Approximate eigenfunctions for some aperiodic crystals
Abstract
In this paper, we consider a broad class of continuum two-scale Hamiltonians Hε:=T(−i∇x+A(x,ε x))+V(x,ε x), x∈ ℝd where T represents either a Dirac operator or a Schrödinger operator, and x↦ A(x,X) and x↦ V(x,X) are 𝕃-periodic with respect to some lattice 𝕃⊂ℝd. No periodicity assumption is imposed on the second variable X. Let ℝd× ℝd∋ (k,X) ↦ h(k,X):=T(−i∇x+k+A(x,X))+V(x,X) be a family of operators acting on L2(ℝd/𝕃) with periodic boundary conditions. We assume only local spectral information near one point (k0,X0): an isolated J-dimensional Bloch bands at energy e0, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree m∈{1,2}. From these data we derive an effective Hamiltonian h. Then every sufficiently localized eigenpair (v⃗,μ) of h gives an approximate eigenfunction Θε such that for ε small enough, |Θε|L2(ℝd)=|Ω|−1/2+O(ε1/2), |(Hε−e0−εm/2μ)Θε|L2(ℝd) =O(εm/2+1/4).
Related
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- Boundary approximate controllability of some linear parabolic systems
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- Strict Log-Concavity of First Dirichlet Eigenfunctions under Symmetry in ℂℙn
- Hypercube eigenfunctions with two strong nodal domains
BibTeX
@misc{meng2026approximate,
title = {{Approximate eigenfunctions for some aperiodic crystals}},
author = {Long Meng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08320},
eprint = {2607.08320},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08320},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximate eigenfunctions for some aperiodic crystals
AU - Long Meng
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2607.08320
UR - https://arxiv.org/abs/2607.08320
AB - In this paper, we consider a broad class of continuum two-scale Hamiltonians H_ε:=T(−i∇_x+A(x,ε x))+V(x,ε x), x∈ ℝ^d where T represents either a Dirac operator or a Schrödinger operator, and x↦ A(x,X) and x↦ V(x,X) are 𝕃-periodic with respect to some lattice 𝕃⊂ℝ^d. No periodicity assumption is imposed on the second variable X. Let ℝ^d× ℝ^d∋ (k,X) ↦ h(k,X):=T(−i∇_x+k+A(x,X))+V(x,X) be a family of operators acting on L^2(ℝ^d/𝕃) with periodic boundary conditions. We assume only local spectral information near one point (k_0,X_0): an isolated J-dimensional Bloch bands at energy e_0, possibly with internal crossings, and a Hermitian matrix-valued homogeneous function of degree m∈{1,2}. From these data we derive an effective Hamiltonian h. Then every sufficiently localized eigenpair (v⃗,μ) of h gives an approximate eigenfunction Θ_ε such that for ε small enough, |Θ_ε|_L^2(ℝ^d)=|Ω|^−1/2+O(ε^1/2), |(H_ε−e_0−ε^m/2μ)Θ_ε|_L^2(ℝ^d) =O(ε^m/2+1/4).
ER -