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We say $H$ is of generic type if $0$ is not an eigenvalue nor a resonance of $H$ and of exceptional type if otherwise. We assume that $V$ satisfies $\\Fg(\\ax^{-2\\s}V) \\in L^{m_\\ast}$ for some $\\s\u003e\\frac{1}{m_\\ast}$, $m_\\ast=\\frac{m-1}{m-2}$. We show that $W_\\pm$ are bounded in $L^p(\\R^m)$ for all $1\\leq p \\leq \\infty$ if $V$ satisfies in addition $|V(x)|\\leq C \\ax^{-m-2-\\ep}$ for some $\\ep\u003e0$ and if $H$ is of generic type; and that $W_\\pm$ are bounded in $L^p(\\R^m)$ for all $p$ between $\\frac{m}{m-2}$ and $\\frac{m}{2}$ but not for $p$ outside the closed interval $[\\frac{m}{m-2}, \\frac{m}{2}]$ if $V$ satisfies $|V(x)|\\leq C \\ax^{-m-3-\\ep}$ and if $H$ is of exceptional type. This in particular implies that the continuous part of the propagator satisfies the $L^p$-$L^q$ estimates $\\|e^{-itH}P_c(H)u \\|_p \\leq C |t|^{\\frac{1}{m}\\left(\\frac12-\\frac{1}{q}\\right)}\\|u\\|_q$ for the dual exponents $\\frac{1}{p}+\\frac1{q}=1$ such that $1\\leq q\\leq 2 \\leq p\\leq \\infty$ if $H$ is of generic type, and for $\\frac{m}{m-2}\u003c q\\leq 2 \\leq p \u003c \\frac{m}{2}$, $m \\geq 5$, or $\\frac32\u003cq\\leq 2 \\leq p\u003c3$, $m=3$, if $H$ of exceptional type.", "subitem_description_type": "Abstract"}]}, "item_4_publisher_20": {"attribute_name": "出版者", "attribute_value_mlt": [{"subitem_publisher": "Graduate School of Mathematical Sciences, The University of Tokyo"}]}, "item_4_source_id_10": {"attribute_name": "書誌レコードID", "attribute_value_mlt": [{"subitem_source_identifier": "AA11021653", "subitem_source_identifier_type": "NCID"}]}, "item_4_source_id_8": {"attribute_name": "ISSN", "attribute_value_mlt": [{"subitem_source_identifier": "13405705", "subitem_source_identifier_type": "ISSN"}]}, "item_4_subject_15": {"attribute_name": "日本十進分類法", "attribute_value_mlt": [{"subitem_subject": "415", "subitem_subject_scheme": "NDC"}]}, "item_4_text_17": {"attribute_name": "Mathmatical Subject Classification", "attribute_value_mlt": [{"subitem_text_value": "35P25(MSC1991)"}, {"subitem_text_value": "35J10(MSC1991)"}, {"subitem_text_value": "47A40(MSC1991)"}, {"subitem_text_value": "47F05(MSC1991)"}, {"subitem_text_value": "47N50(MSC1991)"}, {"subitem_text_value": "81U50(MSC1991)"}]}, "item_4_text_33": {"attribute_name": "原稿受領日", "attribute_value_mlt": [{"subitem_text_value": "2005-12-26"}]}, "item_4_text_34": {"attribute_name": "資源タイプ", "attribute_value_mlt": [{"subitem_text_value": "Departmental Bulletin Paper"}]}, "item_creator": {"attribute_name": "著者", "attribute_type": "creator", "attribute_value_mlt": [{"creatorNames": [{"creatorName": "Yajima, Kenji"}], "nameIdentifiers": [{"nameIdentifier": "138706", "nameIdentifierScheme": "WEKO"}]}]}, "item_files": {"attribute_name": "ファイル情報", "attribute_type": "file", "attribute_value_mlt": [{"accessrole": "open_date", "date": [{"dateType": "Available", "dateValue": "2017-06-27"}], "displaytype": "detail", "download_preview_message": "", "file_order": 0, "filename": "jms130103.pdf", "filesize": [{"value": "387.4 kB"}], "format": "application/pdf", "future_date_message": "", "is_thumbnail": false, "licensetype": "license_free", "mimetype": "application/pdf", "size": 387400.0, "url": {"label": "jms130103.pdf", "url": "https://repository.dl.itc.u-tokyo.ac.jp/record/40125/files/jms130103.pdf"}, "version_id": "80cdca66-3054-474f-887a-91f170990661"}]}, "item_language": {"attribute_name": "言語", "attribute_value_mlt": [{"subitem_language": "eng"}]}, "item_resource_type": {"attribute_name": "資源タイプ", "attribute_value_mlt": [{"resourcetype": "departmental bulletin paper", "resourceuri": "http://purl.org/coar/resource_type/c_6501"}]}, "item_title": "The L^p Boundedness of Wave Operators for Schrodinger Operators with Threshold Singuralities I. The Odd Dimesional