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{"title":"离散周期Schrödinger算符的费米等谱性","authors":"Wencai Liu","doi":"10.1002/cpa.22161","DOIUrl":null,"url":null,"abstract":"<p>Let <math>\n <semantics>\n <mrow>\n <mi>Γ</mi>\n <mo>=</mo>\n <msub>\n <mi>q</mi>\n <mn>1</mn>\n </msub>\n <mi>Z</mi>\n <mi>⊕</mi>\n <msub>\n <mi>q</mi>\n <mn>2</mn>\n </msub>\n <mi>Z</mi>\n <mi>⊕</mi>\n <mtext>…</mtext>\n <mi>⊕</mi>\n <msub>\n <mi>q</mi>\n <mi>d</mi>\n </msub>\n <mi>Z</mi>\n </mrow>\n <annotation>$\\Gamma =q_1\\mathbb {Z}\\oplus q_2 \\mathbb {Z}\\oplus \\ldots \\oplus q_d\\mathbb {Z}$</annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n <msub>\n <mi>q</mi>\n <mi>l</mi>\n </msub>\n <mo>∈</mo>\n <msub>\n <mi>Z</mi>\n <mo>+</mo>\n </msub>\n </mrow>\n <annotation>$q_l\\in \\mathbb {Z}_+$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>l</mi>\n <mo>=</mo>\n <mn>1</mn>\n <mo>,</mo>\n <mn>2</mn>\n <mo>,</mo>\n <mtext>…</mtext>\n <mo>,</mo>\n <mi>d</mi>\n </mrow>\n <annotation>$l=1,2,\\ldots ,d$</annotation>\n </semantics></math>, are pairwise coprime. Let <math>\n <semantics>\n <mrow>\n <mi>Δ</mi>\n <mo>+</mo>\n <mi>V</mi>\n </mrow>\n <annotation>$\\Delta +V$</annotation>\n </semantics></math> be the discrete Schrödinger operator, where Δ is the discrete Laplacian on <math>\n <semantics>\n <msup>\n <mi>Z</mi>\n <mi>d</mi>\n </msup>\n <annotation>$\\mathbb {Z}^d$</annotation>\n </semantics></math> and the potential <math>\n <semantics>\n <mrow>\n <mi>V</mi>\n <mo>:</mo>\n <msup>\n <mi>Z</mi>\n <mi>d</mi>\n </msup>\n <mo>→</mo>\n <mi>C</mi>\n </mrow>\n <annotation>$V:\\mathbb {Z}^d\\rightarrow \\mathbb {C}$</annotation>\n </semantics></math> is Γ-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>≥</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d\\ge 3$</annotation>\n </semantics></math>: \n\n </p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1126-1146"},"PeriodicalIF":3.1000,"publicationDate":"2023-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"Fermi isospectrality for discrete periodic Schrödinger operators\",\"authors\":\"Wencai Liu\",\"doi\":\"10.1002/cpa.22161\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <math>\\n <semantics>\\n <mrow>\\n <mi>Γ</mi>\\n <mo>=</mo>\\n <msub>\\n <mi>q</mi>\\n <mn>1</mn>\\n </msub>\\n <mi>Z</mi>\\n <mi>⊕</mi>\\n <msub>\\n <mi>q</mi>\\n <mn>2</mn>\\n </msub>\\n <mi>Z</mi>\\n <mi>⊕</mi>\\n <mtext>…</mtext>\\n <mi>⊕</mi>\\n <msub>\\n <mi>q</mi>\\n <mi>d</mi>\\n </msub>\\n <mi>Z</mi>\\n </mrow>\\n <annotation>$\\\\Gamma =q_1\\\\mathbb {Z}\\\\oplus q_2 \\\\mathbb {Z}\\\\oplus \\\\ldots \\\\oplus q_d\\\\mathbb {Z}$</annotation>\\n </semantics></math>, where <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>q</mi>\\n <mi>l</mi>\\n </msub>\\n <mo>∈</mo>\\n <msub>\\n <mi>Z</mi>\\n <mo>+</mo>\\n </msub>\\n </mrow>\\n <annotation>$q_l\\\\in \\\\mathbb {Z}_+$</annotation>\\n </semantics></math>, <math>\\n <semantics>\\n <mrow>\\n <mi>l</mi>\\n <mo>=</mo>\\n <mn>1</mn>\\n <mo>,</mo>\\n <mn>2</mn>\\n <mo>,</mo>\\n <mtext>…</mtext>\\n <mo>,</mo>\\n <mi>d</mi>\\n </mrow>\\n <annotation>$l=1,2,\\\\ldots ,d$</annotation>\\n </semantics></math>, are pairwise coprime. Let <math>\\n <semantics>\\n <mrow>\\n <mi>Δ</mi>\\n <mo>+</mo>\\n <mi>V</mi>\\n </mrow>\\n <annotation>$\\\\Delta +V$</annotation>\\n </semantics></math> be the discrete Schrödinger operator, where Δ is the discrete Laplacian on <math>\\n <semantics>\\n <msup>\\n <mi>Z</mi>\\n <mi>d</mi>\\n </msup>\\n <annotation>$\\\\mathbb {Z}^d$</annotation>\\n </semantics></math> and the potential <math>\\n <semantics>\\n <mrow>\\n <mi>V</mi>\\n <mo>:</mo>\\n <msup>\\n <mi>Z</mi>\\n <mi>d</mi>\\n </msup>\\n <mo>→</mo>\\n <mi>C</mi>\\n </mrow>\\n <annotation>$V:\\\\mathbb {Z}^d\\\\rightarrow \\\\mathbb {C}$</annotation>\\n </semantics></math> is Γ-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension <math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>≥</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d\\\\ge 3$</annotation>\\n </semantics></math>: \\n\\n </p>\",\"PeriodicalId\":10601,\"journal\":{\"name\":\"Communications on Pure and Applied Mathematics\",\"volume\":\"77 2\",\"pages\":\"1126-1146\"},\"PeriodicalIF\":3.1000,\"publicationDate\":\"2023-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Communications on Pure and Applied Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22161\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22161","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 9
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摘要
让 Γ = q 1 Z ⊕ q 2 Z ⊕ ... ⊕ q d Z $Gamma =q_1\mathbb {Z}\oplus q_2 \mathbb {Z}\oplus \ldots \oplus q_d\mathbb {Z}$ ，其中 q l∈ Z + $q_l\in \mathbb {Z}_+$ , l = 1 , 2 , ... , d $l=1,2,\ldots ,d$ , 是成对的共素数。让 Δ + V $\Delta +V$ 是离散薛定谔算子，其中 Δ 是 Z d $\mathbb {Z}^d$ 上的离散拉普拉奇，势 V : Z d → C $V:\mathbb {Z}^d\rightarrow \mathbb {C}$ 是Γ周期的。我们证明了离散周期薛定谔算子在任意维度 d ≥ 3 $d\ge 3$ 的三个刚度定理：
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