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Two theorems on singularly perturbed semigroups with applications to models of applied mathematics
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
_____, Quadratic quantum Hamiltonians revisited, Cubo, 8 (2006), 61-86. |
[10] |
S. De Bièvre and D. Robert, Semiclassical propagation on |logħ| time scales, Int. Math. Res. Not., (2003), 667-696.
doi: 10.1155/S1073792803204268. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
_____, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), 583-626.
doi: 10.1007/s002200100562. |
[19] |
E. J. Heller, Frozen Gaussians: A very simple semiclassical approximation, J. Chem. Phys., 75 (1981), 2923-2931.
doi: 10.1063/1.442382. |
[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. |
[21] |
L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.
doi: 10.1007/BF02572374. |
[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. |
[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. |
[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. |
[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. |
[26] |
R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291.
doi: 10.1016/0370-1573(86)90103-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
_____, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), 1123-1145.
doi: 10.1142/S0129055X1000417X. |
[36] |
V. Rousse, Semiclassical simple initial value representations,, to appear in Ark. för Mat., ().
|
[37] |
C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842 (electronic).
doi: 10.1137/050623759. |
[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. |
[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. |
[40] |
S. Teufel, "Adiabatic Perturbation Theory in Quantum Dynamics," Lecture Notes in Mathematics, 1821, Springer-Verlag, Berlin, 2003. |
[41] |
C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167.
doi: 10.1007/BF02790171. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
_____, Quadratic quantum Hamiltonians revisited, Cubo, 8 (2006), 61-86. |
[10] |
S. De Bièvre and D. Robert, Semiclassical propagation on |logħ| time scales, Int. Math. Res. Not., (2003), 667-696.
doi: 10.1155/S1073792803204268. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[18] |
_____, A time-dependent Born-Oppenheimer approximation with exponentially small error estimates, Comm. Math. Phys., 223 (2001), 583-626.
doi: 10.1007/s002200100562. |
[19] |
E. J. Heller, Frozen Gaussians: A very simple semiclassical approximation, J. Chem. Phys., 75 (1981), 2923-2931.
doi: 10.1063/1.442382. |
[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. |
[21] |
L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z., 219 (1995), 413-449.
doi: 10.1007/BF02572374. |
[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. |
[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. |
[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. |
[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. |
[26] |
R. G. Littlejohn, The semiclassical evolution of wave packets, Phys. Rep., 138 (1986), 193-291.
doi: 10.1016/0370-1573(86)90103-1. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[35] |
_____, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), 1123-1145.
doi: 10.1142/S0129055X1000417X. |
[36] |
V. Rousse, Semiclassical simple initial value representations,, to appear in Ark. för Mat., ().
|
[37] |
C. Sparber, Effective mass theorems for nonlinear Schrödinger equations, SIAM J. Appl. Math., 66 (2006), 820-842 (electronic).
doi: 10.1137/050623759. |
[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. |
[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. |
[40] |
S. Teufel, "Adiabatic Perturbation Theory in Quantum Dynamics," Lecture Notes in Mathematics, 1821, Springer-Verlag, Berlin, 2003. |
[41] |
C. H. Wilcox, Theory of Bloch waves, J. Analyse Math., 33 (1978), 146-167.
doi: 10.1007/BF02790171. |
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