# Exactly solvable PT-symmetric Hamiltonian having no Hermitian counterpart - High Energy Physics - Theory

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Abstract: In a recent paper Bender and Mannheim showed that the unequal-frequencyfourth-order derivative Pais-Uhlenbeck oscillator model has a realization inwhich the energy eigenvalues are real and bounded below, the Hilbert-spaceinner product is positive definite, and time evolution is unitary. Central tothat analysis was the recognition that the Hamiltonian $H { m PU}$ of themodel is PT symmetric. This Hamiltonian was mapped to a conventionalDirac-Hermitian Hamiltonian via a similarity transformation whose form wasfound exactly. The present paper explores the equal-frequency limit of the samemodel. It is shown that in this limit the similarity transform that was usedfor the unequal-frequency case becomes singular and that $H { m PU}$ becomes aJordan-block operator, which is nondiagonalizable and has fewer energyeigenstates than eigenvalues. Such a Hamiltonian has no Hermitian counterpart.Thus, the equal-frequency PT theory emerges as a distinct realization ofquantum mechanics. The quantum mechanics associated with this Jordan-blockHamiltonian can be treated exactly. It is shown that the Hilbert space iscomplete with a set of nonstationary solutions to the Schr\-odinger equationreplacing the missing stationary ones. These nonstationary states are needed toestablish that the Jordan-block Hamiltonian of the equal-frequencyPais-Uhlenbeck model generates unitary time evolution.

Author: ** Carl M. Bender, Philip D. Mannheim**

Source: https://arxiv.org/