Approximate eigenfunctions for some aperiodic crystals

Abstract

In this paper, we consider a broad class of continuum two-scale Hamiltonians \begin{align*} H_\varepsilon:=T(-i\nabla_x+\mathbf A(x,\varepsilon x))+V(x,\varepsilon x),\qquad x\in \mathbb{R}^d \end{align*} where $T$ represents either a Dirac operator or a Schrödinger operator, and $x\mapsto \mathbf A(x,X)$ and $x\mapsto V(x,X)$ are $\mathbb L$-periodic with respect to some lattice $\mathbb L\subset\mathbb{R}^d$. No periodicity assumption is imposed on the second variable $X$. Let \begin{align*} \mathbb{R}^d\times \mathbb{R}^d\ni (k,X) \mapsto h(k,X):=T(-i\nabla_x+k+\mathbf A(x,X))+V(x,X) \end{align*} be a family of operators acting on $L^2(\mathbb{R}^d/\mathbb{L})$ 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 $m\in\{1,2\}$. From these data we derive an effective Hamiltonian $\mathfrak{h}$. Then every sufficiently localized eigenpair $(\vec v,μ)$ of $\mathfrak{h}$ gives an approximate eigenfunction $Θ_\varepsilon$ such that for $\varepsilon$ small enough, \[ \|Θ_\varepsilon\|_{L^2(\mathbb{R}^d)}=|Ω|^{-1/2}+O(\varepsilon^{1/2}),\qquad \|(H_\varepsilon-e_0-\varepsilon^{m/2}μ)Θ_\varepsilon\|_{L^2(\mathbb{R}^d)} =O(\varepsilon^{m/2+1/4}). \]