谱理论：2025–2026 年度综述

An annual survey of the math.SP arXiv category, covering research published from September 2025 – August 2026. Built on a close-reading set of 120 papers and organised into 44 chapters, it maps the year by the field's research sub-directions.

Book framework (share of the year's output):

• General Spectral Theory of Operators and Scattering (21.8%) — This cluster focuses on abstract spectral properties, scattering theory, and localization phenomena for general and long-range operators in various spaces.
• Eigenvalue Bounds for Laplacians on Domains (17.1%) — Research addresses universal inequalities, asymptotic behavior, and optimization of Dirichlet, Neumann, and Robin eigenvalues for Laplacians on geometric domains.
• Spectral Analysis of Discrete and Fractional Operators (11.0%) — This direction investigates spectral invariants, point spectra, and transport properties of discrete Laplacians, fractional operators, and Dirac systems on lattices.
• Spectral Graph Theory and Combinatorial Laplacians (10.0%) — Studies focus on the spectra of random and periodic graphs, Hodge Laplacians on simplicial complexes, and connections between graph eigenvalues and combinatorial properties.
• Spectral Properties of Matrix and Toeplitz Operators (9.5%) — This cluster examines eigenvalue distributions, convergence, and stability in random matrices, Toeplitz structures, and Jacobi systems.
• Self-Adjointness and Spectra of Dirac Operators (7.7%) — Research centers on essential self-adjointness, singular potentials, and spectral analysis of Dirac and non-self-adjoint operators in differential and discrete settings.
• Spectral Gaps and Quantum Ergodicity on Hyperbolic Surfaces (6.5%) — This area explores spectral gaps, quantum mixing, and ergodicity in the Benjamini-Schramm limit for hyperbolic surfaces and random manifolds.
• Schrödinger Operators with Complex and Periodic Potentials (6.2%) — Focuses on Lieb-Thirring inequalities, transport phenomena, and spectral stability for Schrödinger operators with complex, periodic, or quasi-periodic potentials.
• Inverse Spectral Problems for Sturm-Liouville Systems (5.8%) — Addresses uniqueness, stability, and reconstruction methods for inverse spectral problems involving Sturm-Liouville operators and transmission eigenvalues.
• Spectral Analysis on Bounded Domains and Fractals (4.3%) — Investigates spectral transitions, torsional rigidity, and resolvent estimates for operators on bounded planar domains, fractals, and acoustic media.

Each sub-direction is covered across three chapters — how it framed its questions, the strength of its key findings, and its open debates and frontiers — so the reader gets a traceable picture of what this part of the field actually produced over the year.