Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms

Norbert J. Mauser

Journées équations aux dérivées partielles (2002)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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We deal with classical and “semiclassical limits” , i.e. vanishing Planck constant ℏ ≃ ϵ → 0 , eventually combined with a homogenization limit of a crystal lattice, of a class of “weakly nonlinear” NLS. The Schrödinger-Poisson (S-P) system for the wave functions { ψ j ϵ ( t , x ) } is transformed to the Wigner-Poisson (W-P) equation for a “phase space function” f ϵ ( t , x , ξ ) , the Wigner function. The weak limit of f ϵ ( t , x , ξ ) , as ϵ tends to 0 , is called the “Wigner measure” f ( t , x , ξ ) (also called “semiclassical measure” by P. Gérard). The mathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first by P.L. Lions and T. Paul (1993) and, independently, by P.A. Markowich and N.J. Mauser (1993). There the case of the so called “completely mixed state”, i.e. j = 1 , 2 , ⋯ , ∞ , was considered where strong additional assumptions can be posed on the initial data. For the so called “pure state” case where only one (or a finite number) of wave functions { ψ j ϵ ( t , x ) } is considered, recently P. Zhang, Y. Zheng and N.J. Mauser (2002) have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique. For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of “energy bands”, Bloch decomposition of L 2 etc. On the level of the Wigner transform the Wigner function f ϵ ( t , x , ξ ) is replaced by the “Wigner series” f ϵ ( t , x , k ) , where the “kinetic variable” k lives on the torus (“Brioullin zone”) instead of the whole space. Recently P. Bechouche, N.J. Mauser and F. Poupaud (2001) have given the rigorous “semiclassical” limit from S-P in a crystal to the “semiclassical equations”, i.e. the “semiconductor V-P system”, with the assumption of the initial data to be concentrated in isolated bands.

How to cite

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Mauser, Norbert J.. "Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms." Journées équations aux dérivées partielles (2002): 1-12. <http://eudml.org/doc/93422>.

@article{Mauser2002,
abstract = {We deal with classical and “semiclassical limits” , i.e. vanishing Planck constant $\hbar \simeq \epsilon \rightarrow 0$, eventually combined with a homogenization limit of a crystal lattice, of a class of “weakly nonlinear” NLS. The Schrödinger-Poisson (S-P) system for the wave functions $\lbrace \psi ^\epsilon _j(t,x)\rbrace $ is transformed to the Wigner-Poisson (W-P) equation for a “phase space function” $f^\epsilon (t,x,\xi )$, the Wigner function. The weak limit of $f^\epsilon (t,x,\xi )$, as $\epsilon $ tends to $0$, is called the “Wigner measure” $f(t,x,\xi )$ (also called “semiclassical measure” by P. Gérard). The mathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first by P.L. Lions and T. Paul (1993) and, independently, by P.A. Markowich and N.J. Mauser (1993). There the case of the so called “completely mixed state”, i.e. $j=1,2,\dots ,\infty $, was considered where strong additional assumptions can be posed on the initial data. For the so called “pure state” case where only one (or a finite number) of wave functions $\lbrace \psi ^\epsilon _j(t,x)\rbrace $ is considered, recently P. Zhang, Y. Zheng and N.J. Mauser (2002) have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique. For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of “energy bands”, Bloch decomposition of $L^2$ etc. On the level of the Wigner transform the Wigner function $f^\epsilon (t,x,\xi )$ is replaced by the “Wigner series” $f^\epsilon (t,x,k)$, where the “kinetic variable” $k$ lives on the torus (“Brioullin zone”) instead of the whole space. Recently P. Bechouche, N.J. Mauser and F. Poupaud (2001) have given the rigorous “semiclassical” limit from S-P in a crystal to the “semiclassical equations”, i.e. the “semiconductor V-P system”, with the assumption of the initial data to be concentrated in isolated bands.},
author = {Mauser, Norbert J.},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms},
url = {http://eudml.org/doc/93422},
year = {2002},
}

TY - JOUR
AU - Mauser, Norbert J.
TI - Semi-classical limits of Schrödinger-Poisson systems via Wigner transforms
JO - Journées équations aux dérivées partielles
PY - 2002
PB - Université de Nantes
SP - 1
EP - 12
AB - We deal with classical and “semiclassical limits” , i.e. vanishing Planck constant $\hbar \simeq \epsilon \rightarrow 0$, eventually combined with a homogenization limit of a crystal lattice, of a class of “weakly nonlinear” NLS. The Schrödinger-Poisson (S-P) system for the wave functions $\lbrace \psi ^\epsilon _j(t,x)\rbrace $ is transformed to the Wigner-Poisson (W-P) equation for a “phase space function” $f^\epsilon (t,x,\xi )$, the Wigner function. The weak limit of $f^\epsilon (t,x,\xi )$, as $\epsilon $ tends to $0$, is called the “Wigner measure” $f(t,x,\xi )$ (also called “semiclassical measure” by P. Gérard). The mathematically rigorous classical limit from S-P to the Vlasov-Poisson (V-P) system has been solved first by P.L. Lions and T. Paul (1993) and, independently, by P.A. Markowich and N.J. Mauser (1993). There the case of the so called “completely mixed state”, i.e. $j=1,2,\dots ,\infty $, was considered where strong additional assumptions can be posed on the initial data. For the so called “pure state” case where only one (or a finite number) of wave functions $\lbrace \psi ^\epsilon _j(t,x)\rbrace $ is considered, recently P. Zhang, Y. Zheng and N.J. Mauser (2002) have given the limit from S-P to V-P in one space dimension for a very weak class of measure valued solutions of V-P that are not unique. For the setting in a crystal, as it occurs in semiconductor modeling, we consider Schrödinger equations with an additional periodic potential. This allows for the use of the concept of “energy bands”, Bloch decomposition of $L^2$ etc. On the level of the Wigner transform the Wigner function $f^\epsilon (t,x,\xi )$ is replaced by the “Wigner series” $f^\epsilon (t,x,k)$, where the “kinetic variable” $k$ lives on the torus (“Brioullin zone”) instead of the whole space. Recently P. Bechouche, N.J. Mauser and F. Poupaud (2001) have given the rigorous “semiclassical” limit from S-P in a crystal to the “semiclassical equations”, i.e. the “semiconductor V-P system”, with the assumption of the initial data to be concentrated in isolated bands.
LA - eng
UR - http://eudml.org/doc/93422
ER -

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