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 - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689
References:
[1]

L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable,, $2^{nd}$ edition, (1966).   Google Scholar

[2]

V. I. Arnold, Characteristic class entering in quantization condition,, Func. Anal. Appl., 1 (1967), 1.  doi: 10.1007/BF01075861.  Google Scholar

[3]

V. I. Arnold, Geometrical Methods of the Theory of Ordinary Differential Equations,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4612-1037-5.  Google Scholar

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Springer-Verlag, (1989).  doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[5]

C. Bardos, F. Golse, P. Markowich and T. Paul, Hamiltonian evolution of monokinetic measures with rough momentum profile,, Archive for Rational Mechanics and Analysis, 217 (2015), 71.  doi: 10.1007/s00205-014-0829-7.  Google Scholar

[6]

P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limit and Wigner transforms,, Comm. on Pure and App. Math., 50 (1997), 323.  doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar

[7]

V. Guillemin and S. Sternberg, Geometric Asymptotics,, Amer. Math. Soc., (1977).   Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis,, $2^{nd}$ edition, (1990).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[9]

L. Hörmander, The Analysis of Linear Partial Differential Operators II. Differential Operators with Constant Coefficients,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[10]

L. Hörmander, The Analysis of Linear Partial Differential Operators III. Pseudo-differential Operators,, $2^{nd}$ edition, (1994).  doi: 10.1007/978-3-540-49938-1.  Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators,, $2^{nd}$ edition, (1994).  doi: 10.1007/978-3-642-00136-9.  Google Scholar

[12]

A. Laptev and I. Sigal, Global Fourier integral operators and semiclassical asymptotics,, Review of Math. Phys., 12 (2000), 749.  doi: 10.1142/S0129055X00000289.  Google Scholar

[13]

J. Leray, Lagrangian Analysis and Quantum Mechanics,, The MIT Press, (1981).   Google Scholar

[14]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553.  doi: 10.4171/RMI/143.  Google Scholar

[15]

V. P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques,, Dunod, (1972).   Google Scholar

[16]

V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Quantum Mechanics,, Reidel Publishing Company, (1981).   Google Scholar

[17]

J. Milnor, Morse Theory,, Princeton Univ. Press, (1963).   Google Scholar

[18]

D. Serre, Matrices,, $2^{nd}$ edition, (2010).  doi: 10.1007/978-1-4419-7683-3.  Google Scholar

[19]

J.-M. Souriau, Construction explicite de l'indice de Maslov,, in Group Theoretical Methods in Physics (eds. A. Janner, (1976), 117.   Google Scholar

show all references

References:
[1]

L. V. Ahlfors, Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable,, $2^{nd}$ edition, (1966).   Google Scholar

[2]

V. I. Arnold, Characteristic class entering in quantization condition,, Func. Anal. Appl., 1 (1967), 1.  doi: 10.1007/BF01075861.  Google Scholar

[3]

V. I. Arnold, Geometrical Methods of the Theory of Ordinary Differential Equations,, Springer-Verlag, (1988).  doi: 10.1007/978-1-4612-1037-5.  Google Scholar

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics,, Springer-Verlag, (1989).  doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[5]

C. Bardos, F. Golse, P. Markowich and T. Paul, Hamiltonian evolution of monokinetic measures with rough momentum profile,, Archive for Rational Mechanics and Analysis, 217 (2015), 71.  doi: 10.1007/s00205-014-0829-7.  Google Scholar

[6]

P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limit and Wigner transforms,, Comm. on Pure and App. Math., 50 (1997), 323.  doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar

[7]

V. Guillemin and S. Sternberg, Geometric Asymptotics,, Amer. Math. Soc., (1977).   Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis,, $2^{nd}$ edition, (1990).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[9]

L. Hörmander, The Analysis of Linear Partial Differential Operators II. Differential Operators with Constant Coefficients,, Springer-Verlag, (1983).  doi: 10.1007/978-3-642-96750-4.  Google Scholar

[10]

L. Hörmander, The Analysis of Linear Partial Differential Operators III. Pseudo-differential Operators,, $2^{nd}$ edition, (1994).  doi: 10.1007/978-3-540-49938-1.  Google Scholar

[11]

L. Hörmander, The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators,, $2^{nd}$ edition, (1994).  doi: 10.1007/978-3-642-00136-9.  Google Scholar

[12]

A. Laptev and I. Sigal, Global Fourier integral operators and semiclassical asymptotics,, Review of Math. Phys., 12 (2000), 749.  doi: 10.1142/S0129055X00000289.  Google Scholar

[13]

J. Leray, Lagrangian Analysis and Quantum Mechanics,, The MIT Press, (1981).   Google Scholar

[14]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553.  doi: 10.4171/RMI/143.  Google Scholar

[15]

V. P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques,, Dunod, (1972).   Google Scholar

[16]

V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Quantum Mechanics,, Reidel Publishing Company, (1981).   Google Scholar

[17]

J. Milnor, Morse Theory,, Princeton Univ. Press, (1963).   Google Scholar

[18]

D. Serre, Matrices,, $2^{nd}$ edition, (2010).  doi: 10.1007/978-1-4419-7683-3.  Google Scholar

[19]

J.-M. Souriau, Construction explicite de l'indice de Maslov,, in Group Theoretical Methods in Physics (eds. A. Janner, (1976), 117.   Google Scholar

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253

[18]

Woocheol Choi, Yong-Cheol Kim. The Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators with certain potentials. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1993-2010. doi: 10.3934/cpaa.2018095

[19]

Niels Jacob, Feng-Yu Wang. Higher order eigenvalues for non-local Schrödinger operators. Communications on Pure & Applied Analysis, 2018, 17 (1) : 191-208. doi: 10.3934/cpaa.2018012

[20]

Roberta Bosi, Jean Dolbeault, Maria J. Esteban. Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators. Communications on Pure & Applied Analysis, 2008, 7 (3) : 533-562. doi: 10.3934/cpaa.2008.7.533

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (2)

[Back to Top]