Equivalence criterion for two asymptotic formulae

We study the equivalence conditions of two asymptotic formulae for an arbitrary non-decreasing unbounded sequence $ \{\lambda_n \} $. We show that if $g$ is a non-decreasing and unbounded at infinity function, $\{f_n\}$ is a non-decreasing sequence asymptotically inverse to the function $g$, then for each sequence of real numbers $\lambda_n$ satisfying an asymptotic estimate $\lambda_n\sim f_n$, $n\to+\infty,$ the estimate $N(\lambda)\sim g(\lambda)$, $ \lambda\to+\infty$, holds if and only if $g$ is a pseudo-regularly varying function (PRV-function). We find a necessary and sufficient condition for the non-decreasing sequence $\{f_n\}$ and the function $g$, under which the second formula implies the first one. Employing this criterion, we find a non-trivial class of perturbations preserving the asymptotics of the spectrum of an arbitrary closed densely defined in a separable Hilbert space operator possessing at least one ray of the best decay of the resolvent. This result is the first generalization of the a known Keldysh theorem to the case of operators not close to self-adjoint or normal, whose spectra can strongly vary under small perturbations. We also obtain sufficient conditions for a potential ensuring that the spectrum of the Strum-Liouville operator on a curve has the same asymptotics as for the potential with finitely many poles in a convex hull of the curve obeying the trivial monodromy condition. These sufficient conditions are close to necessary ones.