A Discussion on the Properties of Gamow States

We introduce a dynamical evolution operator for dealing with unstable physical process, such as scattering resonances, photon emission, decoherence and particle decay. With that aim, we use the formalism of rigged Hilbert space and represent the time evolution of quantum observables in the Heisenberg picture, in such a way that time evolution is non-unitary. This allows to describe observables that are initially non-commutative, but become commutative after time evolution. In other words, a non-abelian algebra of relevant observables becomes abelian when times goes to infinity. We finally present some relevant examples. * All authors contributed equally to this manuscript.

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“…Correspondingly, in [49], we have shown that there are two possible expansions for the total Hamiltonian H given by the following expressions,…”

Section: A Non-hermitian Hamiltonianmentioning

confidence: 94%

“…In [49], we have defined a pseudometrics for Gamow vectors. The idea of using pseudometrics was discussed heuristically in previous articles [50,51].…”

Section: B Time Evolution Operatormentioning

confidence: 99%

Dynamics of algebras in quantum unstable systems

Losada

Fortín

Gadella

et al. 2018

Int. J. Mod. Phys. A

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“…It is interesting how this formalism may be applied to Gamow states. This discussion has been already given fundamentally in [ 87 ], and here we want to discuss its main points. In order to make the arguments simpler, we shall use a two Hamiltonian model

with the properties given in the previous sections, that means, its absolutely continuous spectrum of

is simple, so that the absolutely continuous part of

admits the following spectral decomposition [ 17 ]:

with

.…”

Section: Gamow States As Functionals On An Algebra Of Operatorsmentioning

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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“…These coherent states satisfy a couple of resolutions of the identity in a sense to be clarified here. A spectral decomposition of the Hamiltonian in terms of the Gamow vectors has the form [32]…”

Section: Appendix 2 Resolutions Of the Identitymentioning

confidence: 99%

Coherent Gamow states for the hyperbolic Pöschl–Teller potential

Civitarese

Gadella

2019

Annals of Physics

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We have defined a pair of families of coherent states using creation and annihilation operators relating the Gamow states, vector states for non-relativistic quantum resonances. We have used an explicit one dimensional model, for which these operators may be explicitly written: the one dimensional Pöschl-Teller Hamiltonian. We have shown that this model may serve as an example for the pseudo-boson formalism.

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A Discussion on the Properties of Gamow States

Cited by 7 publications

References 40 publications

Dynamics of algebras in quantum unstable systems

Dynamics of algebras in quantum unstable systems

Mathematical Models for Unstable Quantum Systems and Gamow States

Coherent Gamow states for the hyperbolic Pöschl–Teller potential

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