Giao Ky Duong , Rupert L. Frank , Thi Minh Thao Le , Phan Thành Nam , Phuoc-Tai Nguyen
{"title":"哈代-薛定谔算子的 Cwikel-Lieb-Rozenblum 型不等式","authors":"Giao Ky Duong , Rupert L. Frank , Thi Minh Thao Le , Phan Thành Nam , Phuoc-Tai Nguyen","doi":"10.1016/j.matpur.2024.103598","DOIUrl":null,"url":null,"abstract":"<div><p>We prove a Cwikel–Lieb–Rozenblum type inequality for the number of negative eigenvalues of the Hardy–Schrödinger operator <span><math><mo>−</mo><mi>Δ</mi><mo>−</mo><msup><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mo>(</mo><mn>4</mn><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>−</mo><mi>W</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. The bound is given in terms of a weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span>-norm of <em>W</em> which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.</p></div>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000965/pdfft?md5=a35059620cbd155cbff55da815180898&pid=1-s2.0-S0021782424000965-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Cwikel–Lieb–Rozenblum type inequalities for Hardy–Schrödinger operator\",\"authors\":\"Giao Ky Duong , Rupert L. Frank , Thi Minh Thao Le , Phan Thành Nam , Phuoc-Tai Nguyen\",\"doi\":\"10.1016/j.matpur.2024.103598\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove a Cwikel–Lieb–Rozenblum type inequality for the number of negative eigenvalues of the Hardy–Schrödinger operator <span><math><mo>−</mo><mi>Δ</mi><mo>−</mo><msup><mrow><mo>(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><mo>(</mo><mn>4</mn><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>−</mo><mi>W</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></math></span>. The bound is given in terms of a weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>d</mi><mo>/</mo><mn>2</mn></mrow></msup></math></span>-norm of <em>W</em> which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.</p></div>\",\"PeriodicalId\":2,\"journal\":{\"name\":\"ACS Applied Bio Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.6000,\"publicationDate\":\"2024-07-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0021782424000965/pdfft?md5=a35059620cbd155cbff55da815180898&pid=1-s2.0-S0021782424000965-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Bio Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0021782424000965\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, BIOMATERIALS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782424000965","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
引用次数: 0
摘要
我们证明了哈代-薛定谔算子-Δ-(d-2)2/(4|x|2)-W(x) 在 L2(Rd) 上负特征值数量的 Cwikel-Lieb-Rozenblum 型不等式。该约束是通过 W 的加权 Ld/2 准则给出的，在大耦合和小耦合情况下都很尖锐。我们还得到了分数拉普拉卡方的类似约束。
Cwikel–Lieb–Rozenblum type inequalities for Hardy–Schrödinger operator
We prove a Cwikel–Lieb–Rozenblum type inequality for the number of negative eigenvalues of the Hardy–Schrödinger operator on . The bound is given in terms of a weighted -norm of W which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.
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