Also asked on MSE (https://math.stackexchange.com/questions/4987654/in-what-sense-is-a-change-of-boundary-conditions-a-finite-rank-perturbation and https://math.stackexchange.com/questions/4875398/how-is-changing-the-boundary-conditions-a-finite-rank-perturbation?noredirect=1&lq=1) with no answers.
I am concerned with differential operators, let's say Schrodinger operators $T = -\Delta + V$ defined on $\mathcal D(T) = \{f \in H^2(\Omega) : V f \in L^2(\Omega)\} \cap \mathcal B(\Omega)$ for suitable $\Omega \subseteq \mathbb R$ and $\mathcal B(\Omega)$ is certain boundary conditions. Often we have results about spectra only for $\Omega = [0, \infty)$. Taking $\widetilde V(x) = V(-x)$ and reflecting your input functions similarly, you can quickly extend these results to $\Omega = (-\infty, 0]$, but it seems non-trivial to step to any conclusions about $\Omega = \mathbb R$. It seems you cannot just take the direct sum of these two operators. Sets like the singular continuous spectrum are notoriously unstable even under rank-one perturbations, so it becomes a reasonable question of how the singular continuous spectrum of the half-line problems will relate to the singular continuous spectrum of the whole line problem.
In the paper "Operators with Singular Continuous Spectrum V: Sparse Potentials" by Simon and Stolz, it is said at the bottom of page 5 that:
Whole-line problems differ from the direct sum of two half-line problems only by adding boundary conditions at 0, a finite rank perturbation
This does not immediately make sense to me, since a finite rank perturbation will be bounded and so shouldn't change the domain of the operator. In general this paper is a bit brief on the relation between the half-line and whole-line problems. Could someone help make this explicit (ie. explicitlyish what perturbation is being taken and why it is finite rank) or give a reference that does so?
