کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4589799 1334909 2015 50 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues I: The odd dimensional case
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Dispersive estimates for higher dimensional Schrödinger operators with threshold eigenvalues I: The odd dimensional case
چکیده انگلیسی

We investigate L1(Rn)→L∞(Rn)L1(Rn)→L∞(Rn) dispersive estimates for the Schrödinger operator H=−Δ+VH=−Δ+V when there is an eigenvalue at zero energy and n≥5n≥5 is odd. In particular, we show that if there is an eigenvalue at zero energy then there is a time dependent, rank one operator FtFt satisfying ‖Ft‖L1→L∞≲|t|2−n2 for |t|>1|t|>1 such that‖eitHPac−Ft‖L1→L∞≲|t|1−n2,for |t|>1. With stronger decay conditions on the potential it is possible to generate an operator-valued expansion for the evolution, taking the formeitHPac(H)=|t|2−n2A−2+|t|1−n2A−1+|t|−n2A0, with A−2A−2 and A−1A−1 finite rank operators mapping L1(Rn)L1(Rn) to L∞(Rn)L∞(Rn) while A0A0 maps weighted L1L1 spaces to weighted L∞L∞ spaces. The leading order terms A−2A−2 and A−1A−1 vanish when certain orthogonality conditions between the potential V   and the zero energy eigenfunctions are satisfied. We show that under the same orthogonality conditions, the remaining |t|−n2A0 term also exists as a map from L1(Rn)L1(Rn) to L∞(Rn)L∞(Rn), hence eitHPac(H)eitHPac(H) satisfies the same dispersive bounds as the free evolution despite the eigenvalue at zero.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Functional Analysis - Volume 269, Issue 3, 1 August 2015, Pages 633–682
نویسندگان
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