American Institute of Mathematical Sciences

Advanced
December  2010, 28(4): 1381-1412. doi: 10.3934/dcds.2010.28.1381

A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators

David Damanik 1, and  Serguei Tcheremchantsev 2,

1. 

Department of Mathematics, Rice University, Houston, TX 77005
2. 

UMR 6628–MAPMO, Université d’Orléans, B.P. 6759, F-45067 Orléans Cedex, France

Received  October 2009 Revised  February 2010 Published  June 2010

We discuss the long-time behavior of solutions to the Schrödinger equation in some separable Hilbert space, with particular emphasis on the spreading over some orthonormal basis. Various ways of studying wavepacket spreading from this perspective are described and their inter-relations investigated. We also state and discuss known results for concrete quantum systems relative to this general framework.
Keywords: Schrödinger equation, Schrödinger operators., Quantum dynamics.
Mathematics Subject Classification: Primary: 81Q10; Secondary: 47B36, 47N5.
Citation: David Damanik, Serguei Tcheremchantsev. A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators. Discrete and Continuous Dynamical Systems, 2010, 28 (4) : 1381-1412. doi: 10.3934/dcds.2010.28.1381
[1]

Jaime Cruz-Sampedro. Schrödinger-like operators and the eikonal equation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 495-510. doi: 10.3934/cpaa.2014.13.495
[2]

José Luis López. A quantum approach to Keller-Segel dynamics via a dissipative nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2601-2617. doi: 10.3934/dcds.2020376
[3]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288
[4]

Takahisa Inui. Global dynamics of solutions with group invariance for the nonlinear schrödinger equation. Communications on Pure and Applied Analysis, 2017, 16 (2) : 557-590. doi: 10.3934/cpaa.2017028
[5]

Hiroshi Isozaki, Hisashi Morioka. A Rellich type theorem for discrete Schrödinger operators. Inverse Problems and Imaging, 2014, 8 (2) : 475-489. doi: 10.3934/ipi.2014.8.475
[6]

Fritz Gesztesy, Roger Nichols. On absence of threshold resonances for Schrödinger and Dirac operators. Discrete and Continuous Dynamical Systems - S, 2020, 13 (12) : 3427-3460. doi: 10.3934/dcdss.2020243
[7]

Fengping Yao. Optimal regularity for parabolic Schrödinger operators. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1407-1414. doi: 10.3934/cpaa.2013.12.1407
[8]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[9]

Jean Bourgain. On random Schrödinger operators on $\mathbb Z^2$. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 1-15. doi: 10.3934/dcds.2002.8.1
[10]

Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control and Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
[11]

Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
[12]

D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563
[13]

Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791
[14]

Shi Jin, Peng Qi. A hybrid Schrödinger/Gaussian beam solver for quantum barriers and surface hopping. Kinetic and Related Models, 2011, 4 (4) : 1097-1120. doi: 10.3934/krm.2011.4.1097
[15]

Roberto de A. Capistrano–Filho, Márcio Cavalcante, Fernando A. Gallego. Forcing operators on star graphs applied for the cubic fourth order Schrödinger equation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3399-3434. doi: 10.3934/dcdsb.2021190
[16]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[17]

Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete and Continuous Dynamical Systems, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807
[18]

Alexander Arbieto, Carlos Matheus. On the periodic Schrödinger-Debye equation. Communications on Pure and Applied Analysis, 2008, 7 (3) : 699-713. doi: 10.3934/cpaa.2008.7.699
[19]

Rossella Bartolo, Anna Maria Candela, Addolorata Salvatore. Infinitely many solutions for a perturbed Schrödinger equation. Conference Publications, 2015, 2015 (special) : 94-102. doi: 10.3934/proc.2015.0094
[20]

Tarek Saanouni. Remarks on the damped nonlinear Schrödinger equation. Evolution Equations and Control Theory, 2020, 9 (3) : 721-732. doi: 10.3934/eect.2020030

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (139)
  • HTML views (0)
  • Cited by (14)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy