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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.
- ItemA Kohn-Sham system at zero temperature(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Cornean, Horia; Hoke, Kurt; Neidhardt, Hagen; Racec, Paul Nicolae; Rehberg, JoachimAn one-dimensional Kohn-Sham system for spin particles is considered which effectively describes semiconductor nanostructures and which is investigated at zero temperature. We prove the existence of solutions and derive a priori estimates. For this purpose we find estimates for eigenvalues of the Schrödinger operator with effective Kohn-Sham potential and obtain $W^1,2$-bounds of the associated particle density operator. Afterwards, compactness and continuity results allow to apply Schauder's fixed point theorem. In case of vanishing exchange-correlation potential uniqueness is shown by monotonicity arguments. Finally, we investigate the behavior of the system if the temperature approaches zero.
- ItemThe Cayley transform applied to non-interacting quantum transport : dedicated to the memory of Markus Büttiker (1950-2013)(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Cornean, Horia D.; Neidhardt, Hagen; Wilhelm, Lukas; Zagrebnov, Valentin A.We extend the Landauer-Büttiker formalism in order to accommodate both unitary and self-adjoint operators which are not bounded from below. We also prove that the pure point and singular continuous subspaces of the decoupled Hamiltonian do not contribute to the steady current. One of the physical applications is a stationary charge current formula for a system with four pseudo-relativistic semi-infinite leads and with an inner sample which is described by a Schrödinger operator defined on a bounded interval with dissipative boundary conditions. Another application is a current formula for electrons described by a one dimensional Dirac operator; here the system consists of two semi-infinite leads coupled through a point interaction at zero.
- ItemA new model for quantum dot light emitting-absorbing devices : dedicated to the memory of Pierre Duclos(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Neidhardt, Hagen; Wilhelm, Lukas; Zagrebnov, Valentin A.; Duclos, PierreMotivated by the Jaynes-Cummings (JC) model, we consider here a quantum dot coupled simultaneously to a reservoir of photons and to two electric leads (free-fermion reservoirs). This Jaynes-Cummings-Leads (JCL) model makes possible that the fermion current through the dot creates a photon flux, which describes a light-emitting device. The same model also describes a transformation of the photon flux into a fermion current, i.e. a quantum dot light-absorbing device. The key tool to obtain these results is an abstract Landauer-Büttiker formula.
- 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.
- 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.
- ItemTrace formulae for dissipative and coupled scattering systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Behrndt, Jussi; Malamud, Mark; Neidhardt, HagenFor scattering systems consisting of a (family of) maximal dissipative extension(s) and a selfadjoint extension of a symmetric operator with finite deficiency indices, the spectral shift function is expressed in terms of an abstract Titchmarsh-Weyl function and a variant of the Birman-Krein formula is proved.
- 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.
- ItemConvergence rate estimates for Trotter product approximations of solution operators for non-autonomous Cauchy problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Neidhardt, Hagen; Stephan, Artur; Zagrebnov, Valentin A.In the present paper we advocate the Howland-Evans approach to solution of the abstract non-autonomous Cauchy problem (non-ACP) in a separable Banach space X. The main idea is to reformulate this problem as an autonomous Cauchy problem (ACP) in a new Banach space Lp(I;X), p 2 [1;1), consisting of X-valued functions on the time-interval I. The fundamental observation is a one-to-one correspondence between solution operators (propagators) for a non-ACP and the corresponding evolution semigroups for ACP in Lp(I;X). We show that the latter also allows to apply a full power of the operatortheoretical methods to scrutinise the non-ACP including the proof of the Trotter product approximation formulae with operator-norm estimate of the rate of convergence. The paper extends and improves some recent results in this direction in particular for Hilbert spaces.
- ItemOn the unitary equivalence of absolutely continuous parts of self-adjoint extensions : dedicated to the memory of M. S. Birman(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Malamud, Mark M.; Neidhardt, Hagen; Birman, M.S.The classical Weyl-von Neumann theorem states that for any self-adjoint operator $A$ in a separable Hilbert space $gotH$ there exists a (non-unique) Hilbert-Schmidt operator $C = C^*$ such that the perturbed operator $A+C$ has purely point spectrum. We are interesting whether this result remains valid for non-additive perturbations by considering self-adjoint extensions of a given densely defined symmetric operator $A$ in $mathfrak H$ and fixing an extension $A_0 = A_0^*$. We show that for a wide class of symmetric operators the absolutely continuous parts of extensions $widetilde A = widetilde A^*$ and $A_0$ are unitarily equivalent provided that their resolvent difference is a compact operator. Namely, we show that this is true whenever the Weyl function $M(cdot)$ of a pair $A,A_0$ admits bounded limits $M(t) := wlim_yto+0M(t+iy)$ for a.e. $t in mathbbR$. This result is applied to direct sums of symmetric operators and Sturm-Liouville operators with operator potentials.
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