Root System of Singular Perturbations of the Harmonic Oscillator Type Operators

Abstract

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space $${\mathcal{H}}$$H, i.e. of the self-adjoint operator with simple positive eigenvalues μk satisfying μk+1 − μk ≥ Δ > 0. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system contains a Riesz basis.