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Talk Abstract
Mathematical Concepts of Open Quantum Boundary Conditions

Anton Arnold
Universitat des Saarlandes
Saarbrucken, Germany

This talk is concerned with quantum mechanical models for semiconductor devices like quantum-waveguides and resonant tunneling diodes. Our main focus will be on the derivation and numerical discretization of physically meaningful quantum boundary conditions (BCs) that are to be imposed at the device contacts. In the talk we will give an overview of recent mathematical results on such "open" BCs for both the Schrödinger and the Wigner formulations of quantum mechanics. We shall discuss modeling questions and present various analytical and numerical results for stationary and transient problems.

While several (analytical) transparent and absorbing BCs for quantum models have been proposed in the last decade, their numerical discretization is still a delicate problem. First we shall discuss discrete dispersion relations for the Schrödinger equation along with the `correct' discretization of the BCs that allows the convergence to the steady state.

For the time-dependent Wigner equation we discuss the construction and well-posedness of absorbing BCs. Finally, we shall discuss the stationary boundary value problem (BVP) for the linear Wigner equation on a slab of the phase space with inflow BCs. In contrast to its classical counterpart - the stationary BVP for the Liouville equation, which may have closed trajectory loops - this quantum problem is uniquely solvable. In the talk we shall illustrate this situation for a discrete velocity model.


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