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.