Proof of geometric Borg's Theorem in arbitrary dimensions
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915743777423360 |
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| author | Liu, Wencai |
| author_facet | Liu, Wencai |
| contents | Let $Δ+V$ be the discrete Schrödinger operator, where $Δ$ is the discrete Laplacian on $\mathbb{Z}^d$ and potential $V:\mathbb{Z}^d\to \mathbb{C}$ is $Γ$-periodic with $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$. In this study, we establish a comprehensive characterization of complex-valued $Γ$-periodic functions such that the Bloch variety of $Δ+V$ contains a graph of an entire function, in particular, we show that there are exactly $q_1q_2\cdots q_d$ such functions (up to Floquet isospectrality and translation). Moreover, by applying this understanding to real-valued functions $V$, we prove that $V$ is constant if and only if the Bloch variety of $Δ+V$ contains a graph of an entire function, which confirms the conjecture concerning the geometric version of Borg's theorem in arbitrary dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_16412 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Proof of geometric Borg's Theorem in arbitrary dimensions Liu, Wencai Spectral Theory Mathematical Physics Algebraic Geometry Complex Variables Let $Δ+V$ be the discrete Schrödinger operator, where $Δ$ is the discrete Laplacian on $\mathbb{Z}^d$ and potential $V:\mathbb{Z}^d\to \mathbb{C}$ is $Γ$-periodic with $Γ=q_1\mathbb{Z}\oplus q_2 \mathbb{Z}\oplus\cdots\oplus q_d\mathbb{Z}$. In this study, we establish a comprehensive characterization of complex-valued $Γ$-periodic functions such that the Bloch variety of $Δ+V$ contains a graph of an entire function, in particular, we show that there are exactly $q_1q_2\cdots q_d$ such functions (up to Floquet isospectrality and translation). Moreover, by applying this understanding to real-valued functions $V$, we prove that $V$ is constant if and only if the Bloch variety of $Δ+V$ contains a graph of an entire function, which confirms the conjecture concerning the geometric version of Borg's theorem in arbitrary dimensions. |
| title | Proof of geometric Borg's Theorem in arbitrary dimensions |
| topic | Spectral Theory Mathematical Physics Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2306.16412 |