We obtain a simple formula for the first-order trace of a regular differential operator on a segment perturbated by a multiplication operator. The main analytic ingredient of the proof is an improvement of the Tamarkin equiconvergence theorem.
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Alexander I. Nazarov, Dmitriy M. Stolyarov, Pavel B. Zatitskiy, The Tamarkin equiconvergence theorem and a first-order trace formula for regular differential operators revisited. J. Spectr. Theory 4 (2014), no. 2, pp. 365–389DOI 10.4171/JST/73