JournalsjstVol. 7, No. 4pp. 1235–1284

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

  • Ebru Toprak

    University of Illinois, Urbana, USA
A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy cover
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Abstract

We study the two dimensional Schrödinger operator, H=Δ+VH=-\Delta+V, in the weighted L1(R2)L(R2)L^1(\mathbb R^2) \to L^{\infty}(\mathbb R^2) setting when there is a resonance of the first kind at zero energy. In particular, we show that if V(x)x4|V(x)|\lesssim \langle x \rangle ^{-4-} and there is only s-wave resonance at zero of HH, then

w1(eitHPacf1πitFf)Ct(logt)2wf1,t>2,\|w^{-1} (e^{itH}P_{\mathrm {ac}} f - {\frac {1}{\pi it}} F f) \| _{\infty} \leq \frac {C}{|t| (\mathrm {log}|t|)^2} \|wf\|_1,\,\,\,\,\,|t| > 2,

with w(x)=log2(2+x)w(x)=\mathrm {log}^2(2+|x|). Here Ff=14ψψ,fFf=-{\frac {1}{4}} \psi\langle \psi,f \rangle, where ψ\psi is an s-wave resonance function. We also extend this result to wave and matrix Schrödinger equations with potentials under similar conditions.

Cite this article

Ebru Toprak, A weighted estimate for two dimensional Schrödinger, matrix Schrödinger, and wave equations with resonance of the first kind at zero energy. J. Spectr. Theory 7 (2017), no. 4, pp. 1235–1284

DOI 10.4171/JST/189