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Mathematical Properties of Quantum Evolution Equations

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Quantum Transport

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 1946))

Abstract

This chapter focuses on the mathematical analysis of nonlinear quantum transport equations that appear in the modeling of nano-scale semi-conductor devices. We start with a brief introduction on quantum devices like the resonant tunneling diode and quantum waveguides. For the mathematical analysis of quantum evolution equations we shall mostly focus on whole space problems to avoid the technicalities due to boundary conditions. We shall discuss three different quantum descriptions: Schrödinger wave functions, density matrices, and Wigner functions. For the Schrödinger–Poisson analysis (in H 1 and L 2) we present Strichartz inequalities. As for density matrices, we discuss both closed and open quantum systems (in Lindblad form). Their evolution is analyzed in the space of trace class operators and energy subspaces, employing Lieb–Thirring-type inequalities. For the analysis of the Wigner–Poisson–Fokker–Planck system we shall first derive (quantum) kinetic dispersion estimates (for Vlasov–Poisson and Wigner–Poisson). The large-time behavior of the linear Wigner–Fokker–Planck equation is based on the (parabolic) entropy method. Finally, we discuss boundary value problems in the Wigner framework.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8, A-1040, Vienna, Austria

    Anton Arnold

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  1. Anton Arnold
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Editors and Affiliations

  1. Institut de Mathématiques, Université de Toulouse, 118 route de Narbonne, 31062, Toulouse Cedex 9, France

    Naoufel Ben Abdallah

  2. Dipartimento di Matematica Applicata, Università di Firenze, Via S.Marta 3, 50139, Firenze, Italy

    Giovanni Frosali

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Arnold, A. (2008). Mathematical Properties of Quantum Evolution Equations. In: Abdallah, N.B., Frosali, G. (eds) Quantum Transport. Lecture Notes in Mathematics, vol 1946. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79574-2_2

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