SCATTERING FOR THE FRACTIONAL MAGNETIC SCHRÖODINGER OPERATORS

doi:10.1007/s10473-024-0618-9

Abstract

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

CLC Number:

35Q60

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

Trendmd

References

[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