenvalues of the Laplacian. Both inequalities are strict whenever $ψ$ admits no meromorphic continuation to $\mathbb C \setminus \{0\}$. The proof uses quadratic forms, operator monotonicity, and an analysis of equality in resolvent comparisons. Applying the bracketing to intervals and balls gives a unified proof of simplicity of interval eigenvalues and antisymmetry of second eigenfunctions in balls under the same condition on $ψ$. For fractional powers, these recover results of Fall, Ghimenti, Micheletti and Pistoia for the interval, and of Fall, Feulefack, Temgoua and Weth and, independently, Benedikt, Bobkov, Dhara and Girg for the ball. The argument extends these conclusions to a broader class of nonlocal operators.

math.SP↗