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Author: Neidhardt, Hagen × Subject: approximation ×

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Now showing 1 - 2 of 2
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    Spectral Theory of Infinite Quantum Graphs
    (Cham (ZG) : Springer International Publishing AG, 2018) Exner, Pavel; Kostenko, Aleksey; Malamud, Mark; Neidhardt, Hagen
    We 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.
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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.