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Mathematical Properties of Quantum Evolution Equations

Summary: Mathematical Properties of Quantum
Evolution Equations
Anton Arnold
Institut f¨ur Analysis und Scientific Computing, Technische Universit¨at Wien
Wiedner Hauptstr. 8, A-1040 Wien, Austria;
Summary. This chapter focuses on the mathematical analysis of nonlinear quan-
tum 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: Schr¨odinger wave functions,
density matrices, and Wigner functions.
For the Schr¨odinger-Poisson analysis (in H1
and L2
) we present Strichartz in-
equalities. 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


Source: Arnold, Anton - Institut für Analysis und Scientific Computing, Technische Universität Wien


Collections: Mathematics