论文标题
多维准周期施罗丁运营商的贝特·苏默菲尔德猜想和绝对连续的频谱
Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schrödinger Operators
论文作者
论文摘要
我们考虑schrödinger运算符$ h =-Δ+v（{\ mathbf x}）$ in $ {\ mathbb r}^d $，$ d \ geq2 $，带有准期势势$ v（{\ mathbf x}）$。我们证明，通用$ h $的绝对连续频谱包含半轴$ [λ_*，+\ hyfty）$。我们还建立了绝对连续频谱的本征函数家族。这些本征函数是指数的小扰动。证明基于在动量空间中的多尺度分析版本，并在此过程中引入了一些新想法。
We consider Schrödinger operators $H=-Δ+V({\mathbf x})$ in ${\mathbb R}^d$, $d\geq2$, with quasi-periodic potentials $V({\mathbf x})$. We prove that the absolutely continuous spectrum of a generic $H$ contains a semi-axis $[λ_*,+\infty)$. We also construct a family of eigenfunctions of the absolutely continuous spectrum; these eigenfunctions are small perturbations of the exponentials. The proof is based on a version of the multi-scale analysis in the momentum space with several new ideas introduced along the way.
