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arXiv · 1702.03811

Behavior of eigenvalues in a region of broken-PT symmetry

Carl M. Bender·Nima Hassanpour·Daniel W. Hook·S. P. Klevansky·Christoph Sünderhauf·Zichao Wen
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Abstract

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. When $\varepsilon\geq0$, the eigenvalues of this non-Hermitian Hamiltonian are discrete, real, and positive. This portion of parameter space is known as the region of unbroken PT symmetry. In the region of broken PT symmetry $\varepsilon<0$ only a finite number of eigenvalues are real and the remaining eigenvalues appear as complex-conjugate pairs. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry has thus far been unexplored. This paper presents a detailed numerical and analytical examination of the behavior of the eigenvalues for $-4<\varepsilon<0$. In particular, it reports the discovery of an infinite-order exceptional point at $\varepsilon=-1$, a transition from a discrete spectrum to a partially continuous spectrum at $\varepsilon=-2$, a transition at the Coulomb value $\varepsilon=-3$, and the behavior of the eigenvalues as $\varepsilon$ approaches the conformal limit $\varepsilon=-4$.

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Carl M. Bender, Nima Hassanpour, Daniel W. Hook, S. P. Klevansky, Christoph Sünderhauf, Zichao Wen. 2017-02-26. Behavior of eigenvalues in a region of broken-PT symmetry. https://doi.org/10.1103/physreva.95.052113

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