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Journal of Physics A Mathematical and Theoretical

{\cal P}{\cal T} symmetry and its spontaneous breakdown in three dimensions

GÉZA LÉVAI

Published Jun 3, 2008• 14 citations
Publisher

Abstract

The mechanism of introducing non-Hermiticity to non-central -symmetric potentials through both the azimuth and θ polar angles is discussed. Generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials, such as the Scarf I or Rosen–Morse I potentials are used in the eigenvalue equation of the polar component. It is shown that the spontaneous breakdown of symmetry can also occur in these non-central potentials. Several potentials are proposed in the azimuthal eigenvalue equation too, where the use of periodic boundary conditions is essential. Possible generalizations of the results are outlined.

DOI
10.1088/1751-8113/41/24/244015

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The mechanism of introducing non-Hermiticity to non-central -symmetric potentials through both the azimuth and θ polar angles is discussed. Generalizing the results of a previous work it is shown that this can be done also through the polar angle part if appropriate potentials, such as the Scarf I or Rosen–Morse I potentials are used in the eigenvalue equation of the polar component. It is shown that the spontaneous breakdown of symmetry can also occur in these non-central potentials. Several potentials are proposed in the azimuthal eigenvalue equation too, where the use of periodic boundary conditions is essential. Possible generalizations of the results are outlined.
The solution of non-central PT -symmetric potentials is discussed by the separation of the variables in polar and angular coordinates. Conditions are formulated to guarantee the separation of the variables and the PT symmetry of the potential. The original eigenvalue equation is separated into one-dimensional Schrödinger-type differential equations. The importance of the boundary conditions, especially that of the periodic boundary condition of the azimuthal equation is pointed out. Further conditions leading to exact solutions of the whole problem are also formulated. An example combining the harmonic oscillator and the Scarf I potential in the radial and polar equation is discussed in detail, and the bound-state wave functions and the energy eigenvalues are derived. The spectrum exhibits partial degeneracies similar to those observed in the spectrum of the isotropic harmonic oscillator.
We present a general procedure by which solvable non-central potentials can be obtained in 2 and 3 dimensions by the separation of the angular and radial variables. The method is applied to generate solvable non-central PT-symmetric potentials in polar coordinates. General considerations are presented concerning the PT transformation properties of the eigenfunctions, their pseudo-norm and the nature of the energy eigenvalues. It is shown that within the present framework the spontaneous breakdown of PT symmetry can be implemented only in two dimensions.
-symmetric, non-relativistic quantum mechanical potentials are discussed in two and three spatial dimensions. Conditions are formulated under which these potentials are -symmetric and can be solved exactly by the separation of the radial and angular variables. It is found that the angular variables play an essential role in introducing non-Hermiticity via the imaginary potential terms. A simple partially exactly solvable potential is used to demonstrate various aspects of symmetry in both two and three dimensions. Possible generalizations of the results are outlined.
Bound and scattering solutions of the -symmetric Rosen–Morse II potential are investigated. The energy eigenvalues and the corresponding wavefunctions are written in a closed analytic form, and it is shown that this potential always supports at least one bound state. It is found that with increasing non-Hermiticity the real bound-state energy spectrum does not turn into complex conjugate pairs, i.e. the spontaneous breakdown of symmetry does not occur, rather the energy eigenvalues remain real and shift to positive values. Closed expression is found for the pseudo-norm of the bound states, and its sign is found to follow the (−1)n rule. Similarly to the known scattering examples, the reflection coefficients exhibit a handedness effect, while the transmission coefficient picks up a complex phase factor when the direction of the incoming wave is reversed. It is argued that the unusual findings might be caused by the asymptotically non-vanishing, though finite imaginary potential component. Comparison with the real Rosen–Morse II potential is also made.
It is generally believed that Parity-Time (PT)-symmetry breaking occurs when eigenvalues or both eigenvalues and eigenvectors coincide. However, we show that this well-accepted picture of PT-symmetry breaking is incorrect. Instead, we demonstrate that the physical mechanism of PT-symmetry breaking is the resonance between positive- and negative-action modes. It is proved that PT-symmetry breaking occurs when and only when this resonance condition is satisfied, and this mechanism applies to all known PT-symmetry breakings observed in different branches of physics. The result is achieved by proving a remarkable fact that in finite dimensions, a PT-symmetric Hamiltonian is necessarily pseudo-Hermitian, regardless whether it is diagonalizable or not.
The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner’s normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, 𝒫𝒯-symmetric Hamiltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one-or two-dimensional irreducible representations of the symmetry 𝒫𝒯. In this framework, the concept of a spontaneously broken 𝒫𝒯-symmetry is not needed.