A generalized non-Hermitian Pais-Uhlenbeck quantum Hamiltonian, its Hermitian equivalence and position-dependent mass correspondence

Abstract: We explore the Jacobi Last Multiplier as a means for deriving the Lagrangian of a fourth-order differential equation. In particular we consider the classical problem of the Pais-Uhlenbeck oscillator and write down the accompanying Hamiltonian. We then compare such an expression with an alternative derivation of the Hamiltonian that makes use of the Ostrogradski's method and show that a mapping from the one to the other is achievable by variable transformations. Assuming canonical quantization procedure to be valid we go for the operator version of the Hamiltonian that represents a pair of uncoupled oscillators. This motivates us to propose a generalized Pais-Uhlenbeck Hamiltonian in terms of the usual harmonic oscillator creation and annihilation operators by including terms quadratic and linear in them. Such a Hamiltonian turns out to be essentially non-Hermitian but has an equivalent Hermitian representation which is reducible to a typically position-dependent reduced mass form.