论文标题
费米品种对离散的周期性施罗丁运营商和嵌入式特征值的不可约性
Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues
论文作者
论文摘要
令$ h_0 $为$ \ ell^2（\ m athbb {z}^d）$的离散定期schrödinger运营商： $$ H_0 =-Δ+V，$$ 其中$δ$是离散的laplacian和$ v：\ mathbb {z}^d \ to \ mathbb {c} $是周期性的。我们证明，对于任何$ d \ geq3 $，每个能量水平的费米品种都是不可修复的（Modulo周期性）。对于$ d = 2 $，我们证明，除电势的平均值外，每个能级的费米品种都是不可约（Modulo周期性），而在电势平均值的费米品种最多具有两个不可还原的组件（Modulo Pricesiticity）。这是敏锐的，因为对于$ d = 2 $，并且具有恒定的潜在$ v $，$ v $ level的费米品种恰好具有两个不可约组件（Modulo Perikiticity）。我们还证明，对于任何$ d \ geq 2 $，Bloch品种都是不可约（Modulo周期性）。作为应用程序，我们证明，当$ v $是一个实值的周期性功能时，任何光谱频段功能的任何极值的水平集，尤其是光谱频段边缘，对于任何$ d \ geq 3 $，最多都具有$ d-2 $，而$ d \ geq 3 $，以及$ d = 2 $的有限心脏。我们还表明$ h =-Δ+v+v $没有任何嵌入式特征值，只要$ v $超级衰减。
Let $H_0$ be a discrete periodic Schrödinger operator on $\ell^2(\mathbb{Z}^d)$: $$H_0=-Δ+V,$$ where $Δ$ is the discrete Laplacian and $V: \mathbb{Z}^d\to \mathbb{C}$ is periodic. We prove that for any $d\geq3$, the Fermi variety at every energy level is irreducible (modulo periodicity). For $d=2$, we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for $d=2$ and a constant potential $V$, the Fermi variety at $V$-level has exactly two irreducible components (modulo periodicity). We also prove that the Bloch variety is irreducible (modulo periodicity) for any $d\geq 2$. As applications, we prove that when $V$ is a real-valued periodic function, the level set of any extrema of any spectral band functions, spectral band edges in particular, has dimension at most $d-2$ for any $d\geq 3$, and finite cardinality for $d=2$. We also show that $H=-Δ+V+v$ does not have any embedded eigenvalues provided that $v$ decays super-exponentially.
