Algebraic treatment of the Pais-Uhlenbeck oscillator and its PT-variant

TL;DR

This paper explores the algebraic approach to analyzing the spectrum of quadratic Hamiltonians, exemplified through the Pais-Uhlenbeck oscillator and its PT-variant, revealing properties like exceptional points and Jordan form.

Contribution

It applies the algebraic method to the Pais-Uhlenbeck oscillator, demonstrating how to analyze its spectral properties and exceptional points in a systematic way.

Findings

Identification of exceptional points where the matrix becomes defective

Representation of the Hamiltonian in Jordan canonical form

Analysis of the fourth-order differential equation in quantum variables

Abstract

The algebraic method enables one to study the properties of the spectrum of a quadratic Hamiltonian through the mathematical properties of a matrix representation called regular or adjoint. This matrix exhibits exceptional points where it becomes defective and can be written in canonical Jordan form. It is shown that any quadratic function of $K$ coordinates and $K$ momenta leads to a $2K$ differential equation for those dynamical variables. We illustrate all these features of the algebraic method by means of the Pais-Uhlenbeck oscillator and its PT-variant. We also consider a trivial quantization of the fourth-order differential equation for the quantum-mechanical dynamical variables.