 99428 R. Mennicken (Regensburg), A. K. Motovilov (Dubna)
 Operator interpretation of resonances arising
in spectral problems for 2 x 2 operator matrices
(620K, PostScript)
Nov 12, 99

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Abstract. We consider operator matrices
$
\bH=\left(\matrix{
A_0 & B_{01} \cr
B_{10} & A_{1}
}\right)
$
with selfadjoint entries $A_i$, $i=0,1,$ and bounded
$B_{01}=B_{10}^*$, acting in the orthogonal sum
\mbox{${\cal H}={\cal H}_0\oplus{\cal H}_1$}
of Hilbert spaces ${\cal H}_0$ and ${\cal H}_1$.
We are especially interested in the case where the
spectrum of, say, $A_1$ is partly or totally embedded into the
continuous spectrum of $A_0$ and the transfer function
$M_1(z)=A_1z+V_1(z)$, where $V_1(z)=B_{10}(zA_0)^{1}B_{01}$,
admits analytic continuation (as an operatorvalued
function) through the cuts along branches of the continuous
spectrum of the entry $A_0$ into the unphysical sheet(s) of the
spectral parameter plane. The values of $z$ in the unphysical
sheets where $M_1^{1}(z)$ and consequently the resolvent
$(Hz)^{1}$ have poles are usually called resonances. A main
goal of the present work is to find nonselfadjoint operators
whose spectra include the resonances as well as to
study the completeness and basis properties of the resonance
eigenvectors of $M_1(z)$ in ${\cal H}_1$. To this end we
first construct an operatorvalued function $V_1(Y)$ on the space
of operators in ${\cal H}_1$ possessing the property:
$V_1(Y)\psi_1=V_1(z)\psi_1$ for any eigenvector $\psi_1$ of $Y$
corresponding to an eigenvalue $z$ and then study the equation
$
H_1=A_1+V_1(H_1).
$
We prove the solvability of this equation even in the
case where the spectra of $A_0$ and $A_1$ overlap.
Using the fact that the root vectors of the solutions $H_1$
are at the same time such vectors for $M_1(z)$, we prove
completeness and even basis properties for the root vectors
(including those for the resonances).
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