The Asymmetric Quantum Rabi Model And Generalised P(?)schl-teller Potentials

In 1936,in order to describe the effect of a rapidly varying weak magnetic field on an oriented atom possessing nuclear spin,I.I.Rabi introduced the Rabi model.In the next following 80 years,this model has been extensively investigated,including in different experimental settings.At the beginning of this thesis,we take a review on recent years' developments and some background.I took investigators quite an effort on obtaining exact analytical solutions;for instance,Edwin Jaynes and Fred Cummings proposed a “Rotating –Wave Approximation” version of the quantum Rabi model,as the so called J-C model.Then in the review part we have the regular part and exceptional part of eigenspectrum of the quantum Rabi model in discussion.After that we are briefly having reviews on achievements in quasi exact sovable models,including Clare Dunning's work on some spectral equivalence between Schr(?)dinger operators,Ko?'s work on connection from the symmetric rabi model to P(?)schl-Teller Schr(?)dinger potentials.After that,statements of the author's work will figure prominently,which refers to the establishment at a connection between the asymmetric quantum Rabi model(AQRM)and generalised P(?)schl-Teller potentials.This new aspect has the following sections:In section ?,based on AQRM's algebraic equations,in analogy to Kús' s work on symmetric quantum Rabi model,we derived the constraint polynomials in the framework of finite-dimentional irreducible representations in the confluent Heun picture of the AQRM,and accordingly made the connection between the AQRM and Schr(?)dinger equations.In section ?,after comparing the general form of algebraic Bethe ansatz equations and equations on the quasi exactly sovable(QES)sector of AQRM,we obtained the corresponding parameters of canonical form in the Bethe ansatz equations,furthermore derived the Schr(?)dinger potential expression and established the spectral equivalence between the QES sector of the AQRM and the QES hyperbolic Schr(?)dinger potentials.Meanwhile,we have the spectral equivalence between the two systems in the figure of enegy levels as a function of the light-matter coupling parameter.After that,we introduced the existence of QES sector in terms of a second order differential equation and some related calculations.In section ?,we extended the spectral equivalence between the QES energies of the AQRM and hyperbolic Schr(?)dinger potentials to the whole spectrum by applying some change of variables and transformation of wave functions.Then illustrations were made on the equivalence for the special case of the symmetric quantum Rabi model when =ò 0.In addition,some comparison were made from our work to previous work,including the connection and difference of result after different transformations.Finally we briefly discussed some related experiments from AQRM to P(?)schl-Teller potentials and possible developments in this area.