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Author: Malamud, Mark × End date: 2019 × Start date: 2010 × Type: Text × WGL Institute: WIAS ×

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Now showing 1 - 3 of 3
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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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    Sturm-Liouville boundary value problems with operator potentials and unitary equivalence
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Malamud, Mark; Neidhardt, Hagen
    Consider 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
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    Trace formulas for singular perturbations
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2014) Malamud, Mark; Neidhardt, Hagen
    Trace 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.