American Institute of Mathematical Sciences

Advanced
July  2000, 6(3): 705-722. doi: 10.3934/dcds.2000.6.705

The Schrödinger equation with singular time-dependent potentials

Holger Teismann 1,

1. 

Department of Mathematics and Statistics, University of Victoria, P.O. BOX 3045, Victoria, B.C., Canada

Received  August 1999 Revised  March 2000 Published  April 2000

The aim of this note is to extend the theory of (linear) Schrödinger equations with time-dependent potentials developed by K. Yajima [26, 27] to slightly more singular potentials. This is done by proving that the well-known Strichartz estimates for the Schrödinger group remain valid if the usual Lebesgue spaces$^1$ are replaced by the Lorentz spaces $L^{p,2}$. Moreover, the regularity of the solutions can be described more precisely by utilizing a generalized Leibniz rule for fractional derivatives.
Keywords: Strichartz estimates, well-posedeness, generalized Leibnitz rule., time-dependent potentials, Schrödinger equation, generalized Sobolev spaces, Lorentz spaces, evolution systems, fractional derivatives.
Mathematics Subject Classification: 35Q40, 47D06, 35J1.
Citation: Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete and Continuous Dynamical Systems, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[1]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210
[2]

Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265
[3]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks and Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1
[4]

P. Cerejeiras, U. Kähler, M. M. Rodrigues, N. Vieira. Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253
[5]

Vagif S. Guliyev, Ramin V. Guliyev, Mehriban N. Omarova, Maria Alessandra Ragusa. Schrödinger type operators on local generalized Morrey spaces related to certain nonnegative potentials. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 671-690. doi: 10.3934/dcdsb.2019260
[6]

Akio Ito, Noriaki Yamazaki, Nobuyuki Kenmochi. Attractors of nonlinear evolution systems generated by time-dependent subdifferentials in Hilbert spaces. Conference Publications, 1998, 1998 (Special) : 327-349. doi: 10.3934/proc.1998.1998.327
[7]

Yavar Kian, Alexander Tetlow. Hölder-stable recovery of time-dependent electromagnetic potentials appearing in a dynamical anisotropic Schrödinger equation. Inverse Problems and Imaging, 2020, 14 (5) : 819-839. doi: 10.3934/ipi.2020038
[8]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521
[9]

Mourad Bellassoued, Oumaima Ben Fraj. Stability estimates for time-dependent coefficients appearing in the magnetic Schrödinger equation from arbitrary boundary measurements. Inverse Problems and Imaging, 2020, 14 (5) : 841-865. doi: 10.3934/ipi.2020039
[10]

Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723
[11]

Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143
[12]

Valerii Los, Vladimir Mikhailets, Aleksandr Murach. Parabolic problems in generalized Sobolev spaces. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3605-3636. doi: 10.3934/cpaa.2021123
[13]

Kazuhiro Ishige, Yujiro Tateishi. Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 369-401. doi: 10.3934/dcds.2021121
[14]

Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4361-4395. doi: 10.3934/dcdsb.2018168
[15]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905
[16]

Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations and Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009
[17]

Haruya Mizutani. Strichartz estimates for Schrödinger equations with variable coefficients and unbounded potentials II. Superquadratic potentials. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2177-2210. doi: 10.3934/cpaa.2014.13.2177
[18]

Michael S. Jolly, Anuj Kumar, Vincent R. Martinez. On local well-posedness of logarithmic inviscid regularizations of generalized SQG equations in borderline Sobolev spaces. Communications on Pure and Applied Analysis, 2022, 21 (1) : 101-120. doi: 10.3934/cpaa.2021169
[19]

Russell Johnson, Luca Zampogni. Some examples of generalized reflectionless Schrödinger potentials. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1149-1170. doi: 10.3934/dcdss.2016046
[20]

Nakao Hayashi, Elena I. Kaikina, Pavel I. Naumkin. Large time behavior of solutions to the generalized derivative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 93-106. doi: 10.3934/dcds.1999.5.93

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy