Mathematical and Physical Meaning of the Crossings of Energy Levels in $${\mathscr {PT}}$$ PT -Symmetric Systems

Борисов, Д. И.; Znojil, Miloslav

doi:10.1007/978-3-319-31356-6_13

Cited by 6 publications

(7 citation statements)

References 20 publications

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“…In more interesting models namely the scattering potential wells [14] and other [15], Dirichlet spectrum may itself have two branches (identified by quasi parity [20]) and by putting them together one observes coalescing of eigenvalues at the exceptional point(s) of the complex potential. In even more interesting models like Scarf II [21] and shifted harmonic oscillator [22], in addition to the coalescing one may observe crossing(s) of eigenvalues in one dimension! however, the corresponding eigenstates are linearly dependent [21] shunning degeneracy in one-dimension.…”

Section: Resultsmentioning

confidence: 99%

Dirichlet spectra of the paradigm model of complex PT-symmetric potential:V(x)=−(ix)N

Ahmed¹,

Kumar²,

Sharma³

2017

Annals of Physics

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So far the spectra E n (N ) of the paradigm model of complex PT(Parity-Time)-symmetric potential V BB (x, N ) = −(ix) N is known to be analytically continued for N > 4. Consequently, the well known eigenvalues of the Hermitian cases (N = 6, 10) cannot be recovered. Here, we illustrate Kato's theorem that even if a Hamiltonian H(λ) is an analytic function of a real parameter λ, its eigenvalues E n (λ) may not be analytic at finite number of Isolated Points (IPs). In this light, we present the Dirichlet spectra E n (N ) of V BB (x, N ) for 2 ≤ N < 12 using the numerical integration of Schrödinger equation with ψ(x = ±∞) = 0 and the diagonalization of H = p 2 /2µ + V BB (x, N ) in the harmonic oscillator basis. We show that these real discrete spectra are consistent with the most simple two-turning point CWKB (C refers to complex turning points) method provided we choose the maximal turning points (MxTP) [−a + ib, a + ib, a, b ∈ R] such that |a| is the largest for a given energy among all (multiple) turning points. We find that E n (N ) are continuous function of N but non-analytic (their first derivative is discontinuous) at IPs N = 4, 8; where the Dirichlet spectrum is null (as V BB becomes a Hermitian flat-top potential barrier). At N = 6 and 10, V BB (x, N ) becomes a Hermitian well and we recover its well known eigenvalues. * Electronic address: 1:zahmed@barc.gov.in, 2:sachinv@barc.gov.in, 3

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Section: Resultsmentioning

confidence: 99%

Dirichlet spectra of the paradigm model of complex PT-symmetric potential:V(x)=−(ix)N

Ahmed¹,

Kumar²,

Sharma³

2017

Annals of Physics

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“…Their occurrence splits the interval of into separate subintervals. The consequences for the quantum phenomenology are remarkable, for example, for the reasons which were discussed, recently, in [13][14][15][16].…”

Section: Ep Degeneracies At (Ep)mentioning

confidence: 98%

“…Their occurrence and -dependence are illustrated here in Figure 8. Naturally, for < 1 there emerge also the singularities at (EP) (second kind) = ±1 (see the dedicated references [13][14][15][16] for a more thorough explanation of this terminology).…”

Section: Advances In High Energy Physicsmentioning

confidence: 99%

“…In such a setting the stationarity of the theory represents a serious obstacle for experimentalists, mainly because the adiabatic changes and tuning of the parameter-dependence of the observables may lead to multiple counterintuitive nogo theorems [3,8,9]. At the same time, the new degree of the kinematical freedom represented by the nontrivial metric Θ ̸ = may find its efficient use, say, in the manipulations leading to the experimental realizations of various quantum phase transitions in the theory [6,[10][11][12][13][14][15][16] as well as in the laboratory [17].…”

Section: Introductionmentioning

confidence: 99%

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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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“…At the same time, paradoxically, what also survived, until now, was the present author's reply, counterargument and prediction saying, basically, that from the point of view of phenomenology the crucial differences between the textbook and P T −symmetric quantum theories will emerge in the dynamical regime in which the P T −symmetry gets lost during a new form of a phase transition (cf., e.g., [28]). In the context of abstract quantum theory we already mentioned above, that and how one of the productive confirmations of the latter prediction emerged, later on, in the context of the quantum theory of catastrophes in which the Kato's purely mathematical concept of exceptional points was put in direct contact with the phenomenologically crucial instant of the spontaneous breakdown of P T −symmetry.…”

Section: Paraxial Approximation and Anomalous Diffusion Phenomenamentioning

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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Mathematical and Physical Meaning of the Crossings of Energy Levels in $${\mathscr {PT}}$$ PT -Symmetric Systems

Cited by 6 publications

References 20 publications

Dirichlet spectra of the paradigm model of complex PT-symmetric potential:V(x)=−(ix)N

Dirichlet spectra of the paradigm model of complex PT-symmetric potential:V(x)=−(ix)N

Hermitian–Non-Hermitian Interfaces in Quantum Theory

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

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