{"title":"Dunkl 型 Segal-Bargmann 变换及其在某些偏微分方程中的应用","authors":"Fethi Soltani, Meriem Nenni","doi":"10.1515/gmj-2024-2031","DOIUrl":null,"url":null,"abstract":"In this paper, we give some applications of the Dunkl-type Segal–Bargmann transform <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"script\">ℬ</m:mi> <m:mi>α</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2031_eq_0181.png\"/> <jats:tex-math>{\\mathscr{B}_{\\alpha}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in the field of partial differential equations, such as the time-dependent Dunkl–Dirac Laplacian equation and the time-dependent Dunkl–Schrödinger equation. The resolution of these types of problems is based on the techniques of the transmutation operators on the Dunkl-type Fock space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi mathvariant=\"script\">ℱ</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:msup> <m:mi>ℂ</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2031_eq_0190.png\"/> <jats:tex-math>{\\mathscr{F}_{\\alpha}(\\mathbb{C}^{d})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Dunkl-type Segal–Bargmann transform and its applications to some partial differential equations\",\"authors\":\"Fethi Soltani, Meriem Nenni\",\"doi\":\"10.1515/gmj-2024-2031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we give some applications of the Dunkl-type Segal–Bargmann transform <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:msub> <m:mi mathvariant=\\\"script\\\">ℬ</m:mi> <m:mi>α</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2031_eq_0181.png\\\"/> <jats:tex-math>{\\\\mathscr{B}_{\\\\alpha}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> in the field of partial differential equations, such as the time-dependent Dunkl–Dirac Laplacian equation and the time-dependent Dunkl–Schrödinger equation. The resolution of these types of problems is based on the techniques of the transmutation operators on the Dunkl-type Fock space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\\\"http://www.w3.org/1998/Math/MathML\\\"> <m:mrow> <m:msub> <m:mi mathvariant=\\\"script\\\">ℱ</m:mi> <m:mi>α</m:mi> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\\\"false\\\">(</m:mo> <m:msup> <m:mi>ℂ</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo stretchy=\\\"false\\\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\" xlink:href=\\\"graphic/j_gmj-2024-2031_eq_0190.png\\\"/> <jats:tex-math>{\\\\mathscr{F}_{\\\\alpha}(\\\\mathbb{C}^{d})}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-25\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/gmj-2024-2031\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2031","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Dunkl-type Segal–Bargmann transform and its applications to some partial differential equations
In this paper, we give some applications of the Dunkl-type Segal–Bargmann transform ℬα{\mathscr{B}_{\alpha}} in the field of partial differential equations, such as the time-dependent Dunkl–Dirac Laplacian equation and the time-dependent Dunkl–Schrödinger equation. The resolution of these types of problems is based on the techniques of the transmutation operators on the Dunkl-type Fock space ℱα(ℂd){\mathscr{F}_{\alpha}(\mathbb{C}^{d})}.
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