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

TL;DR

This paper establishes decay estimates for the two-dimensional Schrödinger operator with a first-kind resonance at zero energy, extending results to matrix Schrödinger and wave equations under similar conditions.

Contribution

It provides a weighted decay estimate for the evolution operator in 2D Schrödinger equations with a zero-energy resonance, including extensions to matrix Schrödinger and wave equations.

Findings

Decay rate of 1/(|t| (log|t|)^2) for the evolution operator

Extension of results to matrix Schrödinger and wave equations

Conditions on potential decay and resonance type

Abstract

We study the two dimensional Schr\"odinger 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 \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.