Abstract
For the Sturm–Liouville operator \(T\) on a smooth curve with a potential meromorphic in some neighborhood of this curve, we study the class of perturbations under which the generalized trace invariance principle holds. The operator \(T\) is non-self-adjoint and not even close to self-adjoint: the Keldysh theorem on the stability of the spectrum does not work for it and the system of root vectors does not form a basis. On the other hand, the spectrum of this operator can be divided into separate clusters \(\Lambda_{n}\) , the distance between which grows unlimitedly at infinity. We have found sufficient conditions for the perturbation that are close to necessary and under which the classical formula for a regularized trace is valid. The main idea of the proof of this formula is the transition to a ‘‘diagonal’’ sequence during summation.
