Bounds on variation of spectral subspaces under J-self-adjoint perturbations

TL;DR

This paper establishes optimal conditions and sharp bounds for how spectral subspaces of a self-adjoint operator change under specific J-self-adjoint perturbations, with applications to PT-symmetric quantum systems.

Contribution

It provides new criteria ensuring the similarity of perturbed operators to self-adjoint operators and derives precise bounds on spectral subspace variations under J-self-adjoint perturbations.

Findings

Optimal conditions for similarity to self-adjoint operators.

Sharp bounds on spectral subspace variation.

Application to PT-symmetric quantum harmonic oscillator.

Abstract

Let $A$ be a self-adjoint operator on a Hilbert space $\fH$. Assume that the spectrum of $A$ consists of two disjoint components $\sigma_0$ and $\sigma_1$. Let $V$ be a bounded operator on $\fH$, off-diagonal and $J$-self-adjoint with respect to the orthogonal decomposition $\fH=\fH_0\oplus\fH_1$ where $\fH_0$ and $\fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $\sigma_0$ and $\sigma_1$, respectively. We find (optimal) conditions on $V$ guaranteeing that the perturbed operator $L=A+V$ is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on variation of the spectral subspaces of $A$ under the perturbation $V$. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT-symmetric perturbation is discussed.