Operator interpretation of resonances arising in spectral problems for 2 x 2 operator matrices

TL;DR

This paper investigates the spectral properties of 2x2 operator matrices, focusing on resonances in spectral problems, and constructs non-selfadjoint operators to analyze the completeness and basis properties of resonance eigenvectors.

Contribution

It introduces a novel operator-valued function V_1(Y) and studies the equation H_1=A_1+V_1(H_1), demonstrating the existence of operators with spectra including resonances and establishing their eigenvector properties.

Findings

Constructed operator-valued function V_1(Y) with spectral invariance.

Proved solvability of the operator equation H_1=A_1+V_1(H_1).

Established completeness and basis properties of resonance eigenvectors.

Abstract

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)\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).