December  2015, 35(12): 5689-5709. doi: 10.3934/dcds.2015.35.5689

On the classical limit of the Schrödinger equation

1. 

Université Paris-Diderot, Laboratoire J.-L. Lions, BP187, 4 place Jussieu, 75252 Paris Cedex 05, France

2. 

Ecole polytechnique, CMLS, 91128 Palaiseau Cedex, France, France

3. 

King Abdullah University of Science and Technology, MCSE Division, Thuwal 23955-6900, Saudi Arabia

Received  April 2014 Published  May 2015

This paper provides an elementary proof of the classical limit of the Schrödinger equation with WKB type initial data and over arbitrary long finite time intervals. We use only the stationary phase method and the Laptev-Sigal simple and elegant construction of a parametrix for Schrödinger type equations [A. Laptev, I. Sigal, Review of Math. Phys. 12 (2000), 749--766]. We also explain in detail how the phase shifts across caustics obtained when using the Laptev-Sigal parametrix are related to the Maslov index.
Citation: Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
References:
[1]

$2^{nd}$ edition, McGraw Hill, New York, 1966.  Google Scholar

[2]

Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.  Google Scholar

[3]

Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1037-5.  Google Scholar

[4]

Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[5]

Archive for Rational Mechanics and Analysis, 217 (2015), 71-111. doi: 10.1007/s00205-014-0829-7.  Google Scholar

[6]

Comm. on Pure and App. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar

[7]

Amer. Math. Soc., Providence, 1977.  Google Scholar

[8]

$2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[9]

Springer-Verlag, Berlin, Heidelberg, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[10]

$2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-540-49938-1.  Google Scholar

[11]

$2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-00136-9.  Google Scholar

[12]

Review of Math. Phys., 12 (2000), 749-766. doi: 10.1142/S0129055X00000289.  Google Scholar

[13]

The MIT Press, Cambridge, Mass., 1981.  Google Scholar

[14]

Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.  Google Scholar

[15]

Dunod, Paris, 1972. Google Scholar

[16]

Reidel Publishing Company, Dordrecht, 1981.  Google Scholar

[17]

Princeton Univ. Press, Princeton NJ, 1963.  Google Scholar

[18]

$2^{nd}$ edition, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-7683-3.  Google Scholar

[19]

in Group Theoretical Methods in Physics (eds. A. Janner, T. Janssen and M. Boon), Lecture Notes in Phys. 50, Springer-Verlag, 1976, 117-148.  Google Scholar

show all references

References:
[1]

$2^{nd}$ edition, McGraw Hill, New York, 1966.  Google Scholar

[2]

Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.  Google Scholar

[3]

Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4612-1037-5.  Google Scholar

[4]

Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[5]

Archive for Rational Mechanics and Analysis, 217 (2015), 71-111. doi: 10.1007/s00205-014-0829-7.  Google Scholar

[6]

Comm. on Pure and App. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar

[7]

Amer. Math. Soc., Providence, 1977.  Google Scholar

[8]

$2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 1990. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[9]

Springer-Verlag, Berlin, Heidelberg, 1983. doi: 10.1007/978-3-642-96750-4.  Google Scholar

[10]

$2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-540-49938-1.  Google Scholar

[11]

$2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 1994. doi: 10.1007/978-3-642-00136-9.  Google Scholar

[12]

Review of Math. Phys., 12 (2000), 749-766. doi: 10.1142/S0129055X00000289.  Google Scholar

[13]

The MIT Press, Cambridge, Mass., 1981.  Google Scholar

[14]

Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.  Google Scholar

[15]

Dunod, Paris, 1972. Google Scholar

[16]

Reidel Publishing Company, Dordrecht, 1981.  Google Scholar

[17]

Princeton Univ. Press, Princeton NJ, 1963.  Google Scholar

[18]

$2^{nd}$ edition, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-7683-3.  Google Scholar

[19]

in Group Theoretical Methods in Physics (eds. A. Janner, T. Janssen and M. Boon), Lecture Notes in Phys. 50, Springer-Verlag, 1976, 117-148.  Google Scholar

[1]

César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067

[2]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7

[3]

Elena Cordero, Fabio Nicola, Luigi Rodino. Time-frequency analysis of fourier integral operators. Communications on Pure & Applied Analysis, 2010, 9 (1) : 1-21. doi: 10.3934/cpaa.2010.9.1

[4]

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

[5]

David Damanik, Zheng Gan. Spectral properties of limit-periodic Schrödinger operators. Communications on Pure & Applied Analysis, 2011, 10 (3) : 859-871. doi: 10.3934/cpaa.2011.10.859

[6]

Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415

[7]

Lihui Chai, Shi Jin, Qin Li. Semi-classical models for the Schrödinger equation with periodic potentials and band crossings. Kinetic & Related Models, 2013, 6 (3) : 505-532. doi: 10.3934/krm.2013.6.505

[8]

Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018

[9]

Juhi Jang, Ning Jiang. Acoustic limit of the Boltzmann equation: Classical solutions. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 869-882. doi: 10.3934/dcds.2009.25.869

[10]

Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419

[20]

M. Burak Erdoǧan, William R. Green. Dispersive estimates for matrix Schrödinger operators in dimension two. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4473-4495. doi: 10.3934/dcds.2013.33.4473

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (2)

[Back to Top]