Approximate eigenfunctions for some aperiodic crystals

In this paper, we consider Hamiltonians for aperiodic crystals of the form \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 operators or a Schr\"odinger operator, and $x\mapsto {\mathbf A}(x,X)$ and $x\mapsto V(x,X)$ are $\mathbb L$-periodic with respect to some lattice $\mathbb L\subset{\mathbb R}^d$. Let \begin{align*} (k,X)\ni {\mathbb R}^d\times {\mathbb R}^d\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 $L^2_{\rm per}(\mathbb{R}^d/\mathbb{L})$ with periodic boundary conditions. We show that, under some suitable assumptions on the family of operators $(h(k,X))_{k,X}$ around an energy level $e_0\in {\mathbb R}$ and some points $(k_0,X_0)\in {\mathbb R}^d\times {\mathbb R}^d$, one can construct localized approximate eigenfunctions $\Phi_\varepsilon\in L^2({\mathbb R}^d)$ of the operator $H_\varepsilon$ such that for $\varepsilon$ small enough and for some $m\in \{1,2\}$ and $\mu\in {\mathbb R}$, \begin{align}\label{eq:abstract} \|(H_\varepsilon-e_0-\varepsilon^{\frac{m}{2}}\mu)\Phi_\varepsilon\|_{L^2({\mathbb R}^d)}={\mathcal O}(\varepsilon^{\frac{m}{2}+\frac{1}{4}}). \end{align} with \begin{align*} \|\Phi_\varepsilon\|_{L^2({\mathbb R}^d)}=\frac{1}{|{\mathbb R}^d/\mathbb L|^{1/2}}+{\mathcal O}(\sqrt{\varepsilon}). \end{align*}