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Quantum Systems Obeying to Generalized Exclusion-Inclusion Principle

In this work is studied a many body system obeying to a generalized Exclusion-Inclusion Principle (EIP) originated by collective effect, the dynamics, in mean field approximation, being ruled by a nonlinear Schroedinger equation. The EIP is introduced by a judicious generalization of the particle current. By means of variational principle is obtained a canonical nonlinear Schroedinger equation. We study the Lagrangian and Hamiltonian formulation. We study the symmetries of the nonlinear Schroedinger equation obeying to EIP and by means of Noether theorem we obtain and discuss conserved quantities. Successively, EIP-Schroedinger equation is coupled in a minimal way to an abelian gauge field which dynamics is described by the Maxwell-Chern-Simons Lagrangian. We show that the anyonic statistic behavior ascribed to the system by the Chern-Simons Lagrangian is not destroyed by the presence of EIP potential. Finally, we study special solutions of the system. Applications on the Bose-Einstein condensation and vortex-like solution in the picture of Chern-Simons interaction are considered.