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- ItemPoint contacts and boundary triples(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Lotoreichik, Vladimir; Neidhardt, Hagen; Popov, Igor Yu.We suggest an abstract approach for point contact problems in the framework of boundary triples. Using this approach we obtain the perturbation series for a simple eigenvalue in the discrete spectrum of the model self-adjoint extension with weak point coupling.
- 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.
- ItemTrace formulas for singular perturbations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Malamud, Mark; Neidhardt, HagenTrace formulas for pairs of self-adjoint, maximal dissipative and other types of resolvent comparable operators are obtained. In particular, the existence of a complex-valued spectral shift function for a resolvent comparable pair H', H of maximal dissipative operators is proved. We also investigate the existence of a real-valued spectral shift function. Moreover, we treat in detail the case of additive trace class perturbations. Assuming that H and H'=H+V are maximal dissipative and V is of trace class, we prove the existence of a summable complex-valued spectral shift function. We also obtain trace formulas for a pair {A, A*} assuming only that A and A* are resolvent comparable. In this case the determinant of a characteristic function of A is involved in the trace formula. In the case of singular perturbations we apply the technique of boundary triplets. It allows to express the spectral shift function of a pair of extensions in terms of abstract Weyl function and boundary operator. We improve and generalize certain classical results of M.G. Krein for pairs of self-adjoint and dissipative operators, the results of A. Rybkin for such pairs, as well as the results of V. Adamyan, B. Pavlov, and M. Krein for pairs {A, A*} with a maximal dissipative operator A.
- ItemPerturbation determinants for singular perturbations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Malamud, Mark M.; Neidhardt, HagenFor 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.
- 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.
- ItemScattering matrices and Dirichlet-to-Neumann maps(Amsterdam [u.a.] : Elsevier, 2017) Behrndt, Jussi; Malamud, Mark M.; Neidhardt, HagenA general representation formula for the scattering matrix of a scattering system consisting of two self-adjoint operators in terms of an abstract operator valued Titchmarsh–Weyl m-function is proved. This result is applied to scattering problems for different self-adjoint realizations of Schrödinger operators on unbounded domains, Schrödinger operators with singular potentials supported on hypersurfaces, and orthogonal couplings of Schrödinger operators. In these applications the scattering matrix is expressed in an explicit form with the help of Dirichlet-to-Neumann maps.
- ItemOn the spectrum of the lattice spin-boson Hamiltonian for any coupling: 1D case(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Muminov, Mukhiddin; Neidhardt, Hagen; Rasulov, TulkinA lattice model of radiative decay (so-called spin-boson model) of a two level atom and at most two photons is considered. The location of the essential spectrum is described. For any coupling constant the finiteness of the number of eigenvalues below the bottom of its essential spectrum is proved. The results are obtained by considering a more general model H for which the lower bound of its essential spectrum is estimated. Conditions which guarantee the finiteness of the number of eigenvalues of H below the bottom of its essential spectrum are found. It is shown that the discrete spectrum might be infinite if the parameter functions are chosen in a special form.
- ItemSturm-Liouville boundary value problems with operator potentials and unitary equivalence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Malamud, Mark; Neidhardt, HagenConsider the minimal Sturm-Liouville operator A = A_rm min generated by the differential expression A := -fracd^2dt^2 + T in the Hilbert space L^2(R_+,cH) where T = T^*ge 0 in cH. We investigate the absolutely continuous parts of different self-adjoint realizations of cA. In particular, we show that Dirichlet and Neumann realizations, A^D and A^N, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if infsigma_ess(T) = infgs(T) ge 0, then the part wt A^acE_wt A(gs(A^D)) of any self-adjoint realization wt A of cA is unitarily equivalent to A^D. In addition, we prove that the absolutely continuous part wt A^ac of any realization wt A is unitarily equivalent to A^D provided that the resolvent difference (wt A
- ItemSpectral Theory of Infinite Quantum Graphs(Cham (ZG) : Springer International Publishing AG, 2018) Exner, Pavel; Kostenko, Aleksey; Malamud, Mark; Neidhardt, HagenWe investigate quantum graphs with infinitely many vertices and edges without the common restriction on the geometry of the underlying metric graph that there is a positive lower bound on the lengths of its edges. Our central result is a close connection between spectral properties of a quantum graph and the corresponding properties of a certain weighted discrete Laplacian on the underlying discrete graph. Using this connection together with spectral theory of (unbounded) discrete Laplacians on infinite graphs, we prove a number of new results on spectral properties of quantum graphs. Namely, we prove several self-adjointness results including a Gaffney-type theorem. We investigate the problem of lower semiboundedness, prove several spectral estimates (bounds for the bottom of spectra and essential spectra of quantum graphs, CLR-type estimates) and study spectral types.