Root system of singular perturbations of the harmonic oscillator type operators

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues ${\mu }_k$ satisfying ${\mu }_{k+1}-{\mu }_k\ge \mathrm{\Delta }>0$. 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 forms a Riesz basis.