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

Abstract

We study the two dimensional Schrödinger operator, $H=-\Delta +V$, in the weighted L^1(\R^2) \rightarrow L^{\infty}(\R^2) 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)=\log^2(2+|x|). Here Ff=c ψ\la f,ψ\ra, where ψis an s-wave resonance function. We also extend this result to matrix Schrödinger and wave equations with potentials under similar conditions.

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Cited by in corpus (8)

• On the Fourth order Schrödinger equation in three dimensions: dispersive estimates and zero energy resonances
• On the Fourth order Schrödinger equation in four dimensions: dispersive estimates and zero energy resonances
• The Dirac equation in two dimensions: Dispersive estimates and classification of threshold obstructions
• On the $L^p$ boundedness of wave operators for two-dimensional Schrödinger operators with threshold obstructions
• Dispersive estimates for massive Dirac operators in dimension two
• On the one dimensional Dirac equation with potential
• The Massless Dirac Equation in Two Dimensions: Zero-Energy Obstructions and Dispersive Estimates
• Scattering operator and wave operators for 2D Schrödinger operators with threshold obstructions