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

The Schrödinger equation with singular time-dependent potentials

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.
Citation: Holger Teismann. The Schrödinger equation with singular time-dependent potentials. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 705-722. doi: 10.3934/dcds.2000.6.705
[1]

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

[2]

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

[3]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & 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 & Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-17. doi: 10.3934/dcdsb.2018168

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Haim Brezis, Petru Mironescu. Composition in fractional Sobolev spaces. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 241-246. doi: 10.3934/dcds.2001.7.241

[15]

Alexander Zlotnik, Ilya Zlotnik. Finite element method with discrete transparent boundary conditions for the time-dependent 1D Schrödinger equation. Kinetic & Related Models, 2012, 5 (3) : 639-667. doi: 10.3934/krm.2012.5.639

[16]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903

[17]

Stephen Anco, Maria Rosa, Maria Luz Gandarias. Conservation laws and symmetries of time-dependent generalized KdV equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 607-615. doi: 10.3934/dcdss.2018035

[18]

Zhidong Zhang. An undetermined time-dependent coefficient in a fractional diffusion equation. Inverse Problems & Imaging, 2017, 11 (5) : 875-900. doi: 10.3934/ipi.2017041

[19]

Russell Johnson, Luca Zampogni. Remarks on a paper of Kotani concerning generalized reflectionless Schrödinger potentials. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 559-586. doi: 10.3934/dcdsb.2010.14.559

[20]

Antonio Azzollini, Pietro d’Avenia, Valeria Luisi. Generalized Schrödinger-Poisson type systems. Communications on Pure & Applied Analysis, 2013, 12 (2) : 867-879. doi: 10.3934/cpaa.2013.12.867

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]