Local form-subordination condition and Riesz basisness of root systems

TL;DR

This paper introduces a form-local subordination condition that ensures the root system of certain perturbed operators forms a Riesz basis, with applications demonstrated on Schrödinger operators with complex potentials.

Contribution

It establishes a new criterion linking eigenvalue gaps and perturbation strength to guarantee Riesz basisness in non-symmetric operator perturbations.

Findings

Root systems form Riesz bases under the new condition.

Asymptotic formulas for eigenvalues and eigenvectors are valid.

Applicable to Schrödinger operators with complex potentials.

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 Schr\"odinger operators with possibly unbounded or singular complex potential perturbations.