Completeness relations, Mittag-Leffler expansions and the perturbation theory of resonant states

Berggren, Tore

doi:10.1016/0375-9474(82)90519-x

Cited by 40 publications

(41 citation statements)

References 23 publications

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“…We have intentionally avoided discussing the completeness relation and the resonant-state expansion [58,59,60,61,62,63,64,65]. We are planning to report discussions on the topic elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Some Properties of the Resonant State in Quantum Mechanics and Its Computation

Hatano¹,

Sasada²,

Nakamura³

et al. 2008

Progress of Theoretical Physics

160

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The resonant state of the open quantum system is studied from the viewpoint of the outgoing momentum flux. We show that the number of particles is conserved for a resonant state, if we use an expanding volume of integration in order to take account of the outgoing momentum flux; the number of particles would decay exponentially in a fixed volume of integration. Moreover, we introduce new numerical methods of treating the resonant state with the use of the effective potential. We first give a numerical method of finding a resonance pole in the complex energy plane. The method seeks an energy eigenvalue iteratively. We found that our method leads to a super-convergence, the convergence exponential with respect to the iteration step. The present method is completely independent of commonly used complex scaling. We also give a numerical trick for computing the time evolution of the resonant state in a limited spatial area. Since the wave function of the resonant state is diverging away from the scattering potential, it has been previously difficult to follow its time evolution numerically in a finite area.

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Section: Discussionmentioning

confidence: 99%

Some Properties of the Resonant State in Quantum Mechanics and Its Computation

Hatano¹,

Sasada²,

Nakamura³

et al. 2008

Progress of Theoretical Physics

160

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“…2 Several methods have been proposed since the publication of Gamow's work, 4 particularly in connection with the normalizability of Green's functions in the presence of GS and in the treatment of completeness relations. 5 The mathematical equivalence between some of these methods has been discussed recently and the correspondence between Bergreen's and Mittag-Lefler's representations has been explored at length. 6 Presently a rich literature is available regarding the application of these concepts to nuclear reactions and to nuclear structure problems.…”

Section: Introductionmentioning

confidence: 99%

“…15 However, and with reference to explicit numerical applications, the use of these techniques does not guarantee the stability of the results since the onset of the exponential dominance of the GS can manifest itself at physical scales. 5 Among the recent references on GS we shall mention the work of T. Berggren, 16 where the possibility of defining expectation values of operators in a resonant state is considered. In the present work we shall focus our attention on the mathematical aspects of representations which include Gamow states.…”

Section: Introductionmentioning

confidence: 99%

Gamow states as continuous linear functionals over analytical test functions

Bollini

Civitarese

Paoli

et al. 1996

Journal of Mathematical Physics

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The space of analytical test functions , rapidly decreasing on the real axis ͑i.e., Schwartz test functions of the type S on the real axis͒, is used to construct the rigged Hilbert space ͑RHS͒ ͑,H,Ј͒. Gamow states ͑GS͒ can be defined in RHS starting from Dirac's formula. It is shown that the expectation value of a selfadjoint operator acting on a GS is real. We have computed exactly the probability of finding a system in a GS and found that it is finite. The validity of recently proposed approximations to calculate the expectation value of self-adjoint operators in a GS is discussed.

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“…The GSM is a complex-momentum generalization of the traditional shell model through the use of the Berggren basis [40,41]. The Berggren basis is defined for each partial wave c = ( , j) for which the continuum is expected to be important in the problem at hand, and is made of singleparticle bound states, decaying resonances and nonresonant scattering states.…”

Section: Theoretical Analysismentioning

confidence: 99%