Siegert pseudostate formulation of scattering theory: One-channel case

Tolstikhin, Oleg I.; Ostrovsky, V N; Nakamura, Hiroki

doi:10.1103/physreva.58.2077

Cited by 120 publications

(115 citation statements)

References 84 publications

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“…For our analysis it will be convenient to decompose the Green's function (E − H eff (E)) −1 into a sum of simpler terms. As shown in [11] (see also [95]), the generalized eigenvalue problem formalism introduced previously in Sec. II gives a decomposition in terms of all the discrete eigenvalues of the Hamiltonian.…”

Section: Model I: Survival Probability Near the Ep2amentioning

confidence: 99%

“…(69) through (73) for Model I and A P (t) is the contribution from the residue at the λ =λ B pole given in Eq. (95). Note that in contrast to the EP2A case in Model I, here we have to include contributions associated with both the lower and upper edges of the energy band, because the (pure imaginary) EP2B eigenvalue lies directly in the middle of the band on the real energy axis.…”

Section: Inverse Power Law Evolution On Long Timescales Near the Ep2bmentioning

confidence: 99%

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Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Garmon

Ordóñez

2017

Journal of Mathematical Physics

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It has been reported in the literature that the survival probability P (t) near an exceptional point where two eigenstates coalesce should generally exhibit an evolution P (t) ∼ t 2 e −Γt , in which Γ is the decay rate of the coalesced eigenstate; this has been verified in a microwave billiard experiment [B. Dietz, et al., Phys. Rev. E 75, 027201 (2007)]. However, the heuristic effective Hamiltonian that is usually employed to obtain this result ignores the possible influence of the continuum threshold on the dynamics. By contrast, in this work we employ an analytical approach starting from the microscopic Hamiltonian representing two simple models in order to show that the continuum threshold has a strong influence on the dynamics near exceptional points in a variety of circumstances. To report our results, we divide the exceptional points in Hermitian open quantum systems into two cases: at an EP2A two virtual bound states coalesce before forming a resonance, anti-resonance pair with complex conjugate eigenvalues, while at an EP2B two resonances coalesce before forming two different resonances. For the EP2B, which is the case studied in the microwave billiard experiment, we verify the survival probability exhibits the previously reported modified exponential decay on intermediate timescales, but this is replaced with an inverse power law on very long timescales. Meanwhile, for the EP2A the influence from the continuum threshold is so strong that the evolution is non-exponential on all timescales and the heuristic approach fails completely. When the EP2A appears very near the threshold we obtain the novel evolution P (t) ∼ 1 − C1 √ t on intermediate timescales, while further away the parabolic decay (Zeno dynamics) on short timescales is enhanced.

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Section: Model I: Survival Probability Near the Ep2amentioning

confidence: 99%

Section: Inverse Power Law Evolution On Long Timescales Near the Ep2bmentioning

confidence: 99%

Characteristic dynamics near two coalescing eigenvalues incorporating continuum threshold effects

Garmon

Ordóñez

2017

Journal of Mathematical Physics

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“…All of these calculations, except the LiH + 2 and HeH + cases, used Siegert pseudostates [12,13] for the nuclear vibrational basis and they all exploited the following two-step procedure:…”

Section: Introductionmentioning

confidence: 99%

Dissociative recombination by frame transformation to Siegert pseudostates: A comparison with a numerically solvable model

et al. 2018

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We present a simple two-dimensional model of the indirect dissociative recombination process. The model has one electronic and one nuclear degree of freedom and it can be solved to high precision, without making any physically motivated approximations, by employing the exterior complex scaling method together with the finite-elements method and discrete variable representation. The approach is applied to solve a model for dissociative recombination of H + 2 and the results serve as a benchmark to test validity of several physical approximations commonly used in the computational modeling of dissociative recombination for real molecular targets. The second, approximate, set of calculations employs a combination of multi-channel quantum defect theory and frame transformation into a basis of Siegert pseudostates. The cross sections computed with the two methods are compared in detail for collision energies from 0 to 2 eV.

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“…More recently, masking functions, repetitive projection and complex rotation methods, and Siegert pseudostates are common theoretical tools. These techniques are discussed by Yoshida, Watanabe, Reinhold, and Burgdörfer in [94] and by Tolstikhin, Ostrovsky, and Nakamura in [95]. A scaling transformation method that eliminates the rapid phase variation and wavepacket expansion and requires no matching at infinity has been presented by Sidky and Esry in [96].…”

Section: Asymptotic Radiationmentioning

confidence: 99%