On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case

TL;DR

This paper investigates invariant subspaces of a J-self-adjoint operator matrix in the Feshbach case, analyzing spectral properties, operator roots, and Riccati equations, with an explicit example illustrating the theory.

Contribution

It introduces conditions for operator roots of the Schur complement and explores invariant subspaces via Riccati equations in the Feshbach spectral context.

Findings

Operator roots can reproduce parts of the spectrum including resonances.

Bounded solutions to Riccati equations lead to graph representations of invariant subspaces.

An explicit example demonstrates the theoretical results.

Abstract

We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of one of the Schur complements of L to the unphysical sheets of the spectral parameter plane. We present the conditions under which the continued Schur complement has operator roots, in the sense of Markus-Matsaev. The operator roots reproduce (parts of) the spectrum of the Schur complement, including the resonances. We then discuss the case where there are no resonances and the associated Riccati equations have bounded solutions allowing the graph representations for the corresponding J-orthogonal invariant subspaces of L. The presentation ends with an explicitly solvable example.