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May  2012, 17(3): 759-774. doi: 10.3934/dcdsb.2012.17.759

## Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials

 1 CNRS & Univ. Montpellier 2, Mathématiques, CC 051, 34095 Montpellier, France 2 Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 851 South Morgan Street, Chicago, Illinois 60607, United States

Received  March 2011 Revised  November 2011 Published  January 2012

We consider semiclassically scaled Schrödinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrödinger equation with time-dependent effective mass tensor. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales.
Citation: Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759
##### References:
 [1] G. Allaire and M. Palombaro, Localization for the Schrödinger equation in a locally periodic medium, SIAM J. Math. Anal., 38 (2006), 127-142 (electronic). doi: 10.1137/050635572.  Google Scholar [2] G. Allaire and A. Piatnitski, Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys., 258 (2005), 1-22. doi: 10.1007/s00220-005-1329-2.  Google Scholar [3] A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar [4] J. M. Bily and D. Robert, The semi-classical Van Vleck formula. Application to the Aharonov-Bohm effect, in "Long Time Behaviour of Classical and Quantum Systems" (Bologna, 1999), Ser. Concr. Appl. Math., 1, World Sci. Publ., River Edge, NJ, (2001), 89-106.  Google Scholar [5] A. Bouzouina and D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., 111 (2002), 223-252. doi: 10.1215/S0012-7094-02-11122-3.  Google Scholar [6] R. Carles and C. Fermanian-Kammerer, Nonlinear coherent states and Ehrenfest time for Schrödinger equation, Comm. Math. Phys., 301 (2011), 443-472. doi: 10.1007/s00220-010-1154-0.  Google Scholar [7] R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Statist. Phys., 117 (2004), 343-375. doi: 10.1023/B:JOSS.0000044070.34410.17.  Google Scholar [8] M. Combescure and D. Robert, Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptot. Anal., 14 (1997), 377-404.  Google Scholar [9] _____, Quadratic quantum Hamiltonians revisited, Cubo, 8 (2006), 61-86.  Google Scholar [10] S. De Bièvre and D. Robert, Semiclassical propagation on |logħ| time scales, Int. Math. Res. Not., (2003), 667-696. doi: 10.1155/S1073792803204268.  Google Scholar [11] M. Dimassi, J.-C. Guillot and J. Ralston, Gaussian beam construction for adiabatic perturbations, Math. Phys. Anal. Geom., 9 (2006), 187-201. doi: 10.1007/s11040-006-9009-9.  Google Scholar [12] E. Faou, V. Gradinaru and C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput., 31 (2009), 3027-3041. doi: 10.1137/080729724.  Google Scholar [13] P. Gérard, Mesures semi-classiques et ondes de Bloch, in "Séminaire sur les Équations aux Dérivées Partielles," 1990-1991, Exp. No. XVI, École Polytech., Palaiseau, (1991), 19 pp.  Google Scholar [14] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar [15] J.-C. Guillot, J. Ralston and E. Trubowitz, Semiclassical asymptotics in solid state physics, Comm. Math. Phys., 116 (1988), 401-415. doi: 10.1007/BF01229201.  Google Scholar [16] G. A. Hagedorn, Semiclassical quantum mechanics. I. The ħ $\rightarrow 0$ limit for coherent states, Comm. Math. Phys., 71 (1980), 77-93. doi: 10.1007/BF01230088.  Google Scholar [17] G. A. Hagedorn and A. Joye, Exponentially accurate semiclassical dynamics: Propagation, localization, Ehrenfest times, scattering, and more general states, Ann. Henri Poincaré, 1 (2000), 837-883. doi: 10.1007/PL00001017.  Google Scholar [18] _____, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), 583-626. doi: 10.1007/s002200100562.  Google Scholar [19] E. J. Heller, Frozen Gaussians: A very simple semiclassical approximation, J. Chem. Phys., 75 (1981), 2923-2931. doi: 10.1063/1.442382.  Google Scholar [20] M. A. Hoefer and M. I. Weinstein, Defect modes and homogenization of periodic Schrödinger operators, SIAM J. Math. Anal., 43 (2011), 971-996. doi: 10.1137/100807302.  Google Scholar [21] L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449. doi: 10.1007/BF02572374.  Google Scholar [22] F. Hövermann, H. Spohn and S. Teufel, Semiclassical limit for the Schrödinger equation with a short scale periodic potential, Comm. Math. Phys., 215 (2001), 609-629. doi: 10.1007/s002200000314.  Google Scholar [23] B. Ilan and M. I. Weinstein, Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations, and effective media, Multiscale Model. Simul., 8 (2010), 1055-1101. doi: 10.1137/090769417.  Google Scholar [24] H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 193-226.  Google Scholar [25] H. Kitada and H. Kumano-go, A family of Fourier integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math., 18 (1981), 291-360.  Google Scholar [26] R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291. doi: 10.1016/0370-1573(86)90103-1.  Google Scholar [27] J. Lu, W. E and X. Yang, Asymptotic analysis of the quantum dynamics: Bloch-Wigner transform and Bloch dynamics, to appear in Acta Math. Appl. Sin. Engl. Ser., 2011. Google Scholar [28] G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond, Comm. Math. Phys., 242 (2003), 547-578. doi: 10.1007/s00220-003-0950-1.  Google Scholar [29] T. Paul, Semi-classical methods with emphasis on coherent states, in "Quasiclassical Methods" (Minneapolis, MN, 1995), IMA Vol. Math. Appl., 95, Springer, New York, (1997), 51-88.  Google Scholar [30] _____, Échelles de temps pour l'évolution quantique à petite constante de Planck, in "Séminaire: Équations aux Dérivées Partielles," 2007-2008, Exp. No. IV, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2009), 21 pp.  Google Scholar [31] J. V. Ralston, On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys., 51 (1976), 219-242. doi: 10.1007/BF01617921.  Google Scholar [32] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar [33] D. Robert, Long time propagation results in quantum mechanics, in "Mathematical Results in Quantum Mechanics" (Taxco, 2001), Contemp. Math., 307, Amer. Math. Soc., Providence, RI, (2002), 255-264.  Google Scholar [34] _____, Revivals of wave packets and Bohr-Sommerfeld quantization rules, in "Adventures in Mathematical Physics," Contemp. Math., 447, Amer. Math. Soc., Providence, RI, (2007), 219-235.  Google Scholar [35] _____, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), 1123-1145. doi: 10.1142/S0129055X1000417X.  Google Scholar [36] V. Rousse, Semiclassical simple initial value representations,, to appear in Ark. för Mat., ().   Google Scholar [37] C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842 (electronic). doi: 10.1137/050623759.  Google Scholar [38] H. Spohn, Long time asymptotics for quantum particles in a periodic potential, Phys. Rev. Lett., 77 (1996), 1198-1201. doi: 10.1103/PhysRevLett.77.1198.  Google Scholar [39] T. Swart and V. Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys., 286 (2009), 725-750. doi: 10.1007/s00220-008-0681-4.  Google Scholar [40] S. Teufel, "Adiabatic Perturbation Theory in Quantum Dynamics," Lecture Notes in Mathematics, 1821, Springer-Verlag, Berlin, 2003.  Google Scholar [41] C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167. doi: 10.1007/BF02790171.  Google Scholar

