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  • 1
    Electronic Resource
    Electronic Resource
    Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit (1999)
    Springer
    Numerische Mathematik 81 (1999), S. 595-630 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65M12
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. We apply Wigner-transform techniques to the analysis of difference methods for Schrödinger-type equations in the case of a small Planck constant. In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not. Numerical test examples are presented to help interpret the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050406
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  • 2
    Electronic Resource
    Electronic Resource
    The pseudo-differential approach¶to finite differences revisited (1999)
    Springer
    Calcolo 36 (1999), S. 161-186 
    Details
    ISSN: 1126-5434
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract: We systematically derive arbitrary-order finite difference discretizations of linear (pseudo-) differential problems by appropriately approximating the symbols of the (pseudo) differential operators in frequency space. The convergence (i.e., stability and consistency) analysis of the difference schemes is based directly on micro-local properties of the exact and approximating symbols. The analytical approach presented here sheds new light on the general philosophy of finite differences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s100920050029
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  • 3
    Electronic Resource
    Electronic Resource
    On steady state Euler-Poisson models for semiconductors (1991)
    Springer
    Zeitschrift für angewandte Mathematik und Physik 42 (1991), S. 389-407 
    Details
    ISSN: 1420-9039
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract This paper is concerned with an analysis of the Euler-Poisson model for unipolar semiconductor devices in the steady state isentropic case. In the two-dimensional case we prove the existence of smooth solutions under a smallness assumption on the prescribed outflow velocity (small boundary current) and, additionally, under a smallness assumption on the gradient of the velocity relaxation time. The latter assumption allows a control of the vorticity of the flow and the former guarantees subsonic flow. The main ingredient of the proof is a regularization of the equation for the vorticity. Also, in the irrotational two- and three-dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift-term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the drift-diffusion model as this parameter tends to zero.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00945711
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  • 4
    Electronic Resource
    Electronic Resource
    Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems (2000)
    Springer
    Journal of statistical physics 100 (2000), S. 543-601 
    Details
    ISSN: 1572-9613
    Keywords: Fokker–Planck equation ; Wigner distribution ; scaling limit ; coupled harmonic oscillators
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018667323830
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  • 5
    Electronic Resource
    Electronic Resource
    A steady state potential flow model for semiconductors (1993)
    Springer
    Annali di matematica pura ed applicata 165 (1993), S. 87-98 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present a three-dimensional steady state irrotational flow model for semiconductors which is based on the hydrodynamic equations. We prove existence and local uniqueness of smooth solutions under a smallness assumptions on the data. This assumption implies subsonic flow of electrons in the semiconductors device.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01765842
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  • 6
    Electronic Resource
    Electronic Resource
    A mathematical analysis of quantum transport in three-dimensional crystals (1991)
    Springer
    Annali di matematica pura ed applicata 160 (1991), S. 171-191 
    Details
    ISSN: 1618-1891
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We present an existence and uniqueness result for a quantum transport model in three dimensional crystals. The model consists of a quantum transport (Wigner) equation posed on the phase space consisting of a discrete position variable and a «continuous» wave vector, which is restricted to a bounded domain inR 3 (first Brillouin zone of the crystal). The potential is modeled self-consistently by a discrete Poisson equation (Coulomb interaction). Also we investigate the limits of solutions of this model as the grid spacing tends to zero and show that they converge to the solution of a quantum transport model posed on the «fully continuous» phase space. The transport model derived by this limiting procedure treats the band diagram of the crystal in a semi-classical way and the potential energy term quantum mechanically.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01764126
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  • 7
    Electronic Resource
    Electronic Resource
    On the equivalence of the Schrödinger and the quantum Liouville equations (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 459-469 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We present a semigroup analysis of the quantum Liouville equation, which models the temporal evolution of the (quasi) distribution of an electron ensemble under the action of a scalar potential. By employing the density matrix formulation of quantum physics we prove that the quantum Liouville operator generates a unitary group on L2 if the corresponding Hamiltonian is essentially self-adjoint.Also, we analyse the existence and non-negativity of the particale density and prove that the solutions of the quantum Liouville equation converge to weak solutions of the classical Liouville equation as the Planck constant tends to zero (assuming that the potential is sufficiently smooth).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110404
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  • 8
    Electronic Resource
    Electronic Resource
    The three-dimensional wigner-poisson problem: Existence, uniqueness and approximation (1991)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 14 (1991), S. 35-61 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We prove the existence of a unique, global classical solution of the quantum Vlasov-Poisson problem posed on the phase space ∝x3 × ∝v3. The proof is based on a reformulation of the quantum Vlasov-Poisson problem as a system of countably many Schrödinger equations coupled to a Poisson equation for the potential. The Schrödinger-Poisson problem is first analysed on a bounded domain in ∝x3 and the solution of the whole-space problem is then obtained by a limiting procedure in which the domains ‘tend’ to ∝x3.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670140103
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  • 9
    Electronic Resource
    Electronic Resource
    A non-isentropic Euler-Poisson model for a collisionless plasma (1993)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 16 (1993), S. 409-442 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We analyse transonic solutions of the one-dimensional Euler-Poisson model for a collisionless gas of charged particles in the non-isentropic steady-state case. The model consists of the conservation of mass, momentum and energy equations. The electric field is modelled self-consistently (Coulomb field). Boundary conditions on the particle density and particle temperature are imposed.The analysis is based on representing solutions piecewise as orbits in the particle-density-electric-field phase plane and connecting the orbit segments by the jump and entropy conditions.We characterize the set of all solutions of the Euler-Poisson problem. In particular, we show that, depending upon the length of the interval on which the boundary value problem is posed, fully subsonic, one-shock and (in certain cases) two-shock transonic and smooth transonic solutions exist. Also, numerical computations illustrating the structure of the solutions are reported.
    Additional Material: 21 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670160603
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  • 10
    Book
    Book
    Applied partial differential equations : a visual approach (2007)
    Berlin [u.a.] :Springer,
    Details
    Title: Applied partial differential equations : a visual approach
    Author: Markowich, Peter A.
    Publisher: Berlin [u.a.] :Springer,
    Year of publication: 2007
    Pages: IX, 206 S. : , zahlr. Ill. +
    ISBN: 978-3-540-34645-6 , 3-540-34645-7
    Type of Medium: Book
    Language: English
    URL: http://deposit.d-nb.de/cgi-bin/dokserv?id=2841821&prov=M&dok_var=1&dok_ext=htm
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