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Summer 2010 Spectral multipliers for Schrödinger operators
Shijun Zheng
Illinois J. Math. 54(2): 621-647 (Summer 2010). DOI: 10.1215/ijm/1318598675

Abstract

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.

Citation

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Shijun Zheng. "Spectral multipliers for Schrödinger operators." Illinois J. Math. 54 (2) 621 - 647, Summer 2010. https://doi.org/10.1215/ijm/1318598675

Information

Published: Summer 2010
First available in Project Euclid: 14 October 2011

zbMATH: 1235.42008
MathSciNet: MR2846476
Digital Object Identifier: 10.1215/ijm/1318598675

Subjects:
Primary: 35J10, 42B15

Rights: Copyright © 2010 University of Illinois at Urbana-Champaign

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Vol.54 • No. 2 • Summer 2010
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