论文标题
离散周期性施罗丁格运算符的Fermi等光谱具有可分离的电位
Fermi isospectrality of discrete periodic Schrödinger operators with separable potentials on $\mathbb{Z}^2$
论文作者
论文摘要
令$γ= Q_1 \ Mathbb {Z} \ Oplus Q_2 \ Mathbb {Z} $，$ Q_1 \ in \ Mathbb {Z} _+$和$ Q_2 \ in \ Mathbb {Z} _} _+$+$。令$Δ+x $为$ \ mathbb {z}^2 $上的离散定期schrödinger运算符，其中$δ$是离散的laplacian和$ x：\ m athbb {z}^2 \ to \ mathbb {c} $ is $γ$ - periodic。在本文中，我们开发了从复杂分析到研究离散周期性施罗丁运营商的等光谱的工具。 We prove that if two $Γ$-periodic potentials $X$ and $Y$ are Fermi isospectral and both $X=X_1\oplus X_2$ and $Y= Y_1\oplus Y_2$ are separable functions, then, up to a constant, one dimensional potentials $X_j$ and $Y_j$ are Floquet isospectral, $j=1,2$.这使我们能够证明，对于任何可分开的实价$γ$周期的潜力，费米品种$f_λ（v）/\ mathbb {z}^2 $对于任何\ Mathbb {c} $ in \ Mathbb {C} $不可估量，这部分确认了giesekere的早期和knementers and knere and knere and kner。
Let $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z} $ with $q_1\in \mathbb{Z}_+$ and $q_2\in\mathbb{Z}_+$. Let $Δ+X$ be the discrete periodic Schrödinger operator on $\mathbb{Z}^2$, where $Δ$ is the discrete Laplacian and $X:\mathbb{Z}^2\to \mathbb{C}$ is $Γ$-periodic. In this paper, we develop tools from complex analysis to study the isospectrality of discrete periodic Schrödinger operators. We prove that if two $Γ$-periodic potentials $X$ and $Y$ are Fermi isospectral and both $X=X_1\oplus X_2$ and $Y= Y_1\oplus Y_2$ are separable functions, then, up to a constant, one dimensional potentials $X_j$ and $Y_j$ are Floquet isospectral, $j=1,2$. This allows us to prove that for any non-constant separable real-valued $Γ$-periodic potential, the Fermi variety $F_λ(V)/\mathbb{Z}^2$ is irreducible for any $λ\in \mathbb{C}$, which partially confirms a conjecture of Gieseker, Knörrer and Trubowitz in the early 1990s.
