Coherent Gamow states for the hyperbolic Pöschl–Teller potential

Civitarese, O.; Gadella, M.

doi:10.1016/j.aop.2019.04.005

Cited by 8 publications

(10 citation statements)

References 26 publications

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“…In previous works we have studied such possibility by analyzing the resonances of the initial structure [16,47,57]. It is viable to construct the supersymmetric partners using resonances of the initial system [31], a technique implemented also in the cosh-like case [52,53] and for soliton-like models [58]. However, the transformation of resonances and/or using resonances is elaborated, so it will be analyzed elsewhere.…”

Section: Discussion Of Results and Conclusionmentioning

confidence: 99%

“…Potentials V(x) = −n 0 (x, m) form the subset of transparent Pöschl-Teller potentials in quantum mechanics. The solutions of the Schrödinger equation for the entire family are well known [18,50,51], including resonances and anti-bound states [52,53]. Our interest in n 0 (x, m) obeys the fact that this refractive index admits exactly m guided TE modes, defined by the quadratic rule [18,50,51]…”

Section: Manipulating a Set Of Guided Modes At Oncementioning

confidence: 99%

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Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

2021

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The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux–Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case.

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Section: Discussion Of Results and Conclusionmentioning

confidence: 99%

Section: Manipulating a Set Of Guided Modes At Oncementioning

confidence: 99%

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

2021

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“…In a recent article [ 107 ], coherent Gamow states have been introduced for the one dimensional hyperbolic Pöschl-Teller potential [ 108 , 109 ]. Both trigonometric and hyperbolic Pöschl-Teller potentials are part of a list of one dimensional exactly solvable potentials [ 110 ], which are used as a manageable approximation to other potentials.…”

Section: Coherent Gamow Statesmentioning

confidence: 99%

Mathematical Models for Unstable Quantum Systems and Gamow States

Gadella

Fortín

Jorge

et al. 2022

Entropy

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We review some results in the theory of non-relativistic quantum unstable systems. We account for the most important definitions of quantum resonances that we identify with unstable quantum systems. Then, we recall the properties and construction of Gamow states as vectors in some extensions of Hilbert spaces, called Rigged Hilbert Spaces. Gamow states account for the purely exponential decaying part of a resonance; the experimental exponential decay for long periods of time physically characterizes a resonance. We briefly discuss one of the most usual models for resonances: the Friedrichs model. Using an algebraic formalism for states and observables, we show that Gamow states cannot be pure states or mixtures from a standard view point. We discuss some additional properties of Gamow states, such as the possibility of obtaining mean values of certain observables on Gamow states. A modification of the time evolution law for the linear space spanned by Gamow shows that some non-commuting observables on this space become commuting for large values of time. We apply Gamow states for a possible explanation of the Loschmidt echo.

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“…This can be implemented for some exactly solvable models for resonances, like the high barrier Pöschl-Teller one dimensional model [65,66].…”

Section: Initial Conditionsmentioning

confidence: 99%

Gamow vectors formalism applied to the Loschmidt echo

Fortin,

Gadella,

Holik

et al. 2020

Preprint

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Gamow vectors have been developed in order to give a mathematical description for quantum decay phenomena. Mainly, they have been applied to radioactive phenomena, scattering and to some decoherence models. They play a crucial role in the description of quantum irreversible processes, and in the formulation of time asymmetry in quantum mechanics. In this paper, we use this formalism to describe a well-known phenomenon of irreversibility: the Loschmidt echo. The standard approach considers that the irreversibility of this phenomenon is the result of an additional term in the backward Hamiltonian. Here, we use the non-Hermitian formalism, where the time evolution is non-unitary. Additionally, we compare the characteristic decay times of this phenomenon with the decoherence ones. We conclude that the Loschmidt echo and the decoherence can be considered as two aspects of the same phenomenon, and that there is a mathematical relationship between their corresponding characteristic times.

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Coherent Gamow states for the hyperbolic Pöschl–Teller potential

Cited by 8 publications

References 26 publications

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

Balanced Gain-and-Loss Optical Waveguides: Exact Solutions for Guided Modes in Susy-QM

Mathematical Models for Unstable Quantum Systems and Gamow States

Gamow vectors formalism applied to the Loschmidt echo

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