Analytic perturbation techniques for the Friedrichs model : Intermediate Operator

B. Pavlov and I. Antoniou


techniques of an intermediate operator is developed for the
Friedrichs model obtained as a finite-dimensional perturbation
{cal P} longrightarrow {cal P}_{_{varepsilon}} = {cal P} + varepsilon A
of the momentum operator ${cal P} = frac{1}{i},,frac{d}{dx}$,
defined by the corresponding operator-extension procedure. This
technique permits to observe a creation of the resonance at the
given point $k_{_{0}}$ via presenting the Scattering matrix for
the above pair as a product of the non-analytic at
$left(varepsilon,,kright) = left(0,,k_{_0}right)$ factor
$S^{^{varepsilon}}_{_{0}} left(k right) $ which is the Scattering
matrix to the pair $ left{{cal P}^{^{varepsilon}}_{_{0}},,{cal
of the momentum with a local intermediate operator $
{cal P}^{^{varepsilon}}_{_{0}}$, and an analytic factor $ Sleft({cal P} +
varepsilon A,,{cal P}^{^{varepsilon}}_{_{ 0}} right) $ of both
variables $left(varepsilon,, kright)$ near the point
$left(0,, k_{_0}right)$ which is the Scattering matrix of the
pair $left({cal P}_{_{varepsilon}},{cal P}^{^{varepsilon}}_{_{
0}}right)$. The corresponding representation is valid also for eigenfunctions
of the perturbed operator.} vskip

Operator Extention , Blaschke function

Math Review Classification
Primary pacs{~03.65.Nk} ; Secondary pacs{~ 02.30.Tb}

Last Updated
10 December 2003

22 pages

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