Weak convergence of spectral shift functions revisited

TL;DR

This paper proves that the spectral shift functions for finite interval Schr"odinger operators with coupled boundary conditions converge weakly to those of the full-line operators as the interval length increases, using a Krein-type resolvent identity.

Contribution

It revisits and extends the understanding of spectral shift function convergence for Schr"odinger operators with coupled boundary conditions on expanding intervals.

Findings

Spectral shift functions for finite interval operators converge weakly to the full-line case.

The convergence holds under coupled boundary conditions at the interval endpoints.

A Krein-type resolvent identity is used to establish the convergence.

Abstract

We study convergence of the spectral shift function for the finite interval restrictions of a pair of full-line Schr\"odinger operators to an interval of the form $(-\ell,\ell)$ with coupled boundary conditions at the endpoints as $\ell\to \infty$ in the case when the finite interval restrictions are relatively prime to those with Dirichlet boundary conditions. Using a Krein-type resolvent identity we show that the spectral shift function for the finite interval restrictions converges weakly to that for the pair of full-line Schr\"odinger operators as the length of the interval tends to infinity.