Faddeev differential equations as a spectral problem for a nonsymmetric operator

Yakovlev, S. L.

doi:10.1007/bf02070389

Cited by 13 publications

(9 citation statements)

References 9 publications

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“…But this auxiliary spectrum is nothing but that of H 0 . More precisely, σ (H F ) = σ (H) ∪ σ (H 0 ) (for a proof see, e.g., [24]). The spectrum σ (H 0 ) is usually called the spurious spectrum of the Faddeev operator, although it is quite not "dangerous", since it is assumed that one knows everything about the unperturbed Hamiltonian H 0 .…”

Section: Challengesmentioning

confidence: 99%

Progress in methods to solve the Faddeev and Yakubovsky differential equations

Мотовилов

2008

Few-Body Syst

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Section: Challengesmentioning

confidence: 99%

Progress in methods to solve the Faddeev and Yakubovsky differential equations

Мотовилов

2008

Few-Body Syst

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“…Thus all the nice properties of the original Faddeev equations established for short-range interactions remain valid also for the case of Coulomb-like potentials. We note that the triad of Lippmann-Schwinger equations and the set of Faddeev equations describe the same physics, the equations have identical spectra and in fact, the Faddeev equations are the adjoint representations of the triad of Lippmann-Schwinger equations [10]. Utilizing the properties of the Faddeev components the matrix elements in (35) can be rewritten in a form better suited for numerical calculations…”

Section: Faddeev-merkuriev Integral Equations For the Wave Functiomentioning

confidence: 99%

Three-potential formalism for the three-body scattering problem with attractive Coulomb interactions

Papp

Hlousek

et al. 2001

Phys. Rev. A

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A three-body scattering process in the presence of Coulomb interaction can be decomposed formally into a two-body single channel, a two-body multichannel and a genuine three-body scattering. The corresponding integral equations are coupled Lippmann-Schwinger and Faddeev-Merkuriev integral equations. We solve them by applying the Coulomb-Sturmian separable expansion method. We present elastic scattering and reaction cross sections of the e + + H system both below and above the H(n = 2) threshold. We found excellent agreements with previous calculations in most cases. PACS number(s): 34.10.+x, 34.85.+x, 21.45.+v, 03.65.Nk, 02.30.Rz, 02.60.Nm The three-body Coulomb scattering problem is one of the most challenging long-standing problems of nonrelativistic quantum mechanics. The source of the difficulties is related to the long-range character of the Coulomb potential. In the standard scattering theory it is supposed that the particles move freely asymptotically. That is not the case if Coulombic interactions are involved. As a result the fundamental equations of the three-body problems, the Faddeev-equations, become illbehaved if they are applied for Coulomb potentials in a straightforward manner.The first, and formally exact, approach was proposed by Noble [1]. His formulation was designed for solving the nuclear three-body Coulomb problem, where all Coulomb interactions are repulsive. The interactions were split into short-range and long-range Coulomb-like parts and the long-range parts were formally included in the "free" Green's operator. Therefore the corresponding Faddeev-Noble equations become mathematically wellbehaved and in the absence of Coulomb interaction they fall back to the standard equations. However, the associated Green's operator is not known. This formalism, as presented at that time, was not suitable for practical calculations.In Noble's approach the separation of the Coulomb-like potential into short-range and long-range parts were carried out in the two-body configuration space. Merkuriev extended the idea of Noble by performing the splitting in the three-body configuration space. This was a crucial development since it made possible to treat attractive Coulomb interactions on an equal footing as repulsive ones. This theory has been developed using integral equations with connected (compact) kernels and transformed into configuration-space differential equations with asymptotic boundary conditions [2]. In practical calculations, so far only the latter version of the theory has been considered. The primary reason is that the more complicated structure of the Green's operators in the kernels of the Faddeev-Merkuriev integral equations has not yet allowed any direct solution. However, use of integral equations is a very appealing approach since no boundary conditions are required.Recently, one of us has developed a novel method for treating the three-body problem with repulsive Coulomb interactions in three-potential picture [3]. In this approach a three-body Coulomb scattering process can ...

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“…The approach should be based on a physically correct and mathematically sound representation of the problem. The Faddeev equations formalism [26][27][28] fulfills all of these requirements. Clear separation of asymptotic channels corresponding to different clusterisations of the system is one of the main advantageous features of the formalism from the point of view of practical applications.…”

Section: Introductionmentioning

confidence: 99%

Solving the Faddeev-Merkuriev Equations in Total Orbital Momentum Representation via Spline Collocation and Tensor Product Preconditioning

Gradusov¹

2021

CICP

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The computational approach for solving the Faddeev-Merkuriev equations in total orbital momentum representation is presented. These equations describe a system of three quantum charged particles and are widely used in bound state and scattering calculations. The approach is based on the spline collocation method and exploits intensively the tensor product form of discretized operators and preconditioner, which leads to a drastic economy in both computer resources and time.

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Faddeev differential equations as a spectral problem for a nonsymmetric operator

Cited by 13 publications

References 9 publications

Progress in methods to solve the Faddeev and Yakubovsky differential equations

Progress in methods to solve the Faddeev and Yakubovsky differential equations

Three-potential formalism for the three-body scattering problem with attractive Coulomb interactions

Solving the Faddeev-Merkuriev Equations in Total Orbital Momentum Representation via Spline Collocation and Tensor Product Preconditioning

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