发表机构
Center for Interdisciplinary Applied Mathematics & Institute of Fundamental and Transdisciplinary Research, Zhejiang University（浙江大学交叉应用数学中心与基础与跨学科研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了非周期晶体的哈密顿量，通过构造局部近似本征函数来分析其能量特性。
AI 中文摘要
本文考虑了形如H_ε:=T(-i∇x+ A(x,εx))+V(x,εx)，x∈R^d的非周期晶体的哈密顿量，其中T表示狄拉克算子或薛inge算子，而x→A(x,X)和x→V(x,X)关于某个晶格L⊂R^d是L-周期的。令(k,X)∈R^d×R^d↦h(k,X):=T(-i∇x+k+A(x,X))+V(x,X)为作用于L^2_per(R^d/L)上的算子族，具有周期边界条件。我们证明，在某些关于算子族(h(k,X))_{k,X}在能量水平e_0∈R附近的假设以及某些点(k_0,X_0)∈R^d×R^d的情况下，可以构造出H_ε的局部近似本征函数Φ_ε∈L^2(R^d)，使得对于足够小的ε和某些m∈{1,2}和μ∈R，有‖(H_ε-e_0-ε^{m/2}μ)Φ_ε‖_{L^2(R^d)}=O(ε^{m/2+1/4})，且‖Φ_ε‖_{L^2(R^d)}=1/|R^d/L|^{1/2}+O(√ε)。
英文摘要
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, \[ \|Θ_\varepsilon\|_{L^2(\mathbb{R}^d)}=|Ω|^{-1/2}+O(\varepsilon^{1/2}),\qquad \|(H_\varepsilon-e_0-\varepsilon^{m/2}μ)Θ_\varepsilon\|_{L^2(\mathbb{R}^d)} =O(\varepsilon^{m/2+1/4}). \]
Comments51 pages, 1 figures
