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Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons

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Abstract

On gradient shrinking Ricci solitons, we observe that the study of Schrödinger heat kernel seems to be more natural than the classical heat kernel. In this paper we derive sharp Gaussian upper bounds for the Schrödinger heat kernel on complete gradient shrinking Ricci solitons. As applications, we prove sharp upper bounds for the Green’s function of the Schrödinger operator. We also prove sharp lower bounds for eigenvalues of the Schrödinger operator. These sharp cases are all achieved at Euclidean Gaussian shrinking Ricci solitons.

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Acknowledgements

The author thanks Professor Qi S. Zhang for helpful suggestions. The author also thanks the referee for making valuable comments and suggestions and pointing out many errors which helped to improve the presentation of this work. This work was partially supported by NSFS (17ZR1412800) and NSFC (11671141).

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  1. Department of Mathematics, Shanghai University, Shanghai, 200444, China

    Jia-Yong Wu

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Wu, JY. Sharp Gaussian upper bounds for Schrödinger heat kernel on gradient shrinking Ricci solitons. manuscripta math. 172, 1109–1132 (2023). https://doi.org/10.1007/s00229-022-01431-w

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