Factorization Theorem for the Transfer Function of a 2×2 Operator Matrix with Unbounded Couplings

We consider the analytic continuation of the transfer function associated with a 2×2 operator matrix having unbounded couplings into unphysical sheets of its Riemann surface. We construct a family of non-selfadjoint operators which factorize the transfer function and reproduce certain parts of its spectrum including the nonreal (resonance) spectrum situated in the unphysical sheets neighboring the physical sheet. Introduction In this work we consider 2× 2 operator matrices H0 = ( A0 T01 T10 A1 ) (0.1) acting in the orthogonal sum H = H0 ⊕ H1 of separable Hilbert spaces H0 and H1. The entry A0 : H0 → H0 is assumed to be an unbounded selfadjoint operator with the domain D(A0). We suppose that A0 is semibounded from below, i. e., A0 ≥ α0 for some α0 ∈ R and without loss of generality let α0 > 0. The entry A1 is assumed to be a bounded selfadjoint operator in H1. In contrast to [MM1, MM2], in the present paper we consider unbounded coupling operators Tij : Hj → Hi, i, j = 0, 1, i 6=j. Regarding these operators the following conditions are supposed to be fulfilled: T ∗ 01 = T10 and D(T10) ⊃ D(A 0 ). (0.2) These assumptions are similar to those used in the works by V.M.Adamyan, H.Langer, R. Mennicken and J. Saurer [ALMSa] and by R.Mennicken and A.A. Shkalikov [MS]. Under the conditions (0.2) the matrix (0.1) is a symmetric closable operator in H on the domain D(A0) ⊕ H1 and its closure H = H0 is a Date: December 28, 1999.