On domains of {\cal P}{\cal T} symmetric operators related to −y″(x) + (− 1)nx2ny(x)

Azizov, T. Ya.; Trunk, Carsten

doi:10.1088/1751-8113/43/17/175303

Cited by 12 publications

(4 citation statements)

References 56 publications

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“…There are families of non-hermitian Hamiltonians for which their eigenfunctions and spectrum can not be used to complete the information of H in the whole Hilbert space H. The treatment of these problems involves another tools as generalized Riesz systems [67][68][69], pseudospectrum [14], unbounded metric operators and spectral functions for definitizable operators in Krein spaces [42,66,68,70]. Also, the domain of the spectral functions is a non trivial issue to address [71].…”

Section: Physical Applicationsmentioning

confidence: 99%

Dynamics of finite dimensional non-hermitian systems with indefinite metric

Ramírez¹,

Reboiro

2019

Journal of Mathematical Physics

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We discuss the time evolution of physical finite dimensional systems which are modelled by non-hermitian Hamiltonians. We address both general non-hermitian Hamiltonians and pseudohermitian ones. We apply the theory of Krein Spaces to construct metric operators and well-defined inner products. As an application, we study the stationary behavior of dissipative One Axis Twisting Hamiltonians. We discuss the effect of decoherence under different coupling schemes.

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Section: Physical Applicationsmentioning

confidence: 99%

Dynamics of finite dimensional non-hermitian systems with indefinite metric

Ramírez¹,

Reboiro

2019

Journal of Mathematical Physics

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“…(The difficulties with the existence of different closed extensions, cf. [28], do not arise here since, Re V is trivially bounded from below. )…”

Section: Imaginary Cubic Oscillatormentioning

confidence: 99%

On the metric operator for the imaginary cubic oscillator

2012

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We show that the eigenvectors of the PT-symmetric imaginary cubic oscillator are complete, but do not form a Riesz basis. This results in the existence of a bounded metric operator having intrinsic singularity reflected in the inevitable unboundedness of the inverse. Moreover, the existence of non-trivial pseudospectrum is observed. In other words, there is no quantum-mechanical Hamiltonian associated with it via bounded and boundedly invertible similarity transformations. These results open new directions in physical interpretation of PT-symmetric models with intrinsically singular metric, since their properties are essentially different with respect to self-adjoint Hamiltonians, for instance, due to spectral instabilities.Comment: 7 pages; completely rewritten, new result

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“…While the number of studies of such PT-symmetric systems both in optics [18] and in atomic physics [15,16] is rapidly growing, the volume of related mathematical works is rather limited and mostly constrained to linear problems [6,7,22,33]. It is the purpose of this paper to provide a number of rigorous results on nonlinear stationary states in PT-symmetric discrete systems.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Stationary States in PT-Symmetric Lattices

Kevrekidis¹,

Pelinovsky²,

Tyugin³

2013

SIAM J. Appl. Dyn. Syst.

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In the present work we examine both the linear and nonlinear properties of two related PT-symmetric systems of the discrete nonlinear Schrödinger (dNLS) type.First, we examine the parameter range for which the finite PT-dNLS chains have real eigenvalues and PT-symmetric linear eigenstates. We develop a systematic way of analyzing the nonlinear stationary states with the implicit function theorem at an analogue of the anti-continuum limit for the dNLS equation.Secondly, we consider the case when a finite PT-dNLS chain is embedded as a defect in the infinite dNLS lattice. We show that the stability intervals of the infinite PT-dNLS lattice are wider than in the case of a finite PT-dNLS chain. We also prove existence of localized stationary states (discrete solitons) in the analogue of the anti-continuum limit for the dNLS equation.Numerical computations illustrate the existence of nonlinear stationary states, as well as the stability and saddle-center bifurcations of discrete solitons.

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On domains of {\cal P}{\cal T} symmetric operators related to −y″(x) + (− 1)ⁿx²ⁿy(x)

Cited by 12 publications

References 56 publications

Dynamics of finite dimensional non-hermitian systems with indefinite metric

Dynamics of finite dimensional non-hermitian systems with indefinite metric

On the metric operator for the imaginary cubic oscillator

Nonlinear Stationary States in PT-Symmetric Lattices

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