{"created":"2021-03-01T06:59:42.269999+00:00","id":40125,"links":{},"metadata":{"_buckets":{"deposit":"3f7abe15-6dda-48b5-bf16-ed716b145690"},"_deposit":{"id":"40125","owners":[],"pid":{"revision_id":0,"type":"depid","value":"40125"},"status":"published"},"_oai":{"id":"oai:repository.dl.itc.u-tokyo.ac.jp:00040125","sets":["312:6865:6959:6965","9:504:6868:6961:6966"]},"item_4_biblio_info_7":{"attribute_name":"書誌情報","attribute_value_mlt":[{"bibliographicIssueDates":{"bibliographicIssueDate":"2006-03-21","bibliographicIssueDateType":"Issued"},"bibliographicIssueNumber":"1","bibliographicPageEnd":"93","bibliographicPageStart":"43","bibliographicVolumeNumber":"13","bibliographic_titles":[{"bibliographic_title":"Journal of mathematical sciences, the University of Tokyo"}]}]},"item_4_description_13":{"attribute_name":"フォーマット","attribute_value_mlt":[{"subitem_description":"application/pdf","subitem_description_type":"Other"}]},"item_4_description_5":{"attribute_name":"抄録","attribute_value_mlt":[{"subitem_description":"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