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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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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 |