The Classical Limit of a Self-Consistent Quantum-Vlasov Equation in 3d

Markowich, Peter A.; Mauser, Norbert J.

doi:10.1142/s0218202593000072

Cited by 112 publications

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“…Those results are a rigorous justification of the composition of two different asymptotic processes which already have been independently studied in [8,9] and [5] for the semi-classical limit and, as mentioned above, in [2] for the quasi-neutral limit.…”

Section: Introductionsupporting

confidence: 61%

Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field

Puel

2002

ESAIM: M2AN

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Section: Introductionsupporting

confidence: 61%

Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field

Puel

2002

ESAIM: M2AN

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“…In the results [21], [23] this problem was solved for a "really mixed state" where infinitely many wave functions contribute in a very particular dependence on the Planck constant , as stated in the following assumption…”

Section: Results On Classical Limitsmentioning

confidence: 99%

“…Lions and T. Paul [21] and by P.A. Markowich and N.J. Mauser [23]. Note that the Wigner function f is in general real, but has also negative values, whereas the limit f is a "true", nonnegative distribution function.…”

Section: Introductionmentioning

confidence: 99%

“…Note that the Wigner function f is in general real, but has also negative values, whereas the limit f is a "true", nonnegative distribution function. In [21] and [23], independently, the method of using a smoothed Wigner function, the "Husimi function" (which is closely related to the "coherent states"), was introduced to overcome this problem.…”

Section: Introductionmentioning

confidence: 99%

“…Lions and T. Paul in [21] and, independently, by P.A. Markowich and N.J. Mauser in [23]. There the case of the so called "completely mixed state", i.e.…”

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

Mauser¹

2002

Journées Équations Aux Dérivées Partielles

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We deal with classical and "semiclassical limits" , i.e. vanishing Planck constant → 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 (t, x)} is transformed to the WignerPoisson (W-P) equation for a "phase space function" f (t, x, ξ), the Wigner function. The weak limit of f (t, x, ξ), as 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 in 1993 by P.L. Lions and T. Paul in [21] and, independently, by P.A. Markowich and N.J. Mauser in [23]. 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 (t, x)} is considered, recently P. Zhang, Y. Zheng and N.J. Mauser [33] 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 (t, x, ξ) is replaced by the "Wigner series" f (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 [7] 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.Support by the Austrian START project "Nonlinear Schrödinger and quantum Boltzmann equations" is acknowledged. MSC 2000 : 35L65, 35Q55, 81Q05, 81Q20.

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Semiclassical limit for the Schrödinger‐Poisson equation in a crystal

Bechouche

Mauser

Poupaud

2001

Comm Pure Appl Math

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We give a mathematically rigorous theory for the limit from a weakly nonlinear Schrödinger equation with both periodic and nonperiodic potential to the semiclassical version of the Vlasov equation. To this end we perform simultaneously a classical limit (vanishing Planck constant) and a homogenization limit of the periodic structure (vanishing lattice length taken proportional to the Planck constant).We introduce a new variant of Wigner transforms, namely the "Wigner-Bloch series," as an adaptation of the Wigner series for density matrices related to two different "energy bands." Another essential tool is estimates on the commutators of the projectors into the Floquet subspaces ("band subspaces") and the multiplicative potential operator that destroy the invariance of these band subspaces under the periodic Hamiltonian.We assume the initial data to be concentrated in isolated bands but allow for band-crossing of the other bands, which is the generic situation in more than one space dimension. The nonperiodic potential is obtained from a coupling to the Poisson equation; i.e., we take into account the self-consistent Coulomb interaction. Our results also hold for the easier linear case where this potential is given. We hence give the first rigorous derivation of the (nonlinear) "semiclassical equations" of solid state physics widely used to describe the dynamics of electrons in semiconductors.

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The Classical Limit of a Self-Consistent Quantum-Vlasov Equation in 3d

Cited by 112 publications

References 0 publications

Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field

Convergence of the Schrödinger-Poisson system to the Euler equations under the influence of a large magnetic field

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

Semiclassical limit for the Schrödinger‐Poisson equation in a crystal

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