show all references

##### References:
 [1] G. Allaire and M. Palombaro, Localization for the Schrödinger equation in a locally periodic medium, SIAM J. Math. Anal., 38 (2006), 127-142 (electronic). doi: 10.1137/050635572.  Google Scholar [2] G. Allaire and A. Piatnitski, Homogenization of the Schrödinger equation and effective mass theorems, Comm. Math. Phys., 258 (2005), 1-22. doi: 10.1007/s00220-005-1329-2.  Google Scholar [3] A. Bensoussan, J.-L. Lions and G. Papanicolaou, "Asymptotic Analysis for Periodic Structures," Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar [4] J. M. Bily and D. Robert, The semi-classical Van Vleck formula. Application to the Aharonov-Bohm effect, in "Long Time Behaviour of Classical and Quantum Systems" (Bologna, 1999), Ser. Concr. Appl. Math., 1, World Sci. Publ., River Edge, NJ, (2001), 89-106.  Google Scholar [5] A. Bouzouina and D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., 111 (2002), 223-252. doi: 10.1215/S0012-7094-02-11122-3.  Google Scholar [6] R. Carles and C. Fermanian-Kammerer, Nonlinear coherent states and Ehrenfest time for Schrödinger equation, Comm. Math. Phys., 301 (2011), 443-472. doi: 10.1007/s00220-010-1154-0.  Google Scholar [7] R. Carles, P. A. Markowich and C. Sparber, Semiclassical asymptotics for weakly nonlinear Bloch waves, J. Statist. Phys., 117 (2004), 343-375. doi: 10.1023/B:JOSS.0000044070.34410.17.  Google Scholar [8] M. Combescure and D. Robert, Semiclassical spreading of quantum wave packets and applications near unstable fixed points of the classical flow, Asymptot. Anal., 14 (1997), 377-404.  Google Scholar [9] _____, Quadratic quantum Hamiltonians revisited, Cubo, 8 (2006), 61-86.  Google Scholar [10] S. De Bièvre and D. Robert, Semiclassical propagation on |logħ| time scales, Int. Math. Res. Not., (2003), 667-696. doi: 10.1155/S1073792803204268.  Google Scholar [11] M. Dimassi, J.-C. Guillot and J. Ralston, Gaussian beam construction for adiabatic perturbations, Math. Phys. Anal. Geom., 9 (2006), 187-201. doi: 10.1007/s11040-006-9009-9.  Google Scholar [12] E. Faou, V. Gradinaru and C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput., 31 (2009), 3027-3041. doi: 10.1137/080729724.  Google Scholar [13] P. Gérard, Mesures semi-classiques et ondes de Bloch, in "Séminaire sur les Équations aux Dérivées Partielles," 1990-1991, Exp. No. XVI, École Polytech., Palaiseau, (1991), 19 pp.  Google Scholar [14] P. Gérard, P. A. Markowich, N. J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar [15] J.-C. Guillot, J. Ralston and E. Trubowitz, Semiclassical asymptotics in solid state physics, Comm. Math. Phys., 116 (1988), 401-415. doi: 10.1007/BF01229201.  Google Scholar [16] G. A. Hagedorn, Semiclassical quantum mechanics. I. The ħ $\rightarrow 0$ limit for coherent states, Comm. Math. Phys., 71 (1980), 77-93. doi: 10.1007/BF01230088.  Google Scholar [17] G. A. Hagedorn and A. Joye, Exponentially accurate semiclassical dynamics: Propagation, localization, Ehrenfest times, scattering, and more general states, Ann. Henri Poincaré, 1 (2000), 837-883. doi: 10.1007/PL00001017.  Google Scholar [18] _____, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), 583-626. doi: 10.1007/s002200100562.  Google Scholar [19] E. J. Heller, Frozen Gaussians: A very simple semiclassical approximation, J. Chem. Phys., 75 (1981), 2923-2931. doi: 10.1063/1.442382.  Google Scholar [20] M. A. Hoefer and M. I. Weinstein, Defect modes and homogenization of periodic Schrödinger operators, SIAM J. Math. Anal., 43 (2011), 971-996. doi: 10.1137/100807302.  