SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS*

doi:10.1007/s10473-023-0522-8

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 2309-2319.doi: 10.1007/s10473-023-0522-8

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^*

Yali PAN

School of Mathematics and Big Data, Chaohu University, Hefei 238024, China

收稿日期:2021-09-28 修回日期:2023-04-21 出版日期:2023-10-26 发布日期:2023-10-25

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^*

Yali PAN

School of Mathematics and Big Data, Chaohu University, Hefei 238024, China

Received:2021-09-28 Revised:2023-04-21 Online:2023-10-26 Published:2023-10-25
About author:Yali PAN, E-mail: yalipan@zjnu.edu.cn
Supported by:
National Natural Science Foundation of China (12071437), the Natural Science Foundation from the Education Department of Anhui Province (KJ2020A0044), the Research Fund Project of Chaohu University (KYQD-2023016), the High Level Scientific Research Achievement Award Cultivation Project of Chaohu University (kj20zkjp04) and the Key Construction Discipline of Chaohu University (kj22zdjsxk01).

摘要/Abstract

摘要： Let $\mathcal{L}$ be the Laplace-Beltrami operator. On an $n$-dimensional $ (n\geq 2)$, complete, noncompact Riemannian manifold $\mathbb{M}$, we prove that if $0<\alpha<1, s>\alpha/2$ and $f \in H^s(\mathbb{M})$, then the fractional Schrödinger propagator ${\rm e}^{{\rm i}t|\mathcal{L}|^{\alpha/2}}(f)(x)\rightarrow f(x)$ a.e. as $t\rightarrow0$. In addition, for when $\mathbb{M}$ is a Lie group, the rate of the convergence is also studied. These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.

关键词: Schrödinger propagator, noncompact manifolds, spectra

Abstract: Let $\mathcal{L}$ be the Laplace-Beltrami operator. On an $n$-dimensional $ (n\geq 2)$, complete, noncompact Riemannian manifold $\mathbb{M}$, we prove that if $0<\alpha<1, s>\alpha/2$ and $f \in H^s(\mathbb{M})$, then the fractional Schrödinger propagator ${\rm e}^{{\rm i}t|\mathcal{L}|^{\alpha/2}}(f)(x)\rightarrow f(x)$ a.e. as $t\rightarrow0$. In addition, for when $\mathbb{M}$ is a Lie group, the rate of the convergence is also studied. These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.

Key words: Schrödinger propagator, noncompact manifolds, spectra

中图分类号:

58J05

Yali PAN. SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^*[J]. 数学物理学报, 2023, 43(5): 2309-2319.

Yali PAN. SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^*[J]. Acta mathematica scientia,Series A, 2023, 43(5): 2309-2319.

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SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^*

SOME CONVERGENCE PROBLEMS REGARDING THE FRACTIONAL SCHRÖDINGER PROPAGATOR ON NONCOMPACT MANIFOLDS^*

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