Sergej A. Choroszavin ( sergius@pve.vsu.ru )
  An Interaction of An Oscillator with 
  An One-Dimensional Scalar Field.
  Simple Exactly Solvable Models based on  
  Finite Rank Perturbations Methods. 
  II: Resolvents formulae 
(66K, LaTeX 2.09)

ABSTRACT.   This paper is an electronic application to my set of lectures, 
 subject:`Formal methods in solving differential equations and 
 constructing models of physical phenomena'. Addressed, mainly: 
 postgraduates and related readers. 
 Content: a very detailed discussion of the simple model of interaction based 
 on the equation array: 
 z q +\Omega^2 q -\Omega^2 =w_1 ,
 z u +4\gamma_c\delta_{\alpha,x_0}q -Bu 
         +4\gamma_c\delta_{\alpha,x_0} =w_2. 
 Besides, less detailed discussion of related models.
 Central mathematical points: 
 Finite Rank Perturbations Methods, Resolvents formulae,  
 Donoghue-like models, Friedrichs-like models.
 Central physical points: phenomenon of Resonance 
 and notion of Second Sheet. 
 Hereafter I use a P.A.M. Dirac's ``bra-ket'' syntax and suppose that 
$B$ stands for an abstract linear operator,
$l$ for a linear functional, $u, w_2, \delta_{\alpha,x_0}$ 
 for abstract elements; $q, w_1 z, \Omega, \gamma_c$ stand for numbers. 
$q, u$ are objects to be found, the others are arbitrarily given.