An Interaction of An Oscillator with An One-Dimensional Scalar Field. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. II: Resolvents formulae
Sergej A. Choroszavin
备注：
Latex 2.09, minor changes of the style, bibliography corrected
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, $u, w_2, δ_{α,x_0}$ for abstract elements; $q, w_1 z, Ω, γ_c$ stand for numbers. $q, u$ are objects to be found, the others are arbitrarily given.
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
1D Particle, 1D Field, 1D Interaction. Simple Exactly Solvable Models based on Finite Rank Perturbations Methods. III. Linear Friction as Radiation Reaction
Generalized quantum isotonic nonlinear oscillator in d dimensions
We present a supersymmetric analysis for the d-dimensional Schroedinger equation with the generalized isotonic nonlinear-oscillator potential V(r)={B^2}/{r^{2}}+ω^{2} r^{2}+2g{(r^{2}-a^{2})}/{(r^{2}+a^{2})^{2}}, B \geq 0. We show that the eigenequation for this potential is exactly solvable provided g=2 and (ωa^2)^2 = B^2 +( \ell +(d-2)/2)^2. Under these conditions, we obtain explicit formulae for all the energies and normalized bound-state wavefunctions.
Long-time asymptotics of the integrable defocusing Wadati-Konno-Ichikawa equation with a finite-genus algebro-geometric background
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；School of Mathematics and Information Science, Zhongyuan University of Technology（中原工学院数学与信息科学学院）
