#### Title

Spectral Multipliers for Schrödinger Operators

#### Document Type

Article

#### Publication Date

2010

#### Publication Title

Illinois Journal of Mathematics

#### ISSN

0019-2082

#### Abstract

We prove a sharp Hörmander multiplier theorem for Schrödinger operators *H* = −Δ + *V* on ℝ*n*. The result is obtained under certain condition on a weighted *L*^{∞} 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 *∫*(1 + |*x*|)|*V*(*x*)| *dx* is finite and *H* has no resonance at zero. In the resonance case, we assume *∫*(1 + |*x*|^{2})|*V*(*x*)| *dx* is finite.

#### Recommended Citation

Zheng, Shijun.
2010.
"Spectral Multipliers for Schrödinger Operators."
*Illinois Journal of Mathematics*, 54 (2): 621-647.
source: http://projecteuclid.org/euclid.ijm/1318598675

https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/259