Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart

Carl M. Bender and Philip D. Mannheim
Phys. Rev. D 78, 025022 – Published 16 July 2008

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 HPU 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 HPU 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 Schrödinger 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.

  • Received 25 April 2008

DOI:https://doi.org/10.1103/PhysRevD.78.025022

©2008 American Physical Society

Authors & Affiliations

Carl M. Bender1,* and Philip D. Mannheim2,†

  • 1Physics Department, Washington University, St. Louis, Missouri 63130, USA
  • 2Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA

  • *cmb@wustl.edu
  • philip.mannheim@uconn.edu

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Issue

Vol. 78, Iss. 2 — 15 July 2008

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