Exact solution of the two-axis countertwisting Hamiltonian for the half-integer J case

Bethe ansatz solutions of the two-axis countertwisting Hamiltonian for any (integer and half-integer) J are derived based on the Jordan-Schwinger (differential) boson realization of the SU(2) algebra after desired Euler rotations, where J is the total angular momentum quantum number of the system. It is shown that solutions to the Bethe ansatz equations can be obtained as zeros of the extended Heine-Stieltjes polynomials. Two sets of solutions, with solution number being J+1 and J respectively when J is an integer and J+1/2 each when J is a half-integer, are obtained. Properties of the zeros of the related extended Heine-Stieltjes polynomials for half-integer J cases are discussed. It is clearly shown that double degenerate level energies for half-integer J are symmetric with respect to the E=0 axis. It is also shown that the excitation energies of the `yrast' and other `yrare' bands can all be asymptotically given by quadratic functions of J, especially when J is large.