| The idea that the space-time coordinates do not commutate is old. But noncommutative geometry cannot attract the physicist's interest for a long time. In the recent years, more and more physical problems in the noncom-mutative background are regarded as important by people with the study of the string and quantum Hall Effect. Since the relation between the stringtheory and noncommutative theory was revealled, the study of the solitonsolution in noncommutative field has attracted the theoretical physicist's at-tention. Soliton solutions in noncommutative field and the string theory often provide important clue for the behavior of nonperturbative and strong couple of the string theory. So the study of the soliton solutions in all kinds of space is very important.In this paper,we obtain the quantum states and soliton solutions on noncommutative orbifold R2n/G. The method is obtaining the ground state of the Hamiltonian which is constructed by general quadratic form of generalized coordinates and momenta. Consequently we can obtain eigenstates of any energy level. Invariant quadratic form with respect to the rotation subgroup G of SO(2n) in noncommutative space R2ncan be regularized as this kind of Hamiltonian. so we get the invariant ground state under rotation G. If G is a finite subgroup of SO(2n), we will get the eigenvectors on noncommutative Orbifold R2n/G. Based on this, we may obtain the soliton solution on it.In addition, there exists the high dimensional oscillator in physics, In its Hamiltonian coordinates and momenta are coupled. For instance, quantumHall system does so. We provide the general method of diagonalizationing this kind of Hamiltonian as standard form. |
