Periodic Toda lattice in quantum mechanics

The quantum mechanical periodic Toda lattice is studied by the direct diagonalization of the Hamiltonian. The eigenstates are classified according to the irreducible representations of the dihedral group DN. It is shown that Gutzwiller's quantization conditions are satisfied and they have a one-to-one correspondence to the irreducible representation of the DN group. The authors have also carried out the semiclassical quantization of the periodic Toda lattice by the EBK formulation. The eigenvalues of the semiclassical quantization have a one-to-one correspondence to the integer quantum numbers, and those quantum numbers also have a close relationship to the symmetry of the state. Numerical calculations have been done for N = 3, 4, 5, and 6 particle periodic Toda lattices. The distributions of the eigenvalues are systematic and distinguished by the symmetry of the state. As illustrative examples, amplitudes of the wave functions and density distributions are shown. 14 refs., 8 figs., 11 tabs