An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. II: Resolvents formulae

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<l|_2 u> =w_1, z u +4\gamma_c\delta_{\alpha,x_0}q -Bu +4\gamma_c\delta_{\alpha,x_0}<l|_2 u> =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.