A. Arnold:

"Mathematical Properties of Quantum Evolution Equations";

in: "ASC Report 21/2007", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2007, ISBN: 978-3-902627-00-1.

This chapter focuses on the mathematical analysis of nonlinear quantum

transport equations that appear in the modeling of nano-scale semiconductor

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: Schršodinger wave functions,

density matrices, and Wigner functions.

For the Schršodinger-Poisson analysis (in H1 and L2) 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

quantum transport, partial differential equations, well-posedness,

http://www.asc.tuwien.ac.at/preprint/2007/asc21x2007.pdf

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