Renormalization Group Approximate Solution For A Kind Of(N+1)-Level Systems With Near Resonance Control

The open-loop control problem of(N+1)-level system has a wide range of applications in quantum physics.How to solve the long time approximate solution of this kind of system is a hot issue for scientists.The CGO renormalization group method developed by Chen,Goldenfeld and Oono is an effective tool for studying the long time approximate solution of the system.On this basis,Geng and Zu proposed a higher-order renormalization group method for the near resonance control problem of(N+1)-level system,and obtained an analytical long time approximate solution.They expanded the open-loop control system according to small parameters,solved the renormalization group equation based on the resonance part,and then combined the non-resonance perturbation part to obtain the long-term approximate solution of the openloop control system.Different from the traditional CGO renormalization group method,the expansion coefficient in the work of Geng and Zu contains small parameters,and the scope of application is broader.The strict error estimation between the approximate solution and the exact solution ensures the validity of the approximate solution.However,Geng and Zu need to assume additional non-degenerate conditions in the article.In this thesis,we overcome the limitation of non-degenerate conditions for several special cases.First,without the restriction of non-degenerate conditions,the work of Geng and Zu is extended to the resonance control case.Secondly,considering the near resonance control problem of(N+1)-level system,this thesis develops the renormalization group method,which overcomes the nondegenerate condition for a special case,and obtains the analytical long time approximate solution.Finally,this thesis proposes a renormalization group method for the near resonance control problem of slowly varying systems,discusses the relationship between the slowly varying parameters and the near resonance parameters,and provides an effective renormalization group approximation solution.