An Analogue of the Gelfand–Levitan Trace Formula for the Sturm–Liouville Operator with a Meromorphic Potential

For the Sturm–Liouville operator on a smooth curve with a potential that is meromorphic in some neighborhood of , we study the question of the influence of the poles of the potential on the distribution of the spectrum. We have shown that if all the poles lying on the segment connecting the ends of satisfy the trivial monodromy condition, then the spectrum of the operator has the same asymptotics as in the case of a potential smooth on . The regularized trace formula differs from the well-known Gelfand–Levitan formula only in a correction term, which is equal to the integral of the potential over the curve . The question of the necessity of the trivial monodromy conditions for spectral stability is also studied. It is shown that if this condition is violated by an addition that is holomorphic near the segment the regularized trace formula has the same form as in the case of the trivial monodromy. Previously, the authors showed that in the case of the Bessel potential, the Gelfand–Levitan formula is valid only if the trivial monodromy condition is satisfied.