Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator

TL;DR

This paper derives optimal spectral bounds and estimates on reducing subspaces for J-self-adjoint operators, extending classical results to PT-symmetric perturbations with applications to quantum harmonic oscillators.

Contribution

It provides the first sharp bounds on spectra and operator angles for J-self-adjoint operators under bounded perturbations, generalizing Davis-Kahan theorems.

Findings

Established optimal spectral bounds for J-self-adjoint operators.

Derived norm bounds on operator angles between invariant subspaces.

Applied results to PT-symmetric quantum harmonic oscillator.

Abstract

Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjoint operator commuting with J and V a bounded J-self-adjoint operator anti-commuting with J. We establish optimal estimates on the position of the spectrum of L with respect to the spectrum of A and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator A and the perturbed operator L. All the bounds are given in terms of the norm of V and the distances between pairs of disjoint spectral sets associated with the operator L and/or the operator A. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed. The sharp norm bounds obtained for the operator angles generalize the celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjoint perturbations.