Spectral Analysis of Differential Operators in Complex Domains
Summary
Spectral analysis of differential operators in complex domains explores how the geometry and boundary conditions of a region influence the distribution of its eigenvalues and the behaviour of associated eigenfunctions. When domains exhibit singularities—such as cusps, peaks, perforations or branching structures—the classical theory on smooth bounded sets must be extended by advanced tools: asymptotic expansions, homogenisation techniques, Floquet theory and functional-analytic frameworks that yield resolvent estimates and Hausdorff convergence of spectra. This field unites questions of localisation (trapped modes near singular features), emergence of spectral gaps, and strong‐coupling limits under varied boundary conditions (Dirichlet, Neumann, Robin, Steklov). Beyond pure mathematics, these investigations inform wave propagation in engineered materials, quantum graph models, acoustic black holes and photonic crystals, where control of band structures and resonance phenomena underpins device design.
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Technical terms
Eigenvalue: A scalar λ such that there exists a nonzero function u satisfying L u=λ u under given boundary conditions for differential operator L.
Eigenfunction localisation: Concentration of the energy of an eigenfunction in small regions of a domain, often near geometrical singularities.
Spectral gap: An interval in the real line that contains no spectrum, separating bands of continuous or discrete eigenvalues.
Robin boundary condition: A linear combination of function value and normal derivative prescribed on the boundary, typically ∂νu=α u.
Steklov boundary condition: A spectral parameter appears in the boundary condition itself, often ∂νu=λ u on part of the boundary.
References
- Asymptotic stability of the spectrum of a parametric family of homogenization problems associated with a perforated waveguide. Mathematische Nachrichten (2023).
- Asymptotics of Robin eigenvalues for non-isotropic peaks. Journal of Mathematical Analysis and Applications (2024).
- Spectral problems with perturbed Steklov conditions in thick junctions with branched structure. Applicable Analysis (2024).
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