Abstract

We analyze the dynamics of expectation values of quantum observables for the time-dependent semiclassical Schrödinger equation. To benefit from the positivity of Husimi functions, we switch between observables obtained from Weyl and anti-Wick quantization. We develop and prove a second order Egorov-type propagation theorem with Husimi functions by establishing transition and commutator rules for Weyl and anti-Wick operators. We provide a discretized version of our theorem and present numerical experiments for Schrödinger equations in dimensions two and six that validate our results.

Keywords

  1. time-dependent Schrödinger equation
  2. expectation values
  3. Husimi functions

MSC codes

  1. 81S30
  2. 81Q20
  3. 81-08
  4. 65D30
  5. 65Z05

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References

1.
H. Ando, and Y. Morimoto, Wick calculus and the Cauchy problem for some dispersive equations, Osaka J. Math., 39 (2002), pp. 123--147.
2.
A. Bouzouina and D. Robert, Uniform semiclassical estimates for the propagation of quantum observables, Duke Math. J., 111 (2002), pp. 223--252.
3.
N. de Bruijn, Uncertainty principles in Fourier analysis, in Symposium on Inequalities (Wright-Patterson Air Force Base, Ohio, 1965), Academic Press, 1967, pp. 57--71,.
4.
E. Faou, V. Gradinaru, and C. Lubich, Computing semiclassical quantum dynamics with Hagedorn wavepackets, SIAM J. Sci. Comput., 31 (2009), pp. 3027--3041.
5.
P. Gérard, P. Markowich, N. Mauser, and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), pp. 323--379.
6.
E. Hairer, C. Lubich, and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, 2nd ed., Springer Ser. Comput. Math. 31, Springer-Verlag, Berlin, 2006.
7.
S. Kube, C. Lasser, and M. Weber, Monte Carlo sampling of Wigner functions and surface hopping quantum dynamics, J. Comput. Phys., 228 (2009), pp. 1947--1962.
8.
N. Lerner, Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators, Pseudo-Differential Operators. Theory Appl. 3, Birkhäuser Verlag, Basel, 2010.
9.
C. Lasser and S. Röblitz, Computing expectation values for molecular quantum dynamics, SIAM J. Sci. Comput., 32 (2010), pp. 1465--1483.
10.
H. Liu, O. Runborg, and N. Tanushev, Error Estimates for Gaussian Beam Superpositions, arXiv:1008.1320, 2010.
11.
J. Lu and X. Yang, Convergence of frozen Gaussian approximation for high-frequency wave propagation, Comm. Pure Appl. Math., 65 (2012), pp. 759--789.
12.
D. Robert, On the Herman-Kluk semiclassical approximation, Rev. Math. Phys., 22 (2010), pp. 1123--1145.
13.
H. Spohn and S. Teufel, Adiabatic decoupling and time-dependent Born-Oppenheimer theory, Comm. Math. Phys., 224 (2001), pp. 113--132.
14.
T. Swart and V. Rousse, A mathematical justification for the Herman-Kluk propagator, Comm. Math. Phys., 286 (2009), pp. 725--750.
15.
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990), pp. 262--268.
16.
P. Zhang, Wigner Measure and Semiclassical Limits of Nonlinear Schrödinger Equations, Courant Lect. Notes Math. 17, AMS, Providence, RI, 2008.
17.
M. Zworski, Semiclassical Analysis, Grad. Stud. Math. 138, AMS, Providence, RI, 2012.

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Information

Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 1557 - 1581
ISSN (online): 1095-712X

History

Submitted: 24 August 2012
Accepted: 21 May 2013
Published online: 18 July 2013

Keywords

  1. time-dependent Schrödinger equation
  2. expectation values
  3. Husimi functions

MSC codes

  1. 81S30
  2. 81Q20
  3. 81-08
  4. 65D30
  5. 65Z05

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