SPONTANEOUS BREAKDOWN OF $\mathcal{PT}$ SYMMETRY IN THE SOLVABLE SQUARE-WELL MODEL

“…The PT -symmetric square-well model provides a simple example for this behavior [4]. It describes a particle moving between reflecting boundaries at x = ±1, in the presence of a piecewise constant complex potential,…”

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confidence: 99%

“…Above the threshold, Z > Z c 0 , at least one pair of complex conjugate eigenvalues E 0 and E * 0 develops. One of the corresponding eigenstates has the form [4] …”

mentioning

confidence: 99%

-symmetry and its spontaneous breakdown explained by anti-linearity

Weigert

2003

J. Opt. B: Quantum Semiclass. Opt.

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The impact of an anti-unitary symmetry on the spectrum of non-hermitean operators is studied. Wigner's normal form of an anti-unitary operator is shown to account for the spectral properties of non-hermitean, PT -symmetric Hamiltonians. Both the occurrence of 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.Deep in their hearts, many quantum physicists will renounce hermiticity of operators only reluctantly. However, non-hermitean Hamiltonians are applied successfully in nuclear physics, biology and condensed matter, often modelling the interaction of a quantum system with its environment in a phenomenological way. Since 1998, nonhermitean Hamiltonians continue to attract interest from a conceptual point of view [1]: surprisingly, the eigenvalues of a one-dimensional harmonic oscillator Hamiltonian remain real when the complex potentialV = ix 3 is added to it. Numerical, semiclassical, and analytic evidence [2] has been accumulated confirming that bound states with real eigenvalues exist for the vast class of complex potentials satisfying V † (x) = V (−x). In addition, pairs of complex conjugate eigenvalues occur systematically.PT -symmetry has been put forward to explain the observed energy spectra. The Hamiltonian operatorsĤ under scrutiny are invariant under the combined action of parity P and time reversal T , [Ĥ, PT ] = 0 .

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“…Certainly, the latter family is not small. Pars pro toto it contains Hamiltonians of relativistic quantum mechanics [41,42], the well-known P -symmetric imaginary cubic oscillator [43][44][45][46] (which appears, after a more detailed scrutiny, strongly nonlocal [31,47]), its power-law generalizations [10][11][12]48] as well as exactly solvable models [49][50][51][52], models with methodical relevance in the context of supersymmetry [53,54], realistic and computation-friendly interacting-boson models of heavy nuclei [2], benchmark candidates for classification of quantum catastrophes [55][56][57], and so forth.…”

Section: A2 Physical Inner Productsmentioning

confidence: 99%

Hermitian–Non-Hermitian Interfaces in Quantum Theory

Znojil

2018

Advances in High Energy Physics

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In the global framework of quantum theory, the individual quantum systems seem clearly separated into two families with the respective manifestly Hermitian and hiddenly Hermitian operators of their Hamiltonian. In the light of certain preliminary studies, these two families seem to have an empty overlap. In this paper, we will show that whenever the interaction potentials are chosen to be weakly nonlocal, the separation of the two families may disappear. The overlaps alias interfaces between the Hermitian and non-Hermitian descriptions of a unitarily evolving quantum system in question may become nonempty. This assertion will be illustrated via a few analytically solvable elementary models.

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SPONTANEOUS BREAKDOWN OF $\mathcal{PT}$ SYMMETRY IN THE SOLVABLE SQUARE-WELL MODEL

Abstract: Apparently, the energy levels merge and disappear in many PT symmetric models. This interpretation is incorrect: In square-well model we demonstrate how the doublets of states in question continue to exist at complex conjugate energies in the strongly non-Hermitian regime.

Cited by 70 publications

References 36 publications

An analysis of the zero energy states in graphene

An analysis of the zero energy states in graphene

-symmetry and its spontaneous breakdown explained by anti-linearity

Hermitian–Non-Hermitian Interfaces in Quantum Theory

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