Resonances and virtual poles in scattering theory (Q1421640)

The author develops the formal scattering theory between a free Hamiltonian \(H_0\) and perturbed operator \(H=H_0+V\), \(V= A^\ast CA\). Both operators have an absolutely continuous spectrum of uniform mulitplicity, possess no singular spectrum and \(H_0\) has only finitely many eigenvalues of finite multiplicity. A key role in this analysis is played by the partial resolvent \(P_0 (z-H)^{-1} P_0\), and the closely related Livsic matrix. Both are operators of finite rank, because \(P_0\) denotes the projection on the bound states of \(H_0\). Additional conditions involving the limit of \(A (\lambda +i\varepsilon -H_0)^{-1} A^\ast\) then guarantee the existence and completeness of the wave operators and the meromorphy of the scattering amplitude, \(T\), whose poles are the resonances. These coincide with the non real poles of \(CA (z-H)A^\ast\). With the aid of the intermediate operator \(H_1=H_0 +P_0^\perp VP^\perp +P_0VP_0\), a Gelfand triple is constructed, in which the resonances have an eigenfunctional representation. No motivation for the results nor examples are given in this paper.