Asymptotic completeness in dissipative scattering theory

Abstract

We consider an abstract pseudo-Hamiltonian for the nuclear optical model, given by a dissipative operator of the form H = HV −iC * C, where HV = H0 +V is self-adjoint and C is a bounded operator. We study the wave operators associated to H and H0. We prove that they are asymptotically complete if and only if H does not have spectral singularities on the real axis. For Schrödinger operators, the spectral singularities correspond to real resonances.

Generic nature of asymptotic completeness in dissipative scattering theory

“…We recall that the unit-sphere in R 3 is denoted by S 2 . We refer to [10,11] for details showing that the abstract Hypotheses 1-8 are indeed satisfied in the case of the nuclear optical model, under the conditions on the potentials imposed in the following theorems. Proof.…”

“…In the case where H = −∆+V −iC * C on L 2 (R 3 ), with V and C bounded and compactly supported potentials, a spectral singularity of H corresponds to a resonance embedded in the essential spectrum [0, ∞) (see, e.g., [7] for the theory of resonances for Schrödinger operators, and [10] for a comparison between the notions of resonances and spectral singularities).…”

Scattering matrices for dissipative quantum systems

“…Our next concern is to find a necessary and sufficient condition for the invertibility of S(λ). We use the notion of spectral singularity introduced in [14]. This notion is closely related to that considered in [6,7,37] within the theory of "spectral operators".…”

“…The notion of spectral singularities in the nuclear optical model (1.1) and its relation with the notion of resonances will be discussed in Section 2.4 (see also [14,Section 6]).…”

“…a state whose probability of absorption is equal to 1). In this respect our results are to be compared to those recently proven in [14].…”