Lax-Phillips scattering theory and well-posed linear systems: a coordinate-free approach

Abstract

Abstract. We give a further elaboration of the fundamental connections between Lax-Phillips scattering, conservative input/state/output linear systems and Sz.-Nagy-Foias model theory for both the discrete-and continuous-time settings. In particular, for the continuous-time setting, we show how to locate a scattering-conservative L 2 -well-posed linear system (in the sense of Staffans and Weiss) embedded in a Lax-Phillips scattering system presented in axiomatic form; conversely, given a scattering-conservative linear system, we show how one can view the space of finite-energy input-state-output trajectories of the system as the ambient space for an associated Lax-Phillips scattering system. We use these connections to give a simple, conceptual proof of the identity of the scattering function of the scattering system with the transfer function of the input-state-output linear system. As an application we show how system-theoretic ideas can be used to arrive at the spectral analysis of the scattering function.

Well-posed systems—The LTI case and beyond

“…To our regret, because of length constraints, we also have to leave out topics that would have been very well suited in this paper. One such topic is the Lax-Phillips semigroup associated to a well-posed system (the connection between well-posed systems theory and scattering theory), for which we refer to [7,9,59,65,67,68]. Another topic that we are compelled to leave out are the timevarying well-posed linear systems, for which we refer to [12,33,58,59] (an incomplete list).…”

Weak Markov processes as linear systems

“…This leads to the term transfer function in control theory. Similar connections between system trajectories and the dynamics of the ambient system have been worked out in [9], see also [8] for a commutative polydisk setting and [4] for a one-variable continuous time setting.…”

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

“…(8.56) With the additional wandering-subspace assumption and orthogonal-decomposition assumption (8.54) given above in place, then the map k → (e(n), h(n), e * (n)) defined by (8.52) gives a one-to-one correspondence between elements k of K and finite-energy U -system trajectories ( e, h, e * ). This last statement is essentially Lemma 2.3 in [9] and is the main ingredient in the coordinate-free approach in embedding a unitary colligation into a (discrete-time) Lax-Phillips scattering system. The reader can check that the situation in Theorem 8.6 meets all these assumptions (with Δ in place of E * and Δ * in place of E); the computation in the proof of Theorem 8.6 exhibits the k = i H0 h 0 + i Δ * ,0 δ * corresponding to the finite-energy system trajectory supported on Z + with initial condition h 0 and impulse input supported at time Vol.…”

“…By a minimal unitary extension of V we mean a unitary operator U on a Hilbert space K containing H 0 as a subspace such that the restriction of U * to D agrees with V and the smallest U-reducing subspace containing H 0 is all of K. From the work of Arov and Grossman [7,8] and Katsnelson et al [26], it is known that there is a special unitary colligation U 0 (called the universal unitary colligation) so that any such unitary extension U * of V arises as the lower feedback connection U * = F (U 0 , U 1 ) of U 0 with a free-parameter unitary colligation U 1 (see Theorem 6.1). A special unitary extension of V is obtained as the unitary dilation U * 0 of the universal unitary colligation U 0 (or, in the language of [9], U 0 is the unitary evolution operator for the Lax-Phillips scattering system in which U 0 is embedded). This special unitary extension U * 0 of V is called the universal unitary extension.…”