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 +Ω^2 q -Ω^2<l|_2 u> =w_1, z u +4γ_cδ_{α,x_0}q -Bu +4γ_cδ_{α,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.

Latex 2.09, minor changes of the style, bibliography corrected

References in corpus (1)

• An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. I: D'Alembert-Kirchhoff-like formulae

Cited by in corpus (1)

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