Singular spectral shift function for Schr\"odinger operators

TL;DR

This paper proves that the spectral shift function for Schrödinger operators can be decomposed into absolutely continuous and singular parts, providing new insights and proofs that connect classical perturbation theory with resonance and invariant concepts.

Contribution

It introduces a natural decomposition of the spectral shift function for Schrödinger operators and offers two distinct proofs, linking classical and modern spectral theory results.

Findings

Spectral shift function decomposes into absolutely continuous and singular parts.

Two proofs demonstrate the equality of the singular SSF with resonance index and μ-invariant.

Reformulation of classical results in perturbation theory of self-adjoint operators.

Abstract

Let $H_0 = -\Delta + V_0(x)$ be a Schroedinger operator on $L_2(\mathbb{R}^\nu),$ $\nu=1,2,$ or 3, where $V_0(x)$ is a bounded measurable real-valued function on $\mathbb{R}^\nu.$ Let $V$ be an operator of multiplication by a bounded integrable real-valued function $V(x)$ and put $H_r = H_0+rV$ for real $r.$ We show that the associated spectral shift function (SSF) $\xi$ admits a natural decomposition into the sum of absolutely continuous $\xi^{(a)}$ and singular $\xi^{(s)}$ 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 $\mu$-invariant.