Approximate resonance states in the semigroup decomposition of resonance evolution

Journal of Mathematical Physics (Impact Factor: 1.24). 01/2007; 47(12). DOI: 10.1063/1.2383069
Source: arXiv


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.

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    • "The recently developed semigroup decomposition formalism for the description of the time evolution of quantum mechanical resonances [28] [29] [27] utilizes two central ingredients, namely the Sz.-Nagy-Foias theory of contraction operators and strongly contractive semigroups on Hilbert space [32] and the contractive nesting of Hilbert spaces, i.e., the embedding of one Hilbert space into another via a contractive quasi-affine transformation [13], in order to decompose the time evolution of resonances in standard, non-relativistic, quantum mechanical scattering problems into a sum of a semigroup part and a non-semigroup part. In this decomposition the semigroup part, given in terms of a Lax-Phillips type semigroup (see for example [27] for the terminology used here), is the resonance term and the non-semigroup part is called the background term. The complex eigenvalues of the generator of the semigroup, providing the typical exponential decay behaviour of the resonance part, are associated with resonance poles of the scattering matrix. "
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