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Mat. Zametki, 2019, Volume 105, Issue 4, Pages 483–506 (Mi mz12061)  

Solvability of the Operator Riccati Equation in the Feshbach Case

S. Albeverioab, A. K. Motovilovcd

a Universität Bonn, Institut für Angewandte Mathematik
b Universität Bonn, Interdisziplinäres Zentrum für Komplexe Systeme
c Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Moscow Region
d University "Dubna", Dubna, Moskow Reg.

Abstract: Let $L$ be a bounded $2\times2$ block operator matrix whose main-diagonal entries are self-adjoint operators. It is assumed that the spectrum of one of these entries is absolutely continuous, being presented by a single finite band, and the spectrum of the other main-diagonal entry is entirely contained in this band. We establish conditions under which the operator matrix $L$ admits a complex deformation and, simultaneously, the operator Riccati equations associated with the deformed $L$ possess bounded solutions. The same conditions also ensure a Markus–Matsaev-type factorization of one of the initial Schur complements analytically continued onto the unphysical sheet(s) of the complex plane of the spectral parameter. We prove that the operator roots of this Schur complement are explicitly expressed through the respective solutions to the deformed Riccati equations.

Keywords: operator Riccati equation, Feshbach case, Friedrichs model, graph subspace, resonance, unphysical sheet.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-51-12389
16-01-00706
Deutsche Forschungsgemeinschaft AL 214/49
Federal Ministry of Education and Research (Germany)
This work was supported by the Heisenberg–Landau Program, the Deutsche Forschungsgemeinschaft (DFG), and the Russian Foundation for Basic Research under grants 15-51-12389 and 16-01-00706.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm12061

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English version:
Mathematical Notes, 2019, 105:4, 485–502

Bibliographic databases:

UDC: 517.983
Received: 08.05.2018

Citation: S. Albeverio, A. K. Motovilov, “Solvability of the Operator Riccati Equation in the Feshbach Case”, Mat. Zametki, 105:4 (2019), 483–506; Math. Notes, 105:4 (2019), 485–502

Citation in format AMSBIB
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\paper Solvability of the Operator Riccati Equation in the Feshbach Case
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