Refine
Language
- English (24) (remove)
Keywords
- Partial differential equations (4)
- dissipative Schroedinger-type operators (4)
- discontinuous coefficients (3)
- nonsmooth domains (3)
- Elliptic transmission problems (2)
- Lax-Phillips scattering theory (2)
- Sturm-Liouville operators (2)
- carrier and current densities (2)
- characteristic function (2)
- density matrices (2)
We show that elliptic second order operators $A$ of divergence type fulfill maximal parabolic regularity on distribution spaces, even if the underlying domain is highly non-smooth and $A$ is complemented with mixed boundary conditions. Applications to quasilinear parabolic equations with non-smooth data are presented.
The paper is devoted to Schroedinger operators on bounded intervals of the real axis with dissipative boundary conditions. In the framework of the Lax-Phillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed, in particular, trace formula and Birman-Krein formula are verified directly. The results are used for dissipative Schroedinger-Poisson systems.
The well known De Giorgi result on Hölder continuity for solutions of the Dirichlet problem is re-established for mixed boundary value problems, provided that the underlying domain is a Lipschitz domain and the border between the Dirichlet and the Neumann boundary part satisfies a very general geometric condition. Implications of this result for optimal control theory are presented.
We regard drift-diffusion equations for semiconductor devices in Lebesgue spaces. To that end we reformulate the (generalized) van Roosbroeck system as an evolution equation for the potentials to the driving forces of the currents of electrons and holes. This evolution equation falls into a class of quasi-linear parabolic systems which allow unique, local in time solution in certain Lebesgue spaces. In particular, it turns out that the divergence of the electron and hole current is an integrable function. Hence, Gauss' theorem applies, and gives the foundation for space discretization of the equations by means of finite volume schemes. Moreover, the strong differentiability of the electron and hole density in time is constitutive for the implicit time discretization scheme. Finite volume discretization of space, and implicit time discretization are accepted custom in engineering and scientific computing. ---This investigation puts special emphasis on non-smooth spatial domains, mixed boundary conditions, and heterogeneous material compositions, as required in electronic device simulation.
Let H be a semi–bounded self–adjoint operator in a separable Hilbert space.
For a certain class of positive, continuous, decreasing, and convex functions
F we show the convexity of trace functionals tr(F (H + U − ε(U ))) − ε(U ),
where U is a bounded self–adjoint operator on H and ε(U ) is a normalizing
real function—the Fermi level—which may be identical zero. If additionally
F is continuously differentiable, then the corresponding trace functional is
Fréchet differentiable and there is an expression of its gradient in terms off
the derivative of F . The proof of the differentiability of the trace functional
is based upon Birman and Solomyak’s theory of double Stieltjes operator
integrals. If, in particular, H is a Schrödinger–type operator and U a real-valued function, then the gradient of the trace functional is the quantum
mechanical expression of the particle density with respect to an equilibrium
distribution function f = −F . Thus, the monotonicity of the particle density
in its dependence on the potential U of Schrödinger’s operator—which has
been understood since the late 1980s—follows as a special case.
We study in detail Schroedinger-type operators on a bounded interval of the real axis with dissipative boundary conditions. The characteristic function of such operators is computed, its minimal self-adjoint dilation is constructed and the generalized eigenfunction expansion for the dilation is developed. The problem is motivated by semiconductor physics.
We describe an embedding of a quantum mechanically described structure into a macroscopic flow. The open quantum system is partly driven by an adjacent macroscopic flow acting on the boundary of the bounded spatial domain designated to quantum mechanics. This leads to an essentially non-selfadjoint Schroedinger-type operator, the spectral properties of which will be investigated.
Using a classical theorem of Sobolevskii on equations of parabolic type in a Banach space and recently obtained results on elliptic operators with discontinuous coefficients including mixed boundary conditions we prove that quasilinear parabolic systems in diagonal form admit a local, classical solution in the space of p-integrable functions, for some p greater than 1, over a bounded two dimensional space domain. As applications we have in mind systems of reaction diffusion equations, e.g. van Roosbroeck's system. The treatment of such equations in a space of integrable functions enables us to define the normal component of the flow across any part of the Dirichlet boundary by Gauss' theorem.