Remark on the Compensation of Singularities in Krein’s Formula

We reduce the spectral problem for an additively perturbed self-adjoint operator H V =H 0−V, to the dual problem of finding zeros of the operator function $$ Sign V - |V|^{1/2} [H_0 - \lambda ]^{ - 1} |V|^{1/2} , $$ and develop the Schmidt perturbation procedure for the resolvent of H v . Based on Rouche theorem for operator-valued analytic functions, we observe, in the Krein’s formula for the perturbed resolvent [H v -λI]-1, the compensation of singularities inherited from H 0, and suggest a convenient algorithm for approximate calculation of the groups of eigenfunctions and eigenvalues of the perturbed operator.