Comments on the Dynamics of the Pais–Uhlenbeck Oscillator

We discuss the quantum dynamics of the PU oscillator, i.e. the system with the Lagrangian
\begin{gather} \label{1} L=\frac12\left[\ddot q^2-(\Omega_1^2+\Omega_2^2)\dot q^2+\Omega_1^2\Omega_2^2 q\right] \quad(+\text{ nonlinear terms}). \end{gather}
When $\Omega_1\neq\Omega_2$, the free PU oscillator has a pure point spectrum that is dense everywhere. When $\Omega_1=\Omega_2$, the spectrum is continuous, $E\in\{-\infty,\infty\}$. The spectrum is not bounded from below, but that is not disastrous as the Hamiltonian is Hermitian and the evolution operator is unitary. Generically, the inclusion of interaction terms breaks unitarity, but in some special cases unitarity is preserved. We discuss also the nonstandard realization of the PU oscillator suggested by Bender and Mannheim, where the spectrum of the free Hamiltonian is positive definite, but wave functions grow exponentially for large real values of canonical coordinates. The free nonstandard PU oscillator is unitary at $\Omega_1\neq\Omega_2$, but unitarity is broken in the equal frequencies limit.