The integrability of quantum systems is charactized by the algebraic notion of dynamical symmetry,hence the nonintegrability of quantum systems should be discussed topologically in reference to corresponding integrabl...The integrability of quantum systems is charactized by the algebraic notion of dynamical symmetry,hence the nonintegrability of quantum systems should be discussed topologically in reference to corresponding integrable ones.A topological approach is proposed.The topological map in general is shown to be constituted of the piecewise diffeomorphic maps interrupted by sudden nondiffeomorphic maps and thus has the similar typical feature of stretching-folding stretching......as the Smale horse-shoe map.展开更多
The basic problem of quantum integrability has been systematically studied with an algebraic approach.With the introduction of the dynamical symmetry algebra the difficulty in transferring the classical integrability ...The basic problem of quantum integrability has been systematically studied with an algebraic approach.With the introduction of the dynamical symmetry algebra the difficulty in transferring the classical integrability condition into quantum mechanics is resolved.Using the superconvergent technique of Kolmogorov,it has been proved that systems with Hamiltonians expressed merely with elements of a definite dynamical symmetry algebra form an integrable class of systems corresponding to that dynamical symmetry.展开更多
A topological approach to the study of global stationary property of quantum systems is carefully formulated.Particular attentions are paid to basic notions such as homeomorphic map,topological-structure-preserving(TS...A topological approach to the study of global stationary property of quantum systems is carefully formulated.Particular attentions are paid to basic notions such as homeomorphic map,topological-structure-preserving(TSP)map,and bifurcation due to local destruction of the TSP condition.展开更多
In order to have a complete description of the spatio-temporal evolution of a quantum state,the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry.Then we h...In order to have a complete description of the spatio-temporal evolution of a quantum state,the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry.Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry.In the former case,we have regular motion with the general character which tends to that of the corresponding classical case in the limit n→0.While in the latter case,we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.展开更多
基金Supported by the National Basic Research Project“Nonlinear Science”of Chinathe National Natural Science Foundation of China under Grant No.19675019Natural Science Foundation of Fujian Province of China.
文摘The integrability of quantum systems is charactized by the algebraic notion of dynamical symmetry,hence the nonintegrability of quantum systems should be discussed topologically in reference to corresponding integrable ones.A topological approach is proposed.The topological map in general is shown to be constituted of the piecewise diffeomorphic maps interrupted by sudden nondiffeomorphic maps and thus has the similar typical feature of stretching-folding stretching......as the Smale horse-shoe map.
基金Supported by the National Basic Research Project“Nonlinear Science”of Chinathe National Natural Science Foundation of China under Grant No.19675019the Natural Science Foundation of Fujian Province of China.
文摘The basic problem of quantum integrability has been systematically studied with an algebraic approach.With the introduction of the dynamical symmetry algebra the difficulty in transferring the classical integrability condition into quantum mechanics is resolved.Using the superconvergent technique of Kolmogorov,it has been proved that systems with Hamiltonians expressed merely with elements of a definite dynamical symmetry algebra form an integrable class of systems corresponding to that dynamical symmetry.
基金Supported by the National Basic Research Project“Nonlinear Science”of Chinathe National Natural Science Foundation of China under Grant No.19675019the Natural Science Foundation of Fujian Province of China.
文摘A topological approach to the study of global stationary property of quantum systems is carefully formulated.Particular attentions are paid to basic notions such as homeomorphic map,topological-structure-preserving(TSP)map,and bifurcation due to local destruction of the TSP condition.
基金Supported by the National Basic Research Project"Nonlinear Science"of Chinathe National Natural Science Foundation of China under Grant No.19675019the Natural Science Foundation of Fujian Province of China.
文摘In order to have a complete description of the spatio-temporal evolution of a quantum state,the initial state should be properly specified as the coherent state corresponding to a definite dynamical symmetry.Then we have qualitatively different situations where the initial coherent state evolves under the action of a system having or not having the same dynamical symmetry.In the former case,we have regular motion with the general character which tends to that of the corresponding classical case in the limit n→0.While in the latter case,we may have energy regions of global stochasticity with characters closely related to the initial dynamical symmetry strongly violated.
