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Singular spectral shift function for resolvent comparable operators
Author(s) -
Azamov Nurulla,
Daniels Tom
Publication year - 2019
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201700293
Subject(s) - mathematics , resolvent , bounded function , operator (biology) , locally integrable function , pure mathematics , invariant (physics) , mathematical analysis , discrete mathematics , integrable system , mathematical physics , biochemistry , chemistry , repressor , transcription factor , gene
Abstract LetH 0 = − Δ + V 0 ( x )be a Schrödinger operator onL 2 ( R ν ) , ν = 1 , 2 , or 3, whereV 0 ( x )is a bounded measurable real‐valued function on R ν . Let V be an operator of multiplication by a bounded integrable real‐valued function V ( x ) and putH r = H 0 + r V for real r . We show that the associated spectral shift function (SSF) ξ admits a natural decomposition into the sum of absolutely continuous and singular SSFs. In particular, the singular SSF is integer‐valued almost everywhere, even within the absolutely continuous spectrum where the same cannot be said of the SSF itself. This is a special case of an analogous result for resolvent comparable pairs of self‐adjoint operators, which generalises the case of a trace class perturbation appearing in [2] while also simplifying its proof. We present two proofs which demonstrate the equality of the singular SSF with two a priori different and intrinsically integer‐valued functions which can be associated with the pair H 0 , V : the total resonance index [3] and the singular μ‐invariant [2].