SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS

doi:10.1007/s10473-024-0618-9

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (6): 2391-2410.doi: 10.1007/s10473-024-0618-9

SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS

Lei WEI, Zhiwen DUAN^†

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

收稿日期:2023-07-04 修回日期:2024-05-01 发布日期:2024-12-06
通讯作者: † Zhiwen DUAN, E-mail: duanzhw@hust.edu.cn
作者简介:Lei WEI, E-mail: weileimath@hust.edu.cn
基金资助:
NSFC (61671009, 12171178).

SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS

Lei WEI, Zhiwen DUAN^†

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2023-07-04 Revised:2024-05-01 Published:2024-12-06
Contact: † Zhiwen DUAN, E-mail: duanzhw@hust.edu.cn
About author:Lei WEI, E-mail: weileimath@hust.edu.cn
Supported by:
NSFC (61671009, 12171178).

摘要/Abstract

摘要： In this paper, we prove the existence of the scattering operator for the fractional magnetic Schrödinger operators. In order to do this, we construct the fractional distorted Fourier transforms with magnetic potentials. Applying the properties of the distorted Fourier transforms, the existence and the asymptotic completeness of the wave operators are obtained. Furthermore, we prove the absence of positive eigenvalues for fractional magnetic Schrödinger operators.

关键词: magnetic Schrödinger operators, fractional, scattering, distorted Fourier transform

Abstract: In this paper, we prove the existence of the scattering operator for the fractional magnetic Schrödinger operators. In order to do this, we construct the fractional distorted Fourier transforms with magnetic potentials. Applying the properties of the distorted Fourier transforms, the existence and the asymptotic completeness of the wave operators are obtained. Furthermore, we prove the absence of positive eigenvalues for fractional magnetic Schrödinger operators.

Key words: magnetic Schrödinger operators, fractional, scattering, distorted Fourier transform

中图分类号:

35Q60

Lei WEI, Zhiwen DUAN. SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS[J]. 数学物理学报(英文版), 2024, 44(6): 2391-2410.

Lei WEI, Zhiwen DUAN. SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS[J]. Acta mathematica scientia,Series B, 2024, 44(6): 2391-2410.

参考文献

[1] Amrein W O, Berthier A M, Georgescu V.$L^{p}$ inequalities for the Lalacian and unique continuation. Ann Inst Fourier (Grenoble), 1981, 31: 153-168
[2] Agmon S.Spectral properties of Schrödinger operators and scattering theory. Ann Sc Norm Sup Pisa Cl Sci, 1975, 2: 151-218
[3] D'Ancona P, Fanelli L. Strichartz and smoothing estimates for dispersive equation with magnetic potentials. Comm Part Diff Equs, 2008, 33: 1082-1112
[4] D'Ancona P, Fanelli L, Vega L, Visciglia N. Endpoint Strichartz estimates for the magnetic Schrödinger equation. J Funct Anal, 2010, 258: 3227-3240
[5] Demuth M, Ouhabaz E M.Scattering for Schrödinger operators with magnetic fields. Math Nachr, 1997, 185: 49-58
[6] Erdoğan M B, Goldberg M, Schlag W.Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in $\mathbb{R}^{3}$. J Eur Math Soc, 2008, 10: 507-531
[7] Feng H L, Wang H, Yao X H.Scattering theory for the defocusing fourth order NLS with potentials. Acta Mathematica Sinica, 2018, 34(4): 773-786
[8] Giere E. Asymptotic completeness for functions of the Laplacian perturbed by potentials and obstacles. Math Nachr, 2004, 263-264: 133-153
[9] Georgiev V, Stefanov A, Tarulli M.Smoothing-Strichartz estimates for the Schrödinger equation with small magnetic potential. Discrete Contin Dyn Syst, 2007, 17: 771-786
[10] Hörmander L.The Analysis of Linear Partial Differential Operators Ⅰ-Ⅳ. Berlin: Springer-Verlag, 1983-1985
[11] Iwatsuka A.Spectral representation for Schrödinger operator with magnetic vector potential. J Math Kyoto Univ, 1982, 22: 223-242
[12] Kitada H.Scattering theory for the fractional power of negative Laplacian. J Abstr Differ Equ Appl, 2010, 1: 1-26
[13] Kato T.Perturbation Theory for Linear Operators. Heidelberg: Springer-Verlag, 1980
[14] Komech A I, Kopylova E A.Dispersive decay for the magnetic Schrödinger equation. J Funct Anal, 2013, 264: 735-751
[15] Koch H, Tataru D.Carleman estimates and absence of embedded eigenvalues. Commun Math Phys, 2006, 267: 419-449
[16] Reed M, Simon B.Methods of Modern Mathematical Physics. Ⅲ: Scattering Theory. New York: Academic Press, 1979
[17] Schlag W.Intertwing wave operators, Fourier restriction, and Wiener theorems. Conference Proceedings Kato Centenntial Meeting, Tokyo, 2018
[18] Zhang R, Huang T, Zheng Q.The scattering of fractional Schrödinger operators with short range potentials. J Funct Anal, 2021, 281: 109033

SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS

SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS

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