Bound states emerging from below the continuum in a solvable<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">PT</mml:mi></mml:math>-symmetric discrete Schrödinger equation

Znojil, Miloslav

doi:10.1103/physreva.96.012127

Cited by 7 publications

(3 citation statements)

References 85 publications

(110 reference statements)

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“…One speaks of broken PT symmetry, and the EP marks the position of the PT symmetry breaking. Since the occurrence of exceptional points is a generic feature of the PT phase transition a large number of works exists for PT -symmetric quantum mechanics [27,[34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], quantum field theories [50,51], electromagnetic waves [52][53][54][55][56][57][58], and electronic devices [59].…”

Section: Introductionmentioning

confidence: 99%

Simple Models of Three Coupled Pt -Symmetric Wave Guides Allowing for Third-Order Exceptional Points

Schnabel¹,

Cartarius²,

Main³

et al. 2017

Acta Polytech

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Abstract.We study theoretical models of three coupled wave guides with a PT -symmetric distribution of gain and loss. A realistic matrix model is developed in terms of a three-mode expansion. By comparing with a previously postulated matrix model it is shown how parameter ranges with good prospects of finding a third-order exceptional point (EP3) in an experimentally feasible arrangement of semiconductors can be determined. In addition it is demonstrated that continuous distributions of exceptional points, which render the discovery of the EP3 difficult, are not only a feature of extended wave guides but appear also in an idealised model of infinitely thin guides shaped by delta functions.

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Section: Introductionmentioning

confidence: 99%

Simple Models of Three Coupled Pt -Symmetric Wave Guides Allowing for Third-Order Exceptional Points

Schnabel¹,

Cartarius²,

Main³

et al. 2017

Acta Polytech

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“…In parallel, one can argue that the sparse-matrix structure of this component of the Hamiltonian makes it very natural to replace also the strictly local (i.e., diagonal-matrix) interaction v( 푗 ) by its weakly nonlocal tridiagonal-matrix generalization [32].…”

Section: Discrete Coordinatesmentioning

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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“…In between the two (viz., real-x and complex-x) extremes, one can find the models in which the coordinates x are discrete (cf., e.g., [32], with further references). This might make the contact with experiments in optics even closer.…”

Section: The Turn Of Attention To Classical Opticsmentioning

confidence: 99%

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

2017

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Schrödinger equations with non-Hermitian, but P T -symmetric quantum potentials V(x) found, recently, a new field of applicability in classical optics. The potential acquired there a new physical role of an "anomalous" refraction index. This turned attention to the nonlinear Schrödinger equations in which the interaction term becomes state-dependent, V(x) → W(ψ(x), x). Here, the state-dependence in W(ψ(x), x) is assumed logarithmic, and some of the necessary mathematical assumptions, as well as some of the potential phenomenological consequences of this choice are described. Firstly, an elementary single-channel version of the nonlinear logarithmic model is outlined in which the complex self-interaction W(ψ(x), x) is regularized via a deformation of the real line of x into a self-consistently constructed complex contour C. The new role played by P T -symmetry is revealed. Secondly, the regularization is sought for a multiplet of equations, coupled via the same nonlinear self-interaction coupling of channels. The resulting mathematical structures are shown to extend the existing range of physics covered by the logarithmic Schrödinger equations.

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Bound states emerging from below the continuum in a solvablePT-symmetric discrete Schrödinger equation

Cited by 7 publications

References 85 publications

Simple Models of Three Coupled Pt -Symmetric Wave Guides Allowing for Third-Order Exceptional Points

Simple Models of Three Coupled Pt -Symmetric Wave Guides Allowing for Third-Order Exceptional Points

Hermitian–Non-Hermitian Interfaces in Quantum Theory

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

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