On the classical limit of the Schrödinger equation

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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

Abstract
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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 and 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.
[2]	V. I. Arnold, Characteristic class entering in quantization condition, Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.
[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.
[4]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.
[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.
[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.
[7]	V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, 1977.
[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.
[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.
[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.
[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.
[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.
[13]	J. Leray, Lagrangian Analysis and Quantum Mechanics, The MIT Press, Cambridge, Mass., 1981.
[14]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.
[15]	V. P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques, Dunod, Paris, 1972.
[16]	V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Quantum Mechanics, Reidel Publishing Company, Dordrecht, 1981.
[17]	J. Milnor, Morse Theory, Princeton Univ. Press, Princeton NJ, 1963.
[18]	D. Serre, Matrices, $2^{nd}$ edition, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-7683-3.
[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.

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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.
[2]	V. I. Arnold, Characteristic class entering in quantization condition, Func. Anal. Appl., 1 (1967), 1-14. doi: 10.1007/BF01075861.
[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.
[4]	V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.
[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.
[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.
[7]	V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, 1977.
[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.
[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.
[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.
[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.
[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.
[13]	J. Leray, Lagrangian Analysis and Quantum Mechanics, The MIT Press, Cambridge, Mass., 1981.
[14]	P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.
[15]	V. P. Maslov, Théorie des Perturbations et Méthodes Asymptotiques, Dunod, Paris, 1972.
[16]	V. P. Maslov and M. V. Fedoryuk, Semiclassical Approximation in Quantum Mechanics, Reidel Publishing Company, Dordrecht, 1981.
[17]	J. Milnor, Morse Theory, Princeton Univ. Press, Princeton NJ, 1963.
[18]	D. Serre, Matrices, $2^{nd}$ edition, Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-7683-3.
[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.

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