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search.filters.applied.f.access: openAccess × Author: Neidhardt, Hagen × Start date: 2000 × Type: Text × Version: publishedVersion × Publisher: Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik × WGL Institute: WIAS ×

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Now showing 1 - 10 of 21
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    The 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.
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    A 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, Pierre
    Motivated 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.
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    Trace formulae for dissipative and coupled scattering systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Behrndt, Jussi; Malamud, Mark; Neidhardt, Hagen
    For 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.
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    Trotter-Kato product formula for unitary groups
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Exner, Pavel; Neidhardt, Hagen
    Let $A$ 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.
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    Convergence 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.
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    On 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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    Perturbation determinants for singular perturbations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Malamud, Mark M.; Neidhardt, Hagen
    For proper extensions of a densely defined closed symmetric operator with trace class resolvent difference the perturbation determinant is studied in the framework of boundary triplet approach to extension theory.
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    Scattering theory for open quantum systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, Hagen; Exner, Pavel
    Quantum 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.
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    R-matrix formalism for electron scattering in two dimensions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Racec, Paul N.; Racec, Roxana; Neidhardt, Hagen
    We 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
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    Scattering matrices and Weyl functions
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, Hagen
    For 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.