Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart

Journal Article · Tue Jul 15 04:00:00 UTC 2008 · Physical Review. D, Particles Fields

DOI:https://doi.org/10.1103/PHYSREVD.78.025022· OSTI ID:21250417

Bender, Carl M ^[1]; Mannheim, Philip D ^[2]

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1. Physics Department, Washington University, St. Louis, Missouri 63130 (United States)
2. Department of Physics, University of Connecticut, Storrs, Connecticut 06269 (United States)

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{sub 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{sub 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 Schroedinger 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.

• OSTI ID: 21250417
• Journal Information: Physical Review. D, Particles Fields, Journal Name: Physical Review. D, Particles Fields Journal Issue: 2 Vol. 78; ISSN PRVDAQ; ISSN 0556-2821
• Country of Publication: United States
• Language: English

Related Subjects

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
EIGENSTATES
EIGENVALUES
EVOLUTION
EXACT SOLUTIONS
HAMILTONIANS
HILBERT SPACE
OSCILLATORS
QUANTUM MECHANICS
SCHROEDINGER EQUATION
SIMULATION
TRANSFORMATIONS