Local form-subordination condition and Riesz basisness of root systems

Abstract

We exploit the so called form-local subordination in the analysis of non-symmetric perturbations of unbounded self-adjoint operators with isolated simple positive eigenvalues. If the proper condition relating the size of gaps between the unperturbed eigenvalues and the strength of perturbation, measured by the form-local subordination, is satisfied, the root system of the perturbed operator contains a Riesz basis and usual asymptotic formulas for perturbed eigenvalues and eigenvectors hold. The power of the abstract perturbation results is demonstrated particularly on Schrödinger operators with possibly unbounded or singular complex potential perturbations.

Pseudomodes for Schrödinger operators with complex potentials

“…Our assumption (3.3) on the allowed size of Re V is optimal, at least for polynomial-like potentials (with ν ± = −1 in assumption (3.2)). Indeed, by completely different methods, it has been established in [14,15] that e.g. for potentials V satisfying Re V (x) = |x| β with β ≥ 1 and…”

“…the eigensystem of H V contains a Riesz basis (and there are possibly only finitely many degenerate eigenvalues) and hence the only non-trivial pseudomodes exist for λ close to the eigenvalues of H V (with known asymptotics, see [14]). In turn, the current results suggest that the condition (1.3) is optimal with respect to the Riesz basis property of H V (which can be indeed concluded if more information about the position of eigenvalues of H V is available) and confirms that the borderline case (potentials with ǫ = 0 in (1.3)) is the most challenging one, see [1,Open Problem 15.1].…”