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Author: Neidhardt, Hagen × End date: 2019 × Start date: 2010 × Version: publishedVersion × Subject: perturbation determinants × WGL Institute: WIAS ×

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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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    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.