Solvability of the operator Riccati equation in the Feshbach case

TL;DR

This paper investigates conditions for the solvability of operator Riccati equations in a specific spectral setting, enabling block diagonalization of certain operator matrices and explicit expression of operator roots via solutions to these equations.

Contribution

It provides new criteria for the existence of bounded solutions to operator Riccati equations in the Feshbach case with continuous spectrum, facilitating operator matrix diagonalization.

Findings

Established conditions for bounded solutions to Riccati equations.

Linked solutions of Riccati equations to the factorization of the Schur complement.

Derived explicit formulas for operator roots of the Schur complement.

Abstract

We consider a bounded block operator matrix of the form $$ L=\left(\begin{array}{cc} A & B \\ C & D \end{array} \right), $$ where the main-diagonal entries $A$ and $D$ are self-adjoint operators on Hilbert spaces $H_{_A}$ and $H_{_D}$, respectively; the coupling $B$ maps $H_{_D}$ to $H_{_A}$ and $C$ is an operator from $H_{_A}$ to $H_{_D}$. It is assumed that the spectrum $\sigma_{_D}$ of $D$ is absolutely continuous and uniform, being presented by a single band $[\alpha,\beta]\subset\mathbb{R}$, $\alpha<\beta$, and the spectrum $\sigma_{_A}$ of $A$ is embedded into $\sigma_{_D}$, that is, $\sigma_{_A}\subset(\alpha,\beta)$. We formulate conditions under which there are bounded solutions to the operator Riccati equations associated with the complexly deformed block operator matrix $L$; in such a case the deformed operator matrix $L$ admits a block diagonalization. The same conditions also ensure the Markus-Matsaev-type factorization of the Schur complement $M_{_A}(z)=A-z-B(D-z)^{-1}C$ analytically continued onto the unphysical sheet(s) of the complex $z$ plane adjacent to the band $[\alpha,\beta]$. We prove that the operator roots of the continued Schur complement $M_{_A}$ are explicitly expressed through the respective solutions to the deformed Riccati equations.