The good, the bad and the ugly coherent states through polynomial   Heisenberg algebras
  Second degree polynomial Heisenberg algebras are realized through the harmonic oscillator Hamiltonian, together with two deformed ladder operators chosen as the third powers of the standard annihilation and creation operators. The corresponding solutions to the Painleve IV equation are easily found. Moreover, three different sets of eigenstates of the deformed annihilation operator are constructed, called the good, the bad and the ugly coherent states. Some physical properties of such states will be as well studied. 
Miguel Castillo-Celeita and David J. Fernandez C
quant-ph hep-th math-ph math.MP