Case", "item_titles": {"attribute_name": "タイトル", "attribute_value_mlt": [{"subitem_title": "The L^p Boundedness of Wave Operators for Schrodinger Operators with Threshold Singuralities I. The Odd Dimesional Case"}]}, "item_type_id": "4", "owner": "1", "path": ["6965", "6966"], "permalink_uri": "http://hdl.handle.net/2261/7529", "pubdate": {"attribute_name": "公開日", "attribute_value": "2008-03-04"}, "publish_date": "2008-03-04", "publish_status": "0", "recid": "40125", "relation": {}, "relation_version_is_last": true, "title": ["The L^p Boundedness of Wave Operators for Schrodinger Operators with Threshold Singuralities I. The Odd Dimesional Case"], "weko_shared_id": null}
The L^p Boundedness of Wave Operators for Schrodinger Operators with Threshold Singuralities I. The Odd Dimesional Case
http://hdl.handle.net/2261/7529
http://hdl.handle.net/2261/7529e73b64e6-e6de-4c3c-935b-07ae2cdbe516
名前 / ファイル | ライセンス | アクション |
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jms130103.pdf (387.4 kB)
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Item type | 紀要論文 / Departmental Bulletin Paper(1) | |||||
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公開日 | 2008-03-04 | |||||
タイトル | ||||||
タイトル | The L^p Boundedness of Wave Operators for Schrodinger Operators with Threshold Singuralities I. The Odd Dimesional Case | |||||
言語 | ||||||
言語 | eng | |||||
資源タイプ | ||||||
資源 | http://purl.org/coar/resource_type/c_6501 | |||||
タイプ | departmental bulletin paper | |||||
著者 |
Yajima, Kenji
× Yajima, Kenji |
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抄録 | ||||||
内容記述タイプ | Abstract | |||||
内容記述 | Let $H=-\lap +V(x)$ be an odd $m$-dimensional Schr\""odinger operator, $m \geq 3$, $H_0=-\lap$, and let ${\ds W_\pm=\lim_{t\to \pm \infty} e^{itH}e^{-itH_0}}$ be the wave operators for the pair $(H, H_0)$. We say $H$ is of generic type if $0$ is not an eigenvalue nor a resonance of $H$ and of exceptional type if otherwise. We assume that $V$ satisfies $\Fg(\ax^{-2\s}V) \in L^{m_\ast}$ for some $\s>\frac{1}{m_\ast}$, $m_\ast=\frac{m-1}{m-2}$. We show that $W_\pm$ are bounded in $L^p(\R^m)$ for all $1\leq p \leq \infty$ if $V$ satisfies in addition $|V(x)|\leq C \ax^{-m-2-\ep}$ for some $\ep>0$ and if $H$ is of generic type; and that $W_\pm$ are bounded in $L^p(\R^m)$ for all $p$ between $\frac{m}{m-2}$ and $\frac{m}{2}$ but not for $p$ outside the closed interval $[\frac{m}{m-2}, \frac{m}{2}]$ if $V$ satisfies $|V(x)|\leq C \ax^{-m-3-\ep}$ and if $H$ is of exceptional type. This in particular implies that the continuous part of the propagator satisfies the $L^p$-$L^q$ estimates $\|e^{-itH}P_c(H)u \|_p \leq C |t|^{\frac{1}{m}\left(\frac12-\frac{1}{q}\right)}\|u\|_q$ for the dual exponents $\frac{1}{p}+\frac1{q}=1$ such that $1\leq q\leq 2 \leq p\leq \infty$ if $H$ is of generic type, and for $\frac{m}{m-2}< q\leq 2 \leq p < \frac{m}{2}$, $m \geq 5$, or $\frac32<q\leq 2 \leq p<3$, $m=3$, if $H$ of exceptional type. | |||||
書誌情報 |
Journal of mathematical sciences, the University of Tokyo 巻 13, 号 1, p. 43-93, 発行日 2006-03-21 |
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ISSN | ||||||
収録物識別子タイプ | ISSN | |||||
収録物識別子 | 13405705 | |||||
書誌レコードID | ||||||
収録物識別子タイプ | NCID | |||||
収録物識別子 | AA11021653 | |||||
フォーマット | ||||||
内容記述タイプ | Other | |||||
内容記述 | application/pdf | |||||
日本十進分類法 | ||||||
主題 | 415 | |||||
主題Scheme | NDC | |||||
Mathmatical Subject Classification | ||||||
35P25(MSC1991) | ||||||
Mathmatical Subject Classification | ||||||
35J10(MSC1991) | ||||||
Mathmatical Subject Classification | ||||||
47A40(MSC1991) | ||||||
Mathmatical Subject Classification | ||||||
47F05(MSC1991) | ||||||
Mathmatical Subject Classification | ||||||
47N50(MSC1991) | ||||||
Mathmatical Subject Classification | ||||||
81U50(MSC1991) | ||||||
出版者 | ||||||
出版者 | Graduate School of Mathematical Sciences, The University of Tokyo | |||||
原稿受領日 | ||||||
2005-12-26 |