A sparse potential is built so that the essential spectrum is [-2,2] and the spectral measure of the boundary vector stays non-Rajchman for every rank-one perturbation.
Poltoratski, Survival probability in rank-one perturbation problems , Commun
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On a Question of Poltoratski
A sparse potential is built so that the essential spectrum is [-2,2] and the spectral measure of the boundary vector stays non-Rajchman for every rank-one perturbation.