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- ItemLinear non-autonomous Cauchy problems and evolution semigroups(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Neidhardt, Hagen; Zagrebnov, Valentin A.The paper is devoted to the problem of existence of propagators for an abstract linear non-autonomous evolution Cauchy problem of hyperbolic type in separable Banach spaces. The problem is solved using the so-called evolution semigroup approach which reduces the existence problem for propagators to a perturbation problem of semigroup generators. The results are specified to abstract linear non-autonomous evolution equations in Hilbert spaces where the assumption is made that the domains of the quadratic forms associated with the generators are independent of time. Finally, these results are applied to time-dependent Schrödinger operators with moving point interactions in 1D.
- ItemR-matrix formalism for electron scattering in two dimensions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Racec, Paul N.; Racec, Roxana; Neidhardt, HagenWe investigate the scattering phenomena in two dimensions produced by a general finite-range nonseparable potential. This situation can appear either in a Cartesian geometry or in a heterostructure with cylindrical symmetry. Increasing the dimensionality of the scattering problem new processes as the scattering between conducting channels and the scattering from conducting to evanescent channels are allowed. For certain values of the energy, called resonance energy, the transmission through the scattering region changes dramatically in comparison with an one-dimensional problem. If the potential has an attractive character even the evanescent channels can be seen as dips of the total transmission. The multi-channel current scattering matrix is determined using its representation in terms of the R-matrix. The resonant transmission peaks are characterized quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances. Our formalism is applied to a quantum dot in a two dimensional electron gas and a conical quantum dot embedded inside a nanowire
- ItemScattering theory for open quantum systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, Hagen; Exner, PavelQuantum systems which interact with their environment are often modeled by maximal dissipative operators or so-called Pseudo-Hamiltonians. In this paper the scattering theory for such open systems is considered. First it is assumed that a single maximal dissipative operator $A_D$ in a Hilbert space $sH$ is used to describe an open quantum system. In this case the minimal self-adjoint dilation $widetilde K$ of $A_D$ can be regarded as the Hamiltonian of a closed system which contains the open system $[A_D,sH]$, but since $widetilde K$ is necessarily not semibounded from below, this model is difficult to interpret from a physical point of view. In the second part of the paper an open quantum system is modeled with a family $[A(mu)]$ of maximal dissipative operators depending on energy $mu$, and it is shown that the open system can be embedded into a closed system where the Hamiltonian is semibounded. Surprisingly it turns out that the corresponding scattering matrix can be completely recovered from scattering matrices of single Pseudo-Hamiltonians as in the first part of the paper. The general results are applied to a class of Sturm-Liouville operators arising in dissipative and quantum transmitting Schrödinger-Poisson systems.
- ItemMonotonicity properties of the quantum mechanical particle density(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Kaiser, Hans-Christoph; Neidhardt, Hagen; Rehberg, JoachimAn elementary proof of the anti-monotonicity of the quantum mechanical particle density with respect to the potential in the Hamiltonian is given for a large class of admissible thermodynamic equilibrium distribution functions. In particular the zero temperature case is included.
- ItemTrotter-Kato product formula for unitary groups(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Exner, Pavel; Neidhardt, HagenLet $A$ and $B$ be non-negative self-adjoint operators in a separable Hilbert space such that its form sum $C$ is densely defined. It is shown that the Trotter product formula holds for imaginary times in the $L^2$-norm. The result remains true for the Trotter-Kato product formula for so-called holomorphic Kato functions; we also derive a canonical representation for any function of this class.
- ItemEvanescent channels and scattering in cylindrical nanowire heterostructures(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Racec, Paul N.; Racec, Roxana; Neidhardt, HagenWe investigate the scattering phenomena produced by a general finite range non-separable potential in a multi-channel two-probe cylindrical nanowire heterostructure. The multi-channel current scattering matrix is efficiently computed using the R-matrix formalism extended for cylindrical coordinates. Considering the contribution of the evanescent channels to the scattering matrix, we are able to put in evidence the specific dips in the tunneling coefficient in the case of an attractive potential. The cylindrical symmetry cancels the ''selection rules'' known for Cartesian coordinates. If the attractive potential is superposed over a non-uniform potential along the nanowire, then resonant transmission peaks appear. We can characterize them quantitatively through the poles of the current scattering matrix. Detailed maps of the localization probability density sustain the physical interpretation of the resonances (dips and peaks). Our formalism is applied to a variety of model systems like a quantum dot, a core/shell quantum ring or a double barrier, embedded into the nano-cylinder.
- ItemClassical solutions of drift-diffusion equations for semiconductor devices: the 2D case(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Kaiser, Hans-Christian; Neidhardt, Hagen; Rehberg, Joachim; Gajewski, Herbert; Gröger, Konrad; Zacharias, KlausWe 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.
- ItemScattering matrices and Weyl functions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, HagenFor a scattering system consisting of two selfadjoint extensions of a symmetric operator A with finite deficiency indices, the scattering matrix and the spectral shift function are calculated in terms of the Weyl function associated with the boundary triplet for A* and a simple proof of the Krein-Birman formula is given. The results are applied to singular Sturm-Liouville operators with scalar- and matrix-valued potentials, to Dirac operators and to Schroedinger operators with point interactions.
- ItemFinite rank perturbations, scattering matrices and inverse problems : dedicated to the memory of our friend Peter Jonas (18.7.1941 - 18.7.2007)(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, Hagen; Jonas, PeterIn this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. The representation results are extended to dissipative scattering systems and an explicit solution of an inverse scattering problem for the Lax-Phillips scattering matrix is presented.
- ItemThe effect of time-dependent coupling on non-equilibirum steady states(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Cornean, Horia D.; Neidhardt, Hagen; Zagrebnov, Valentin A.Consider (for simplicity) two one-dimensional semi-infinite leads coupled to a quantum well via time dependent point interactions. In the remote past the system is decoupled, and each of its components is at thermal equilibrium. In the remote future the system is fully coupled. We define and compute the non equilibrium steady state (NESS) generated by this evolution. We show that when restricted to the subspace of absolute continuity of the fully coupled system, the state does not depend at all on the switching. Moreover, we show that the stationary charge current has the same invariant property, and derive the Landau-Lifschitz and Landauer-Büttiker formulas.