Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

TL;DR

This paper studies how perturbations of a selfadjoint operator with discrete spectrum affect the basis properties of its root vectors, extending previous work on harmonic oscillators and establishing conditions for an unconditional basis.

Contribution

It extends prior results by identifying conditions under which the root vectors of a perturbed operator form an unconditional basis, generalizing from harmonic oscillators to broader classes of operators.

Findings

Root vectors form an unconditional basis under specified spectral gap conditions.

Perturbations with small enough norm relative to spectral gaps preserve basis properties.

Generalization of previous harmonic oscillator results to wider operator classes.

Abstract

We analyze the perturbations $T+B$ of a selfadjoint operator $T$ in a Hilbert space $H$ with discrete spectrum $\{t_k \}$, $T \phi_k = t_k \phi_k$, as an extension of our constructions in arXiv: 0912.2722 where $T$ was a harmonic oscillator operator. In particular, if $t_{k+1}-t_k \geq c k^{\alpha - 1}, \quad \alpha > 1/2$ and $\| B \phi_k \| = o(k^{\alpha - 1})$ then the system of root vectors of $T+B$, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in $H$.