Approximate resonance states in the semigroup decomposition of resonance evolution

Abstract

The semigroup decomposition formalism makes use of the functional model for C ·0 class contractive semigroups for the description of the time evolution of resonances. For a given scattering problem the formalism allows for the association of a definite Hilbert space state with a scattering resonance. This state defines a decomposition of matrix elements of the evolution into a term evolving according to a semigroup law and a background term. We discuss the case of multiple resonances and give a bound on the size of the background term. As an example we treat a simple problem of scattering from a square barrier potential on the half-line.

A modified Lax-Phillips scattering theory for quantum mechanics

“…If in addition to (i)-(ii) we assume that the scattering system satisfies (iii)-(iv) in Section 3, then the results of Refs. [S3,S4,SHV] and Theorem 8 of the present paper imply that to a resonance pole of the scattering matrixŜ QM (·) at a point z = µ, Im µ < 0, there is associated a resonance state ψ res µ ∈ H ac (or an eigensubspace in the more general case) which is an approximate eigenstate of the elements of the approximate Lax-Phillips semigroup {Z app (t)} t≥0 . This last result is the analogue of Theorem 3, a central result of the Lax-Phillips scattering theory associating with each pole of the Lax-Phillips scattering matrix S LP (·) a resonance state (or eigensubspace) in the Lax-Phillips Hilbert space H LP .…”

“…The problem of the definition of appropriate resonance states corresponding to resonance poles of the scattering matrix in quantum mechanical scattering has been addressed in the context of the recent development of the formalism of semigroup decomposition of resonance evolution [S3,S4,SHV] (of course, there are several other formalisms for dealing with the problem of scattering resonances in quantum mechanics, notably complex scaling [AC,BC,Sim1,Sim2,Hun,SZ,HS] and rigged Hilbert spaces [BaSch, Baum, BG, HoSi, PGS]. Here we consider the framework most suitable, in terms of its mathematical constructions, for the development of the formalism introduced in the present paper).…”

Transition Decomposition of Quantum Mechanical Evolution

“…An explicit expression for the approximate resonance state |ψ app μ is provided in [8,9,15]. If we denote by |E − the outgoing solutions of the Lippmann-Schwinger equation for the scattering problem being considered, then |ψ app μ is given by [8,9,15] |ψ app…”

“…If we denote by |E − the outgoing solutions of the Lippmann-Schwinger equation for the scattering problem being considered, then |ψ app μ is given by [8,9,15] |ψ app…”

Resonances of quantum mechanical scattering systems and Lax–Phillips scattering theory

“…A similar procedure (construction of special invariant subspaces of the characteristic semigroup) was recently presented by Strauss et al, 19 where more special assumptions are used (e.g., multiplicity 1, only simple poles of the scattering matrix). They use another approach, starting with methods of Sz.-Nagy and Foias on Harmonic Analysis of Operators on Hilbert spaces.…”