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arXiv:0710.5814

A class of spherically-separable non-Hermitian PphiTphi-symmetric Hamiltonians

Omar Mustafa, S. Habib Mazharimousavi

Published Oct 31, 2007• 0 citations
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Abstract

A family of spherical non-Hermitian potentials is studied. It is shown that the corresponding non-Hermitian Hamiltonians admit some "new" P$phi$T$phi$-symmetry. It is observed that whilst such P$phi$T$phi$-symmetric Hamiltonians just copy the eigenvalues of V(r), the corresponding wavefunctions would rather indulge some new probabilistic interpretations due to the change in the azimuthal part.

DOI
10.48550/arxiv.0710.5814

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A family of spherical non-Hermitian potentials is studied. It is shown that the corresponding non-Hermitian Hamiltonians admit some "new" PphiphiphiTphiphiphi-symmetry. It is observed that whilst such PphiphiphiTphiphiphi-symmetric Hamiltonians just copy the eigenvalues of V(r), the corresponding wavefunctions would rather indulge some new probabilistic interpretations due to the change in the azimuthal part.
Large families of Hamiltonians that are non-Hermitian in the conventional sense have been found to have all eigenvalues real, a fact attributed to an unbroken symmetry. The corresponding quantum theories possess an unconventional scalar product. The eigenvalues are determined by differential equations with boundary conditions imposed in wedges in the complex plane. For a special class of such systems, it is possible to impose the -symmetric boundary conditions on the real axis, which lies on the edges of the wedges. The -symmetric spectrum can then be obtained by imposing the more transparent requirement that the potential be reflectionless.
The spectrum of non-Hermitian Hamiltonians is studied using the point-group symmetries of the Hermitian and non-Hermitian parts of the Hamiltonian. We show that, in principle, the symmetry properties of the Hamiltonian are responsible for the appearance of real eigenvalues in the spectrum of the non-Hermitian Hamiltonian. In practice, however, one must take into account the ``strength'' of the non-Hermitian part of the Hamiltonian. An effective energy-dependent Hermitian Hamiltonian is constructed that has the same real spectrum as the non-Hermitian Hamiltonian. Several properties, including the self-orthogonality of the states at branch points of the spectrum, are proven using this effective representation. An interesting possibility of complex eigenvalues returning to the real axis is exemplified along with several examples of non-Hermitian Hamiltonians with real eigenvalues.
The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of PT\mathrm{PT}PT symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These PT\mathrm{PT}PT symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.
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LINKED PT{\mathcal{PT}}PT-SYMMETRY TO SUPERSYMMETRY IN A CLASS OF NON-HERMITIAN HAMILTONIANS
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A quantum-mechanical theory is -symmetric if it is described by a Hamiltonian that commutes with , where represents space reflection and represents time reversal. Often, but not always, a -symmetric Hamiltonian is non-Hermitian, and thus its eigenvalues may not be all real. However, -symmetric Hamiltonians are interesting because they usually have a parametric region of unbroken symmetry in which the eigenvalues are indeed all real and a region of broken symmetry in which the some of the eigenvalues are complex. These regions are separated by a phase transition that has been repeatedly observed in laboratory experiments. This paper focuses on the properties of a particular -symmetric Hamiltonian, H = p 2 − x 4 , which has an upside-down potential. The spectrum of this -symmetric Hamiltonian is rigorously known to be entirely real, and the purpose of this paper is to present an intuitive explanation of why the spectrum of this Hamiltonian is real, positive, and discrete.