{"title":"具有库仑势的费米子非线性薛定谔系统的基态 II：L^2 美元临界情形","authors":"Bin Chen, Yujin Guo, Shu Zhang","doi":"arxiv-2312.06916","DOIUrl":null,"url":null,"abstract":"As a continuation of \\cite{me}, we consider ground states of the $N$ coupled\nfermionic nonlinear Schr\\\"{o}dinger system with a parameter $a $ and the\nCoulomb potential $V(x)$ in the $L^2$-critical case, where $a>0$ represents the\nattractive strength of the quantum particles. For any given $N\\in\\mathbb{N}^+$,\nwe prove that the system admits ground states, if and only if the attractive\nstrength $a$ satisfies $0<a<a^*_N$, where the critical constant\n$0<a^*_N<\\infty$ is the same as the best constant of a dual finite-rank\nLieb-Thirring inequality. By developing the so-called blow-up analysis of\nmany-body fermionic problems, we also prove the mass concentration behavior of\nground states for the system as $a\\nearrow a_N^*$.","PeriodicalId":501275,"journal":{"name":"arXiv - PHYS - Mathematical Physics","volume":"6 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Ground States of Fermionic Nonlinear Schrödinger Systems with Coulomb Potential II: The $L^2$-Critical Case\",\"authors\":\"Bin Chen, Yujin Guo, Shu Zhang\",\"doi\":\"arxiv-2312.06916\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"As a continuation of \\\\cite{me}, we consider ground states of the $N$ coupled\\nfermionic nonlinear Schr\\\\\\\"{o}dinger system with a parameter $a $ and the\\nCoulomb potential $V(x)$ in the $L^2$-critical case, where $a>0$ represents the\\nattractive strength of the quantum particles. For any given $N\\\\in\\\\mathbb{N}^+$,\\nwe prove that the system admits ground states, if and only if the attractive\\nstrength $a$ satisfies $0<a<a^*_N$, where the critical constant\\n$0<a^*_N<\\\\infty$ is the same as the best constant of a dual finite-rank\\nLieb-Thirring inequality. By developing the so-called blow-up analysis of\\nmany-body fermionic problems, we also prove the mass concentration behavior of\\nground states for the system as $a\\\\nearrow a_N^*$.\",\"PeriodicalId\":501275,\"journal\":{\"name\":\"arXiv - PHYS - Mathematical Physics\",\"volume\":\"6 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - PHYS - Mathematical Physics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2312.06916\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Mathematical Physics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2312.06916","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Ground States of Fermionic Nonlinear Schrödinger Systems with Coulomb Potential II: The $L^2$-Critical Case
As a continuation of \cite{me}, we consider ground states of the $N$ coupled
fermionic nonlinear Schr\"{o}dinger system with a parameter $a $ and the
Coulomb potential $V(x)$ in the $L^2$-critical case, where $a>0$ represents the
attractive strength of the quantum particles. For any given $N\in\mathbb{N}^+$,
we prove that the system admits ground states, if and only if the attractive
strength $a$ satisfies $0
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