2016/08/15 by Nurulla Azamov, Azamov, Nurulla, Tom Daniels +1
Computer Science · Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1608.04184
openalex publication_date 2016/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H0 = -Δ+ V0(x) be a Schroedinger operator on L2(ℝν), ν=1,2, or 3, where V0(x) is a bounded measurable real-valued function on ℝν. Let V be an operator of multiplication by a bounded integrable real-valued function V(x) and put Hr = H0+rV for real r. We show that the associated spectral shift function (SSF) ξ admits a natural decomposition into the sum of absolutely continuous ξ(a) and singular ξ(s) SSFs. This is a special case of an analogous result for resolvent comparable pairs of self-adjoint operators, which generalises the known case of a trace class perturbation while also simplifying its proof. We present two proofs -- one short and one long -- which we consider to have value of their own. The long proof along the way reframes some classical results from the perturbation theory of self-adjoint operators, including the existence and completeness of the wave operators and the Birman-Krein formula relating the scattering matrix and the SSF. The two proofs demonstrate the equality of the singular SSF with two a priori different but intrinsically integer-valued functions: the total resonance index and the singular μ-invariant.