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0/ nC e 1 2 2p 0 ! k/ , with k < n, and their derivatives. Moreover, the inverse Laplace transform can be computed explicitly. n/ is a finite linear combination (depending on ˙ and their derivatives) of terms of the form e p2 =2 Rrs . r; s/ are recursively given by ar D 1, bs D 1 and 1 Kinetic and Hydrodynamic Models for Multi-Band Quantum Transport in Crystals ak D k 1 X bk D ! 126) ! 3/ are computed explicitly, which allows to prove the following. 3. 116) satisfies, up to O. 8S33 S23 / r log nC r log n S32 Ã S12 n S11 C 16˛ 2 S22 log n nC i 46 L.

The classical-quantum analogy of the Wigner function approach allows to describe in a natural way openboundary systems where carrier and energy is exchanged with the environment [51]. Anyway, some criticisms has been raised concerning this point [102]. It is still an 22 L. Barletti et al. open question whether the classical inflow boundary conditions which are generally assumed for the Wigner function (and are mathematically well posed [73]) are always appropriate for modeling the particle reservoirs.

In the following sections we will give the details of the quantum fluid-dynamic approach to multi-band systems, where “multi-band” has to be intended in a wide sense and stands for “quantum systems with multi-component wave function”. Systems of this kind include the envelope and k p models treated in the previous sections but, of course, also include particles with spin or spin-like degrees of freedom. 2 Scalar Quantum Fluid Equations Before treating the multi-band/spinorial case, let us review the single-band/scalar case.

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An Introduction to Numerical Mathematics by Eduard L. Stiefel

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