𝒫𝒯‐Symmetric Operators in Quantum Mechanics: Krein Spaces Methods

“…One of the best known recent samples of such a transfer starts in quantum field theory [33] and ends up in classical electrodynamics [34]. A common mathematical background consists in the requirements of the Krein-space self-adjointness [35] alias parity-times-time-reversal symmetry (PT −symmetry).…”

Section: The Context Of Classical Optical Systems With Gain and Lossmentioning

confidence: 99%

Markov constant and quantum instabilities

Pelantová

Starosta

Znojil

2016

J. Phys. A: Math. Theor.

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For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.

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“…In a historical perspective, the recent enormous growth of interest in the experimental as well as theoretical aspects of optical systems exhibiting various anomalous (typically, complex and space-dependent) forms of refraction index n( x) / ∈ R (mostly in two dimensions, with x ∈ R 2 ) may be traced back to the influential theoretical letter [1] in which Carl Bender with his student Stefan Boettcher recalled the traditional parity-time symmetry alias P T symmetry and transferred its use from relativistic quantum field physics (e.g., [2]) to the Krein-space-related spectral-theory mathematics [3,4]. In this manner Bender with Boettcher opened the Pandora's box of possible applications of the concept of P T symmetry…”

Section: Introductionmentioning

confidence: 99%

Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points

Znojil¹

2016

Symmetry

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For a given operator D(t) of an observable in theoretical parity-time symmetric quantum physics (or for its evolution-generator analogues in the experimental gain-loss classical optics, etc.) the instant t critical of a spontaneous breakdown of the parity-time alias gain-loss symmetry should be given, in the rigorous language of mathematics, the Kato's name of an "exceptional point", t critical = t (EP) . In the majority of conventional applications the exceptional point (EP) values are not real. In our paper, we pay attention to several exactly tractable toy-model evolutions for which at least some of the values of t (EP) become real. These values are interpreted as "instants of a catastrophe", be it classical or quantum. In the classical optical setting the discrete nature of our toy models might make them amenable to simulations. In the latter context the instant of Big Bang is mentioned as an illustrative sample of possible physical meaning of such an EP catastrophe in quantum cosmology.

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𝒫𝒯‐Symmetric Operators in Quantum Mechanics: Krein Spaces Methods

Cited by 12 publications

References 54 publications

On Dual Definite Subspaces in Krein Space

On Dual Definite Subspaces in Krein Space

Markov constant and quantum instabilities

Parity-Time Symmetry and the Toy Models of Gain-Loss Dynamics near the Real Kato’s Exceptional Points

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