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Resonances as eigenvalues in the Gel’fand triplet approach for finite-dimensional Friedrichs models on the positive half line. (English) Zbl 1165.81039

The following finite-dimensional Friedrichs model on the positive half-line \(R_+=(0,\infty)\) is considered. Let \({\mathcal H}_{0,+} = L^2(R_+,{\mathcal K}, d\lambda)\), Were \(\mathcal K\) denotes a multiplicity Hilbert space, \(\text{dim}{\mathcal K} < \infty\). Further, let \(\mathcal E\) be a finite-dimensional Hilbert space, \(\text{dim}{\mathcal E} = \text{dim}{\mathcal K}\) and put \({\mathcal H} := {\mathcal H}_{0,+} \oplus {\mathcal E}\). The projection onto \(\mathcal E\) is denoted by \(P_{\mathcal E}\). \(H_0\) is assumed to be the multiplication operator on \({\mathcal H}_{0,+}\). \(A\) is a self-adjoint operator on \(\mathcal E\) with only positive eiganvalues. The self-adjoint operator \(H\) on \(\mathcal H\) is given by a perturbation of \(H_0\oplus A\) as \(H := (H_0\oplus A) + \Gamma +\Gamma^\star\), where \(\Gamma\) denotes a partial isometry on \(\mathcal H\) with the properties \(\Gamma^\star \Gamma = P_{\mathcal E}, \Gamma^\star \Gamma < 1-P_{\mathcal E}\).
In this paper for finite-dimensional Friedrichs model on the positive half-line resonances (poles of the scattering matrix) are characterized by their their spectral properties with respect to \(H\) directly, i.e. it is shown that they are exactly the eigenvalues of an appropriate extension of \(H\) by a Gel’fand triple. Further the corresponding eigenantilinear forms are used to derive Gamow vectors of resonance, which are eigenvectors of the decay semigroup connected with \(H\).
Conditions are presented such that there are only finitely many resonances and all resonances are simple poles of the scattering matrix.

MSC:

81U20 \(S\)-matrix theory, etc. in quantum theory
47A40 Scattering theory of linear operators
47D06 One-parameter semigroups and linear evolution equations
47N50 Applications of operator theory in the physical sciences
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