Solvability of the operator Riccati equation in the Feshbach case

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.

18 pages, 2 figures

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