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Математические заметки, 2016, том 100, выпуск 6, статья опубликована в англоязычной версии журнала
(Mi mzm11475)
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Эта публикация цитируется в 3 научных статьях (всего в 3 статьях)
Статьи, опубликованные в английской версии журнала
On Invariant Graph Subspaces of a $J$-Self-Adjoint Operator in the Feshbach Case
S. A. Albeverioa, A. K. Motovilovb a Institut für Angewandte Mathematik und HCM, Universität Bonn,
Bonn, Germany
b Joint Institute for Nuclear Research and Dubna State University, Dubna, Russia
Аннотация:
We consider a
$J$-self-adjoint
$2\times2$
block operator matrix
$L$
in the Feshbach spectral case, that is, in the case where the
spectrum of one main-diagonal entry of
$L$
is embedded into the absolutely
continuous spectrum of the other main-diagonal entry.
We work
with the analytic continuation of the Schur complement of a
main-diagonal entry in $L-z$ to the unphysical sheets of the
spectral parameter
$z$
plane.
We present 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.
Ключевые слова:
$J$-self-adjoint operator, subspace perturbation problem,
graph subspace, operator Riccati equation, off-diagonal perturbation, resonance.
Образец цитирования:
S. A. Albeverio, A. K. Motovilov, “On Invariant Graph Subspaces of a $J$-Self-Adjoint Operator in the Feshbach Case”, Math. Notes, 100:6 (2016), 761–773
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