Analyticity of the time dependence of resonance poles: Solving the Friedrichs model with a time-dependent interaction

Civitarese, O.; Gadella, M.

doi:10.1103/physrevc.86.024602

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“…Resonance poles defined with the resolvent and resonance poles as singularities of the function

coincide [ 19 ]. The Friedrichs model has been generalized so as to describe more complex models showing resonances, although not all of the new approaches are exactly solvable due to their increasingly complexity [ 12 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 ]. Even for a construction of resonances in relativistic quantum field theory [ 74 , 75 ], the literature on the subject is far from being complete.…”

Section: Rigged Hilbert Spaces and Gamow Vectorsmentioning

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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“…Resonance poles defined with the resolvent and resonance poles as singularities of the function