{"title":"Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels","authors":"Krzysztof Bogdan , Konstantin Merz","doi":"10.1016/j.matpur.2024.04.003","DOIUrl":null,"url":null,"abstract":"<div><p>Motivated by the study of relativistic atoms, we consider the Hardy operator <span><math><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi><mo>/</mo><mn>2</mn></mrow></msup><mo>−</mo><mi>κ</mi><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>α</mi></mrow></msup></math></span> acting on functions of the form <span><math><mi>u</mi><mo>(</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>ℓ</mi></mrow></msup><msub><mrow><mi>Y</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>/</mo><mo>|</mo><mi>x</mi><mo>|</mo><mo>)</mo></math></span> in <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>, when <span><math><mi>κ</mi><mo>≥</mo><mn>0</mn></math></span> and <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo><mo>∩</mo><mo>(</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>+</mo><mn>2</mn><mi>ℓ</mi><mo>)</mo></math></span>. We give a ground state representation of the corresponding form on the half-line (<span>Theorem 1.5</span>). For the proof we use subordinated Bessel heat kernels.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"186 ","pages":"Pages 176-204"},"PeriodicalIF":2.1000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782424000369/pdfft?md5=cef2b60ae87f0ca0fab3ae618e451f4f&pid=1-s2.0-S0021782424000369-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782424000369","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: 0
Abstract
Motivated by the study of relativistic atoms, we consider the Hardy operator acting on functions of the form in , when and . We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.