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

On the classical limit of the Schrödinger equation

Claude Bardos 1, , François Golse 2, , Peter Markowich 3, and  Thierry Paul 2,

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.
Keywords: WKB expansion, Maslov index., caustic, Lagrangian manifold, classical limit, Schrödinger equation, Fourier integral operators.
Mathematics Subject Classification: Primary: 35Q41, 81Q20; Secondary: 35S30, 53D1.
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]

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

[2]

V. I. Arnold, Characteristic class entering in quantization condition, Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.  Google Scholar

[3]

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

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 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), 71-111. 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), 323-379. 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., Providence, 1977.  Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, $2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 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, Berlin, Heidelberg, 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, Springer-Verlag, Berlin, Heidelberg, 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, Springer-Verlag, Berlin, Heidelberg, 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), 749-766. doi: 10.1142/S0129055X00000289.  Google Scholar

[13]

J. Leray, Lagrangian Analysis and Quantum Mechanics, The MIT Press, Cambridge, Mass., 1981.  Google Scholar

[14]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

D. Serre, Matrices, $2^{nd}$ edition, Springer-Verlag, New York, 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, T. Janssen and M. Boon), Lecture Notes in Phys. 50, Springer-Verlag, 1976, 117-148.  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, McGraw Hill, New York, 1966.  Google Scholar

[2]

V. I. Arnold, Characteristic class entering in quantization condition, Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.  Google Scholar

[3]

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

[4]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 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), 71-111. 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), 323-379. 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., Providence, 1977.  Google Scholar

[8]

L. Hörmander, The Analysis of Linear Partial Differential Operators I. Distribution Theory and Fourier Analysis, $2^{nd}$ edition, Springer-Verlag, Berlin, Heidelberg, 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, Berlin, Heidelberg, 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, Springer-Verlag, Berlin, Heidelberg, 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, Springer-Verlag, Berlin, Heidelberg, 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), 749-766. doi: 10.1142/S0129055X00000289.  Google Scholar

[13]

J. Leray, Lagrangian Analysis and Quantum Mechanics, The MIT Press, Cambridge, Mass., 1981.  Google Scholar

[14]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

D. Serre, Matrices, $2^{nd}$ edition, Springer-Verlag, New York, 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, T. Janssen and M. Boon), Lecture Notes in Phys. 50, Springer-Verlag, 1976, 117-148.  Google Scholar

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