WIAS Preprint No. 1937, (2014)

Trace formulas for singular perturbations



Authors

  • Malamud, Mark M.
  • Neidhardt, Hagen

2010 Mathematics Subject Classification

  • 47A55 47B25 34B44

Keywords

  • Symmetric operators, perturbation determinants, trace formulas, spectral shift function

Abstract

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.

Appeared in

  • Adv. Math., 274 (2015) pp. 736--832.

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