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A weighted estimate for two dimensional Schrodinger, matrix schrodinger and wave equations with resonance of first kind at zero energy (1509.03204v1)

Published 10 Sep 2015 in math.AP

Abstract: We study the two dimensional Schr\"odinger operator, $H=-\Delta+V$, in the weighted L1(\R2) \rightarrow L{\infty}(\R2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if |V(x)|\les \la x \ra {-3-} and there is only s-wave resonance at zero of H, then \big| w{-1} \big( e{itH}P_{ac} f - {\f 1 t } F f \big) \big| _{\infty} \leq \frac {C} {|t| (\log|t|)2 } |wf|_1 |t|>2, with w(x)=\log2(2+|x|). Here Ff=c \psi\la f,\psi \ra, where \psi is an s-wave resonance function. We also extend this result to matrix Schr\"odinger and wave equations with potentials under similar conditions.

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