  We discuss the quantum dynamics of the Pais-Uhlenbeck oscillator. The
Lagrangian of this higher-derivative model depends on two frequencies. When the
frequencies are different, the free PU oscillator has a pure point spectrum
that is dense everywhere. When the frequencies are equal, the spectrum is
continuous. It is not bounded from below, running from minus to plus infinity,
but this is not disastrous as the Hamiltonian is still Hermitian and the
evolution operator is still 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 when the
frequencies are different, but unitarity is broken in the equal frequencies
limit.
