• Corpus ID: 245385798

# Second-order homogenization of periodic Schr\"odinger operators with highly oscillating potentials

@inproceedings{Cances2021SecondorderHO,
title={Second-order homogenization of periodic Schr\"odinger operators with highly oscillating potentials},
author={'Eric Cances and Louis Garrigue and David Gontier},
year={2021}
}
• Published 22 December 2021
• Mathematics
We consider the homogenization at second-order in ε of Lperiodic Schrödinger operators with rapidly oscillating potentials of the form H = −∆+ε−1v(x, ε−1x)+W (x) on L(R), where L is a Bravais lattice of R, v is (L × L)-periodic, W is L-periodic, and ε ∈ N−1. We treat both the linear equation with fixed right-hand side and the eigenvalue problem, as well as the case of physical observables such as the integrated density of states. We illustrate numerically that these corrections to the…
2 Citations

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## References

SHOWING 1-10 OF 19 REFERENCES
First-order expansions for eigenvalues and eigenfunctions in periodic homogenization
• Jinping Zhuge
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2020
Abstract For a family of elliptic operators with periodically oscillating coefficients, $-{\rm div}(A(\cdot /\varepsilon )\nabla )$ with tiny ε > 0, we comprehensively study the first-order
Scattering, Homogenization, and Interface Effects for Oscillatory Potentials with Strong Singularities
• Mathematics
Multiscale Model. Simul.
• 2011
One-dimensional scattering for a decaying potential with rapid periodic oscillations and strong localized singularities is studied, and the Schrodinger equation Heψ≡(-∂x2+V0(x)+q(x,x/e)ψ=k2ψ for k∈ℝ and e≪1 is considered.
Unique continuation and absence of positive eigenvalues for Schrodinger operators
• Mathematics
• 1985
On considere l'operateur de Laplace Δ sur R n et une fonction V(x) sur un sous-ensemble connexe, ouvert Ω de R n . On montre que si n≥3, une propriete de prolongement unique est vraie pour V∈L loc
Third-Order Corrections in Periodic Homogenization for Elliptic Problem
• Mathematics
• 2021
This paper is devoted to the study of the error estimates in the periodic homogenization of elliptic equations in divergence form with Dirichlet boundary conditions. We are interested in the
Boundary layer tails in periodic homogenization
• Mathematics
• 1999
This paper focus on the properties of boundary layers in periodic homogenization in rectangular domains which are either fixed or have an oscillating boundary. Such boundary layers are highly
Estimates of eigenvalues and eigenfunctions in periodic homogenization
• Mathematics
• 2012
For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet
Homogenization of elliptic eigenvalue problems: Part 1
ResumeLe but de cet article est d'étudier l'homogénéisation du problème de valeurs propres pour des opérateurs elliptiques. On prend comme exemple un problème de second-ordre avec des conditions de
The marvels of moiré materials
• Physics
Nature Reviews Materials
• 2021
Moiré systems formed by 2D atomic layers have widely tunable electrical and optical properties and host exotic, strongly correlated and topological phenomena, including superconductivity, correlated