Approximation of Dirac operators with δ-shell potentials in the norm resolvent sense, II: Quantitative results

IF 0.8 3区数学 Q2 MATHEMATICS

Mathematische Nachrichten Pub Date : 2026-04-05 Epub Date: 2026-01-28 DOI:10.1002/mana.70085

Jussi Behrndt, Markus Holzmann, Christian Stelzer-Landauer

{"title":"Approximation of Dirac operators with δ-shell potentials in the norm resolvent sense, II: Quantitative results","authors":"Jussi Behrndt, Markus Holzmann, Christian Stelzer-Landauer","doi":"10.1002/mana.70085","DOIUrl":null,"url":null,"abstract":"This paper is devoted to the approximation of two- and three-dimensional Dirac operators <math>\n <semantics>\n <msub>\n <mi>H</mi>\n <mrow>\n <mover>\n <mi>V</mi>\n <mo>∼</mo>\n </mover>\n <msub>\n <mi>δ</mi>\n <mi>Σ</mi>\n </msub>\n </mrow>\n </msub>\n <annotation>$H_{\\widetilde{V} \\delta _\\Sigma }$</annotation>\n </semantics></math> with combinations of electrostatic and Lorentz scalar <math>\n <semantics>\n <mi>δ</mi>\n <annotation>$\\delta$</annotation>\n </semantics></math>-shell interactions in the norm resolvent sense. Relying on results from Behrndt, Holzmann, and Stelzer-Landauer [Math. Nachr. 298 (2025), 2499–2546], an explicit smallness condition on the coupling parameters is derived so that <math>\n <semantics>\n <msub>\n <mi>H</mi>\n <mrow>\n <mover>\n <mi>V</mi>\n <mo>∼</mo>\n </mover>\n <msub>\n <mi>δ</mi>\n <mi>Σ</mi>\n </msub>\n </mrow>\n </msub>\n <annotation>$H_{\\widetilde{V} \\delta _\\Sigma }$</annotation>\n </semantics></math> is the limit of Dirac operators with scaled electrostatic and Lorentz scalar potentials. Via counterexamples it is shown that this condition is sharp. The approximation of <math>\n <semantics>\n <msub>\n <mi>H</mi>\n <mrow>\n <mover>\n <mi>V</mi>\n <mo>∼</mo>\n </mover>\n <msub>\n <mi>δ</mi>\n <mi>Σ</mi>\n </msub>\n </mrow>\n </msub>\n <annotation>$H_{\\widetilde{V} \\delta _\\Sigma }$</annotation>\n </semantics></math> for larger coupling constants is achieved by adding an additional scaled magnetic term.","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"299 4","pages":"704-763"},"PeriodicalIF":0.8000,"publicationDate":"2026-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70085","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematische Nachrichten","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/mana.70085","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2026/1/28 0:00:00","PubModel":"Epub","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

This paper is devoted to the approximation of two- and three-dimensional Dirac operators $H_{\tilde{V} δ_{Σ}}$ with combinations of electrostatic and Lorentz scalar $δ$ -shell interactions in the norm resolvent sense. Relying on results from Behrndt, Holzmann, and Stelzer-Landauer [Math. Nachr. 298 (2025), 2499–2546], an explicit smallness condition on the coupling parameters is derived so that $H_{\tilde{V} δ_{Σ}}$ is the limit of Dirac operators with scaled electrostatic and Lorentz scalar potentials. Via counterexamples it is shown that this condition is sharp. The approximation of $H_{\tilde{V} δ_{Σ}}$ for larger coupling constants is achieved by adding an additional scaled magnetic term.

查看原文本刊更多论文

范数解析意义上δ壳势Dirac算子的逼近，II：定量结果

本文研究了二维和三维狄拉克算符H V ～ δ Σ $H_{\widetilde{V} \delta _\Sigma }$与静电和洛伦兹标量δ $\delta$在范数解析意义上的壳层相互作用。依靠Behrndt， Holzmann和Stelzer-Landauer[数学]的结果。中华医学杂志，2004 (5),349 - 349 [j]。推导了耦合参数的显式小条件，使得H V ～ δ Σ $H_{\widetilde{V} \delta _\Sigma }$是带缩放的狄拉克算子的极限静电和洛伦兹标量势。通过反例证明了这一条件是尖锐的。对于较大的耦合常数，H V ～ δ Σ $H_{\widetilde{V} \delta _\Sigma }$的近似是通过添加一个额外的缩放磁项来实现的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Mathematische Nachrichten 数学-数学

CiteScore

1.50

自引率

0.00%

发文量

157

审稿时长

4-8 weeks

期刊介绍： Mathematische Nachrichten - Mathematical News publishes original papers on new results and methods that hold prospect for substantial progress in mathematics and its applications. All branches of analysis, algebra, number theory, geometry and topology, flow mechanics and theoretical aspects of stochastics are given special emphasis. Mathematische Nachrichten is indexed/abstracted in Current Contents/Physical, Chemical and Earth Sciences; Mathematical Review; Zentralblatt für Mathematik; Math Database on STN International, INSPEC; Science Citation Index