Operator interpretation of resonances arising in spectral problems for 2 x 2 operator matrices
| dc.creator | Mennicken, R. | |
| dc.creator | Motovilov, A. K. | |
| dc.date | 1997-08-01 | |
| dc.date | 1998-03-17 | |
| dc.date.accessioned | 2026-07-07T08:59:19Z | |
| dc.date.available | 2026-07-07T08:59:19Z | |
| dc.description | We consider operator matrices {\bf H}= (A_0 B_{01} \\ B_{10} A_{1}) with self-adjoint entries A_i, i=0,1, and bounded B_{01}=B_{10}^*, acting in the orthogonal sum {\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)ψ_1=V_1(z)ψ_1 for any eigenvector ψ_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). | |
| dc.description | LaTeX 2e (REVTeX style), 62 pages. A slightly extended version. In particular a figure was added. A number of corrections was made. Submitted to Math. Nachr | |
| dc.identifier | https://arxiv.org/abs/funct-an/9708001 | |
| dc.identifier | http://arxiv.org/abs/funct-an/9708001 | |
| dc.identifier | Mathematische Nachrichten, vol. 201 (1999), pp. 117-181. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/147665 | |
| dc.subject | Functional Analysis | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | Mathematical Physics | |
| dc.subject | Operator Algebras | |
| dc.subject | Spectral Theory | |
| dc.subject | Nuclear Theory | |
| dc.subject | 47A56 47Nxx (Primary) 47N50 47A40 (Secondary) | |
| dc.title | Operator interpretation of resonances arising in spectral problems for 2 x 2 operator matrices | |
| dc.type | text |