99-428 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 self-adjoint 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 $M_1(z)=A_1-z+V_1(z)$, where $V_1(z)=B_{10}(z-A_0)^{-1}B_{01}$, admits analytic continuation (as an operator-valued 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 $(H-z)^{-1}$ have poles are usually called resonances. A main goal of the present work is to find non-selfadjoint 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 operator-valued 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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