Abstract
Consider the Schrödinger operator on , where is a nonnegative and locally bounded potential on so that for all with , with some constants and a nondecreasing and strictly positive function that satisfies for all and . We establish global in time and qualitatively sharp bounds for the heat kernel of the associated Schrödinger semigroup by the probabilistic method. In particular, we can present global in space and time two-sided bounds of heat kernel even when the Schrödinger semigroup is not intrinsically ultracontractive. Furthermore, two-sided estimates for the corresponding Green’s function are also obtained.
Funding Statement
The research of Xin Chen is supported by the National Natural Science Foundation of China (No. 12122111).
The research of Jian Wang is supported by the National Key R&D Program of China (2022YFA1006003) and the National Natural Science Foundation of China (Nos. 11831014, 12071076 and 12225104).
Acknowledgments
We thank the two referees for helpful comments on an earlier version of our paper.
Jian Wang is also affiliated with the School of Mathematics and Statistics & Key Laboratory of Analytical Mathematics and Applications (Ministry of Education) and Fujian Provincial Key Laboratory of Statistics and Artificial Intelligence, Fujian Normal University.
Citation
Xin Chen. Jian Wang. "Two-sided heat kernel estimates for Schrödinger operators with unbounded potentials." Ann. Probab. 52 (3) 1016 - 1047, May 2024. https://doi.org/10.1214/23-AOP1680
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