The eigenvalue problem is solved exactly for the excited states of the Calogero-Sutherland-Moser model. The Young diagrams play an essential role in deriving solutions. Each eigenfunction, which is an irreducible representation of the λ-deformed symmetry group, is characterized by the corresponding Young diagram.
The U(1) Calogero Sutherland Model with anti-periodic boundary condition is studied. The Hamiltonian is reduced to a convenient form by similarity transformation. The matrix representation of the Hamiltonian acting on a partially ordered state space is obtained in an upper triangular form. Consequently the diagonal elements become the energy eigenvalues. The eigenstates are constructed using Young diagram and represented in terms of Jack symmetric polynomials. The eigenstates so obtained are orthonormalized.
Using the collective field technique, we give the description of the spin Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be applicable for arbitrary coupling constant and provides the bosonized Hamiltonian of the spin CSM. The boson Fock space can be identified with the Hilbert space of the spin CSM in the large N limit. We show that the eigenstates corresponding to the Young diagram with a single row or column are represented by the vertex operators. We also derive a dual description of the Hamiltonian and comment on the construction of the general eigenstates. JSPS fellow
A collective field method is extended to obtain all the explicit solutions of the generalized Calogero-Sutherland models that are characterized by the roots of all the classical groups, including the solutions corresponding to spinor representations for BN and DN cases.