Hyperspherical Elliptic Coordinates and Three-Body Coulomb Problem

Tolstikhin, Oleg I.; Watanabe, Shinichi; Matsuzawa, Michio

doi:10.1103/physrevlett.74.3573

Cited by 59 publications

(41 citation statements)

References 16 publications

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“…SVD provides a very powerful method for treating nonadiabatic couplings [8]. The whole computational scheme is highly efficient, and for a variety of three-body Coulomb systems it has been shown to yield an accuracy comparable with the best available variational results [8][9][10].…”

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Siegert Pseudo-States as a Universal Tool: Resonances,SMatrix, Green Function

1997

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The Siegert states have long been recognized as a potentially powerful tool in the formal scattering theory. Here we propose an efficient method to implement this power in practice. Our method yields bound states, complex-energy resonances, and scattering wave functions, i.e., a complete solution of the Schrödinger equation. We also obtain a representation of the Green function suitable for a variety of applications. The method is demonstrated by realistic examples of the eep and dtm three-body Coulomb systems. [S0031-9007(97)03930-6] PACS numbers: 34.10. + x, 03.65.Nk, 34.80.Bm, 36.10.Dr In 1939, in search of a formal derivation of the Breit-Wigner resonance formula, Siegert introduced [1] a class of solutions to the Schrödinger equation which now bear his name. The Siegert states satisfy outgoing wave boundary conditions, and, for the simplest generic scattering problem, they are defined by ∑ 2 1 2These equations can be satisfied simultaneously only for a discrete set of generally complex momenta k n ; thus one should consider Eqs. (1a) and (1b) as an eigenvalue problem defining k n and corresponding eigenfunctions f n ͑r͒. The eigenvalues k n coincide with the poles of the S matrix in the complex k plane. Pure imaginary k n with Im͑k n ͒ . 0 represent bound states of the system. Those lying close to the real k axis correspond to resonances. Others, though not observable directly, complement the set and, at least for finite range potentials, suffice to determine the S matrix in the entire k plane [2]. The significance of Siegert states within a general formalism of the scattering theory and their potential usefulness for computational implementations stem from the fact that they provide a possibility of unified description of bound states, resonances, and continuum spectrum in terms of a purely discrete set of states. For example, by virtue of the Mittag-Leffler expansion theorem (if applicable), one can construct the Green function simply in the form of a sum over the Siegert states, avoiding the annoying integral over the continuum [3]. Of course, there is a price to be paid for such a discretization. First, one must move away from the real energy axis to the complex energy plane. Second, for practical applications one must solve the problem of exponential divergence of f n ͑r͒ at r !`, say, for instance, by making the radial variable r also complex as in the complex rotation method [4]. However, the most important point is that in order to achieve the completeness required for representing the continuum, one has to generate not just one or a few but many solutions of Eqs. (1a) and (1b). This causes an essential practical difficulty, because the nonlinearity of Eqs. (1a) and (1b) with respect to the eigenvalue k prohibits a direct application of the variational methods, which renders the problem tractable only by means of an iterative procedure. In fact, this difficulty was restricting the power of Siegert states to studying only individual resonances [5].Here, we define Siegert pseudo-states by applying the...

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“…The channel functions have been calculated using the hyperspherical elliptic coordinates introduced in [9]. SVD provides a very powerful method for treating nonadiabatic couplings [8].…”

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confidence: 99%

Siegert Pseudo-States as a Universal Tool: Resonances,SMatrix, Green Function

1997

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“…Several approximation schemes have been proposed, such as "potential harmonics" [257] and "adiabatic approximation" [255,259]. An interesting variant is proposed in [260].…”

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confidence: 99%

Stability of few-charge systems in quantum mechanics

2005

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We consider non-relativistic systems in quantum mechanics interacting through the Coulomb potential, and discuss the existence of bound states which are stable against spontaneous dissociation into smaller atoms or ions. We review the studies that have been made of specific mass configurations and also the properties of the domain of stability in the space of masses or inverse masses. These rigorous results are supplemented by numerical investigations using accurate variational methods. A section is devoted to systems of three arbitrary charges and another to molecules in a world with two space-dimensions.

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“…Connections with molecular structure [52,53] arising from these classifications have more recently afforded extending the hyperspherical approach to systems with several heavy particles besides electrons (i.e., to molecules) [54]. In three-body Coulomb problems, hyperspherical methods have become computationally competitive through the "hyperspherical close-coupling method" [55], and more recently in the form of "diabatic-by-sector" numerical methods [56].…”

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confidence: 99%

Geometry and symmetries of multi-particle systems

Fano

Green

Bohn

et al. 1999

J. Phys. B: At. Mol. Opt. Phys.

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Abstract. The quantum dynamical evolution of atomic and molecular aggregates, from their compact to their fragmented states, is parametrized by a single collective radial parameter. Treating all the remaining particle coordinates in d dimensions democratically, as a set of angles orthogonal to this collective radius or by equivalent variables, by-passes all independent-particle approximations. The invariance of the total kinetic energy under arbitrary d-dimensional transformations which preserve the radial parameter gives rise to novel quantum numbers and ladder operators interconnecting its eigenstates at each value of the radial parameter.We develop the systematics and technology of this approach, introducing the relevant mathematics tutorially, by analogy to the familiar theory of angular momentum in three dimensions. The angular basis functions so obtained are treated in a manifestly coordinate-free manner, thus serving as a flexible generalized basis for carrying out detailed studies of wave function evolution in multi-particle systems.

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Hyperspherical Elliptic Coordinates and Three-Body Coulomb Problem

Cited by 59 publications

References 16 publications

Siegert Pseudo-States as a Universal Tool: Resonances,SMatrix, Green Function

Siegert Pseudo-States as a Universal Tool: Resonances,SMatrix, Green Function

Stability of few-charge systems in quantum mechanics

Geometry and symmetries of multi-particle systems

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