Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. Spectral edge case

DOI10.1002/mana.201100272zbMath1257.35080arXiv1110.0225OpenAlexW2962784469MaRDI QIDQ4648246

Andrew S. Raich, Peter A. Kuchment

Publication date: 8 November 2012

Published in: Mathematische Nachrichten (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1110.0225

zbMATH Keywords

asymptoticsGreen's functionelliptic operatorperiodic operatorresolvent estimatespectral edge

Mathematics Subject Classification ID

General topics in linear spectral theory for PDEs (35P05) Spectrum, resolvent (47A10) General theory of partial differential operators (47F05) Second-order elliptic equations (35J15) Applications of operator theory to differential and integral equations (47N20) Green's functions for elliptic equations (35J08)

Related Items

A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term, Global Limit Theorem for Parabolic Equations with a Potential, Green's function asymptotics of periodic elliptic operators on abelian coverings of compact manifolds, Analytic and algebraic properties of dispersion relations (Bloch varieties) and Fermi surfaces. What is known and unknown, High-energy homogenization of a multidimensional nonstationary Schrödinger equation, Degenerate band edges in periodic quantum graphs, Generic properties of dispersion relations for discrete periodic operators, Gap localization of multiple TE‐Modes by arbitrarily weak defects, Spectrum created by line defects in periodic structures, Asymptotic behavior of branching diffusion processes in periodic media, An overview of periodic elliptic operators

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Bethe-Sommerfeld conjecture
• Bethe-Sommerfeld conjecture for periodic operators with strong perturbations
• Comparison theorems for the gap of Schrödinger operators
• Theory of Bloch waves
• Localization at large disorder and at extreme energies: an elementary derivation
• The Mourre theory for analytically fibered operators
• Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients.
• Endpoints of the spectrum of periodic operators are generically simple.
• Second order elliptic operators with periodic coefficients: Criticality theory, perturbations, and positive harmonic functions
• Large deviation and the tangent cone at infinity of a crystal lattice
• Asymptotics of Green functions and the limiting absorption principle for elliptic operators with periodic coefficients
• Liouville theorems and spectral edge behavior on abelian coverings of compact manifolds
• THE SPECTRAL THEORY AND THE INDEX OF ELLIPTIC OPERATORS WITH ALMOST PERIODIC COEFFICIENTS
• Manifolds with group actions and elliptic operators
• Random Walks on Infinite Graphs and Groups
• Integral representations and Liouville theorems for solutions of periodic elliptic equations