Equivalence criterion for two asymptotic formulae

Abstract

We study the equivalence conditions of two asymptotic formulae for an arbitrary non-decreasing unbounded sequence { }. We show that if is a non-decreasing and unbounded at infinity function, { } is a non-decreasing sequence asymptotically inverse to the function , then for each sequence of real numbers satisfying an asymptotic estimate ∼ , → +∞, the estimate () ∼ (), → +∞, holds if and only if is a pseudo-regularly varying function (PRV-function). We find a necessary and sufficient condition for the non-decreasing sequence { } and the function , 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.