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

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

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

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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doi:10.1007/s10473-023-0522-8

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

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 发布日期:2023-10-25

Yali PAN

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

Received:2021-09-28 Revised:2023-04-21 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： 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(英文版), 2023, 43(5): 2309-2319.

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

[1] Bisshop R L, Crittenden R J.Geometry of Manifolds. New York: Academic Press, 1964
[2] Bourgain J. A note on the Schrödinger maximal function. J Anal Math, 2016, 130: 393-396
[3] Cao Z, Fan D, Wang M. The rate of convergence on Schrödinger operator. Illinois Journal of Mathematics, 2018, 62: 365-380
[4] Carleson L.Some analytic problems related to statistical mechanics//Benedetto J. Euclidean Harmonic Analysis. Berlin, Heidelberg: Springer, 1980: 5-45
[5] Chen J, Fan D, Zhao F. On the rate of almost everywhere convergence of combinations and multivariate averages. Potential Anal, 2019, 51(3): 397-423
[6] Chen J, Fan D, Zhao F. Approximation of the fractional Schrödinger propagator on compact manifolds. Acta Math Sinica, 2021, 37(10): 1485-1496
[7] Coifman R, Weiss G. Extensions of Hardy spaces and their use analysis. Bull Amer math Soc, 1977, 83: 569-645
[8] Dahlberg B E J, Kenig C E. A note on the almost everywhere behavior of solutions to the Schrödinger equation//Harmonic Analysis (Minneapolis, Minn, 1981). Lecture Notes in Math, 908. New York: Springer-Verlag, 1982: 205-209
[9] Du X, Guth L, Li X. A sharp Schrödinger maximal estimate in

R^{2}

$\mathbb{R}^2$ . Ann of Math, 2017, 186: 607-640
[10] Du X, Guth L, Li X, Zhang R. Pointwise convergence of Schrödinger solutions and multilinear refined Strichartz estimates. Forum Math Sigma, 2018, 6: e14
[11] Du X, Zhang R. Sharp

L^{2}

$L^2$ estimates of the Schrödinger maximal function in higher dimensions. Ann of Math, 2019, 189: 837-861
[12] Liu H, Zeng H. Local estimate about Schrödinger maximal operator on H-type groups. Acta Math Sci, 2017, 37B(2): 527-538
[13] Marias M.

L^{p}

$L^p$ -boundedness of oscillating spectral multipliers on Riemannian manifolds. Annales Mathématiques, 2003, 10: 133-160
[14] Meaney C Müller D, Prestini E. A.e. convergence of spectral sums on Lie groups. Ann Inst Fourier (Grenoble), 2007, 57(5): 1509-1520
[15] Miao C, Yang J, Zheng J. An improved maximal inequality for 2D fractional order Schrödinger operators. Studia Mathematica, 2015, 230: 121-165
[16] Shubin M A. Special theory of the elliptic operators on noncompact manifolds. Astérisque, 1992, 207: 35-108
[17] Sjölin P. Regularity of solutions to the Schrödinger equation. Duke Math J, 1987, 55(3): 699-715
[18] Sjölin P.

L^{p}

$L^p$ maximal estimates for solutions to the Schrödinger equation. Math Scand, 1997, 81(1): 35-68
[19] Vega L. Schrödinger equations: Pointwise convergence to the initial data. Proc Amer Math Soc, 1988, 102(4): 874-878
[20] Walther B G. Higher integrability for maximal oscillatory Fourier integrals. Ann Acad Sci Fenn Math, 2001, 26(1): 189-204
[21] Walther B G. Global range estimates for maximal oscillatory integrals with radial test functions. Illinois J Math, 2012, 56(2): 521-532
[22] Zhang C. Pointwise convergence of solutions to Schrödinger type equations. Nonlinear Anal, 2014, 109: 180-186

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