Google Scholar [21] L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449. doi: 10.1007/BF02572374.  Google Scholar [22] F. Hövermann, H. Spohn and S. Teufel, Semiclassical limit for the Schrödinger equation with a short scale periodic potential, Comm. Math. Phys., 215 (2001), 609-629. doi: 10.1007/s002200000314.  Google Scholar [23] B. Ilan and M. I. Weinstein, Band-edge solitons, nonlinear Schrödinger/Gross-Pitaevskii equations, and effective media, Multiscale Model. Simul., 8 (2010), 1055-1101. doi: 10.1137/090769417.  Google Scholar [24] H. Kitada, On a construction of the fundamental solution for Schrödinger equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27 (1980), 193-226.  Google Scholar [25] H. Kitada and H. Kumano-go, A family of Fourier integral operators and the fundamental solution for a Schrödinger equation, Osaka J. Math., 18 (1981), 291-360.  Google Scholar [26] R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291. doi: 10.1016/0370-1573(86)90103-1.  Google Scholar [27] J. Lu, W. E and X. Yang, Asymptotic analysis of the quantum dynamics: Bloch-Wigner transform and Bloch dynamics, to appear in Acta Math. Appl. Sin. Engl. Ser., 2011. Google Scholar [28] G. Panati, H. Spohn and S. Teufel, Effective dynamics for Bloch electrons: Peierls substitution and beyond, Comm. Math. Phys., 242 (2003), 547-578. doi: 10.1007/s00220-003-0950-1.  Google Scholar [29] T. Paul, Semi-classical methods with emphasis on coherent states, in "Quasiclassical Methods" (Minneapolis, MN, 1995), IMA Vol. Math. Appl., 95, Springer, New York, (1997), 51-88.  Google Scholar [30] _____, Échelles de temps pour l'évolution quantique à petite constante de Planck, in "Séminaire: Équations aux Dérivées Partielles," 2007-2008, Exp. No. IV, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (2009), 21 pp.  Google Scholar [31] J. V. Ralston, On the construction of quasimodes associated with stable periodic orbits, Comm. Math. Phys., 51 (1976), 219-242. doi: 10.1007/BF01617921.  Google Scholar [32] M. Reed and B. Simon, "Methods of Modern Mathematical Physics. IV. Analysis of Operators," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978.  Google Scholar [33] D. Robert, Long time propagation results in quantum mechanics, in "Mathematical Results in Quantum Mechanics" (Taxco, 2001), Contemp. Math., 307, Amer. Math. Soc., Providence, RI, (2002), 255-264.  Google Scholar [34] _____, Revivals of wave packets and Bohr-Sommerfeld quantization rules, in "Adventures in Mathematical Physics," Contemp. Math., 447, Amer. Math. Soc., Providence, RI, (2007), 219-235.  Google Scholar [35] _____, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), 1123-1145. doi: 10.1142/S0129055X1000417X.  Google Scholar [36] V. Rousse, Semiclassical simple initial value representations,, to appear in Ark. för Mat., ().   Google Scholar [37] C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842 (electronic). doi: 10.1137/050623759.  Google Scholar [38] H. Spohn, Long time asymptotics for quantum particles in a periodic potential, Phys. Rev. Lett., 77 (1996), 1198-1201. doi: 10.1103/PhysRevLett.77.1198.  Google Scholar [39] T. Swart and V. Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys., 286 (2009), 725-750. doi: 10.1007/s00220-008-0681-4.  Google Scholar [40] S. Teufel, "Adiabatic Perturbation Theory in Quantum Dynamics," Lecture Notes in Mathematics, 1821, Springer-Verlag, Berlin, 2003.  Google Scholar [41] C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167. doi: 10.1007/BF02790171.  Google Scholar
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