Illinois Journal of Mathematics

Spectral multipliers for Schrödinger operators

Shijun Zheng

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We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int(1+|x|) |V(x)|\,dx$ is finite and $H$ has no resonance at zero. In the resonance case, we assume $\int(1+|x|^2) |V(x)|\, dx$ is finite.

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Illinois J. Math., Volume 54, Number 2 (2010), 621-647.

First available in Project Euclid: 14 October 2011

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Zentralblatt MATH identifier

Primary: 42B15: Multipliers 35J10: Schrödinger operator [See also 35Pxx]


Zheng, Shijun. Spectral multipliers for Schrödinger operators. Illinois J. Math. 54 (2010), no. 2, 621--647. doi:10.1215/ijm/1318598675.

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