Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart

TL;DR

This paper investigates the equal-frequency limit of a PT-symmetric Pais-Uhlenbeck oscillator, revealing a non-Hermitian Jordan-block Hamiltonian with no Hermitian counterpart, and demonstrates its exact solvability and unitary evolution.

Contribution

It shows that the equal-frequency limit leads to a singular similarity transform, resulting in a Jordan-block Hamiltonian that is exactly solvable and distinct from Hermitian models.

Findings

The Hamiltonian becomes a Jordan-block operator in the equal-frequency limit.

The model's Hilbert space is complete with nonstationary solutions.

Unitary time evolution is maintained despite the non-Hermitian nature.

Abstract

In a recent paper Bender and Mannheim showed that the unequal-frequency fourth-order derivative Pais-Uhlenbeck oscillator model has a realization in which the energy eigenvalues are real and bounded below, the Hilbert-space inner product is positive definite, and time evolution is unitary. Central to that analysis was the recognition that the Hamiltonian $H_{\rm PU}$ of the model is PT symmetric. This Hamiltonian was mapped to a conventional Dirac-Hermitian Hamiltonian via a similarity transformation whose form was found exactly. The present paper explores the equal-frequency limit of the same model. It is shown that in this limit the similarity transform that was used for the unequal-frequency case becomes singular and that $H_{\rm PU}$ becomes a Jordan-block operator, which is nondiagonalizable and has fewer energy eigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart. Thus, the equal-frequency PT theory emerges as a distinct realization of quantum mechanics. The quantum mechanics associated with this Jordan-block Hamiltonian can be treated exactly. It is shown that the Hilbert space is complete with a set of nonstationary solutions to the Schr\"odinger equation replacing the missing stationary ones. These nonstationary states are needed to establish that the Jordan-block Hamiltonian of the equal-frequency Pais-Uhlenbeck model generates unitary time evolution.