June  2013, 6(3): 669-676. doi: 10.3934/dcdss.2013.6.669

Semiclassical limit of Husimi function

1. 

Laboratoire de Mathématiques et Applications (LMA), Téléport 2. B.P. 30179, Boulevard Marie et Pierre Curie, B.P. 179, 86962 Futuroscope-Chasseneuil Cedex, France, France

Received  June 2010 Revised  January 2012 Published  December 2012

We will show that Liouville and quantum Liouville operators $L$ and $L_\hbar$ generate two $C_0$-groups $e^{tL}$ and $e^{tL_h}$ of isometries in $L^2(\mathbb{R}^{2n})$ and $e^{tL_h}$ converges ultraweakly to $e^{tL}$. As a consequence we show that the Gaussian mollifier of the Wigner function, called Husimi function, converges in $L^1(\mathbb{R}^{2n})$ to the solution of the Liouville equation.
Citation: Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669
References:
[1]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels,, Ann. Sci. Éc. Norm. Supér. (4), 3 (1970), 185. Google Scholar

[2]

E. B. Davies, "Quantum Theory of Open Systems,", Academic Press, (1976). Google Scholar

[3]

J. Dixmier, "Les Algèbres d'Opérateurs dans l'Espace Hilbertien,", Gauthier-Villars, (1969). Google Scholar

[4]

H. Emamirad and Ph. Rogeon, An existence family for the Husimi operator,, Transp. Theo. Stat. Phys., 30 (2001), 673. doi: 10.1081/TT-100107422. Google Scholar

[5]

H. Emamirad and Ph. Rogeon, Scattering theory for Wigner equation,, Math. Meth. Appl. Sc., 28 (2005), 947. doi: 10.1002/mma.601. Google Scholar

[6]

K. Husimi, Some formal properties of the density matrix,, Proc. Phys. Math. Soc. Japan (III), 22 (1940), 464. Google Scholar

[7]

P.-L. Lions, Transformèes de Wigner et èquations de Liouville,, RMA Res. Notes Appl. Math., 28 (1994), 539. Google Scholar

[8]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553. doi: 10.4171/RMI/143. Google Scholar

[9]

E.-M. Ouhabaz, "Analysis of Heat Equations on Domains,", Princeton Univ. Press, (2005). Google Scholar

[10]

P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations,, Math. Meth. Appl. Sci., 11 (1989), 459. doi: 10.1002/mma.1670110404. Google Scholar

[11]

P. A. Markowich and C. A. Ringhofer, An analysis of the quantum Liouville equation,, Z. Angew. Math. Mech., 69 (1989), 121. doi: 10.1002/zamm.19890690303. Google Scholar

[12]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics (vol. I : Functional Analysis),", Academic Press, (1975). Google Scholar

show all references

References:
[1]

C. Bardos, Problèmes aux limites pour les équations aux dérivées partielles du premier ordre à coefficients réels,, Ann. Sci. Éc. Norm. Supér. (4), 3 (1970), 185. Google Scholar

[2]

E. B. Davies, "Quantum Theory of Open Systems,", Academic Press, (1976). Google Scholar

[3]

J. Dixmier, "Les Algèbres d'Opérateurs dans l'Espace Hilbertien,", Gauthier-Villars, (1969). Google Scholar

[4]

H. Emamirad and Ph. Rogeon, An existence family for the Husimi operator,, Transp. Theo. Stat. Phys., 30 (2001), 673. doi: 10.1081/TT-100107422. Google Scholar

[5]

H. Emamirad and Ph. Rogeon, Scattering theory for Wigner equation,, Math. Meth. Appl. Sc., 28 (2005), 947. doi: 10.1002/mma.601. Google Scholar

[6]

K. Husimi, Some formal properties of the density matrix,, Proc. Phys. Math. Soc. Japan (III), 22 (1940), 464. Google Scholar

[7]

P.-L. Lions, Transformèes de Wigner et èquations de Liouville,, RMA Res. Notes Appl. Math., 28 (1994), 539. Google Scholar

[8]

P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553. doi: 10.4171/RMI/143. Google Scholar

[9]

E.-M. Ouhabaz, "Analysis of Heat Equations on Domains,", Princeton Univ. Press, (2005). Google Scholar

[10]

P. A. Markowich, On the equivalence of the Schrödinger and the quantum Liouville equations,, Math. Meth. Appl. Sci., 11 (1989), 459. doi: 10.1002/mma.1670110404. Google Scholar

[11]

P. A. Markowich and C. A. Ringhofer, An analysis of the quantum Liouville equation,, Z. Angew. Math. Mech., 69 (1989), 121. doi: 10.1002/zamm.19890690303. Google Scholar

[12]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics (vol. I : Functional Analysis),", Academic Press, (1975). Google Scholar

[1]

Thomas Chen, Ryan Denlinger, Nataša Pavlović. Moments and regularity for a Boltzmann equation via Wigner transform. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4979-5015. doi: 10.3934/dcds.2019204

[2]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006

[3]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325

[4]

Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111

[5]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013

[6]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649

[7]

Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177

[8]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27

[9]

Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243

[10]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801

[11]

Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373

[12]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 709-722. doi: 10.3934/dcdss.2020039

[13]

Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471

[14]

Ingrid Beltiţă, Anders Melin. The quadratic contribution to the backscattering transform in the rotation invariant case. Inverse Problems & Imaging, 2010, 4 (4) : 599-618. doi: 10.3934/ipi.2010.4.599

[15]

Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009

[16]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579

[17]

Juan H. Arredondo, Francisco J. Mendoza, Alfredo Reyes. On the norm continuity of the hk-fourier transform. Electronic Research Announcements, 2018, 25: 36-47. doi: 10.3934/era.2018.25.005

[18]

Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems & Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693

[19]

Paolo Antonelli, Seung-Yeal Ha, Dohyun Kim, Pierangelo Marcati. The Wigner-Lohe model for quantum synchronization and its emergent dynamics. Networks & Heterogeneous Media, 2017, 12 (3) : 403-416. doi: 10.3934/nhm.2017018

[20]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]