On the semigroup decomposition of the time evolution of quantum mechanical resonances

Abstract

A way of utilizing Lax-Phillips type semigroups for the description of time evolution of resonances for scattering problems involving Hamiltonians with a semibounded spectrum was recently introduced by Y. Strauss. In the proposed framework the evolution is decomposed into a background term and an exponentially decaying resonance term evolving according to a semigroup law given by a Lax-Phillips type semigroup; this is called the semigroup decomposition. However, the proposed framework assumes that the S-matrix in the energy representation is the boundary value on the positive real axis of a bounded analytic function in the upper half-plane. This condition puts strong restrictions on possible applications of this formalism. In this paper it is shown that there is a simple way of weakening the assumptions on the S-matrix analyticity while still obtaining the semigroup decomposition of the evolution of a resonance.

Approximate resonance states in the semigroup decomposition of resonance evolution

“…20 we now use assumption (iv) in Section 1. The S-matrixS(·) is then the restriction of its extension S(·) on R + .…”

“…In Section 3 we extend the framework of Ref. 19,20 to the case of multiple resonances and, furthermore, find an estimate on the size of the background term in the expression for the time evolution of the survival probability of a resonance. In Section 4 we analyze a simple but illuminating example involving a one dimensional model of scattering from a square barrier potential.…”

“…An explicit expression for the approximate resonance state ψ µ is provided in Ref. 20. It is shown there that, if we denote by {|E − } E∈R + the set of outgoing solutions of the Lippmann-Schwinger equation (using Dirac's notation), then ψ µ is given by…”

“…20 that there exists a dense set Λ ⊂ H ac (H) and a well defined state ψ µ ∈ H ac (H) such that for any g ∈ Λ and any f ∈ H ac the properties (i)-(iv) above induce, for positive times, a decomposition of matrix elements of the evolution U(t) in the form…”

“…Proposed by one of the authors (Y.S.) of the present article 19,20 , this formalism makes use of the Sz.-Nagy-Foias theory of contraction operators and contractive semigroups on Hilbert space 21 which, from the mathematical point of view, is the fundamental theory underlying the Lax-Phillips construction through the notion of model operators for C ·0 class semigroups (see Section 2).…”