Study Of One-dimensional Integrable Many-body System On Ruijsenaars-Schneider Model

The study of low-dimensional classic and quantum integrable many-body systems continues to be the subject of intensive, high-profile research in the mathematical and physical sciences worldwide. Among these the classical and quantum models with long range interactions of Calogero-Moser (CM) type and their one-parameter deforma- tions. the Ruijsenaars-Schneider(RS) models, have attracted remarkable attention in recent years. In particular, after the renowned works of Ruijsenaars and Schneider in 1986-1981, there have been extensive progress and discussions related to this subject. Not only being a so-called "relativistic generalization" of the Calogero-Moser model, it also includes the Toda model which has nearest neighbor interaction(under certain limit). Despite, as integrable models, the CM and Toda models have their own partic- ular characters, the study of the generic RS model could provide a full and universal understanding of the classical and quantum aspects of finite-dimensional integrable many-body systems in general. This thesis comes in three parts. Part I (chapter I) is a brief introduction of the background for integrable system and newest developments about the CM and RS models. Part II (chapters II and III) mainly deals with the integrable structure of the A RS model. It is shown that the trigonometric s12 Ruijsenaars-Schneider model admits a nondynarnical r-matrix structure under a certain gauge. The corresponding r-matrix is the classical limit of a twisted trigonometric R-matrix. The new factorized classical and quantum L-operators are obtained. Similar to the result that the A1 RS model is related to the Z~ Sklyanin algebra, we find that the elliptic quantum CM Model can be depicted by the sl,., Gaudin algebra. In Part III(chapters JV,V and VI), we are devoted to generalizing the RS model associating with other root system than AN1 one. Using a subtle method of Dirac reduction, we construct the Lax pairs without spectral parameter for the C,., and BC,-, RS models with interaction potential of trigonometric and rational types. The involu- tive Hamiltonians are also given. Afterwards, we present similar results for the more generic elliptic models. It turns out that the spectral curves can be parameterized by the involutive integrals of motion for these models. The relation to the various degenerate limits: the trigonometric, hyperbolic and rational cases, is also remarked. In addition, the Lax pairs without spectral parameter for the trigonometric and ratio. nal types can be obtained as the spectral parameter evaluates at some special points. Furthermore, we attempt to extend these results to the D trigonometric RS system, where a eneraiformfortheLax pairispresented. Forsmalln, aproof is given and Liouville integrability for the corresponding system is revealed.