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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 |
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. |
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. |
[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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