Painlevé IV coherent states

• We use SUSY QM to obtain Hamiltonians with third-order differential ladder operators.
• We show that these systems are related with the Painlevé IV equation.
• We apply different definitions of coherent states to these Hamiltonians using the third-order ladder operators and some linearized ones.
• We construct families of coherent states for such systems, which we called Painlevé IV coherent states.

A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the related third-order ladder operators, and finally as extremal states which are also displaced but now using the so called linearized ladder operators. To each SUSY partner Hamiltonian corresponds two families of coherent states: one inside the infinite subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states created through the SUSY technique.