Hyperspherical adiabatic eigenvalues for zero-range potentials

Gasaneo, Gustavo; Macek, J. H.

doi:10.1088/0953-4075/35/10/304

Cited by 26 publications

(37 citation statements)

References 29 publications

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“…It also agrees with the results of Ref. [17] if λ = 1. For symmetric systems (λ = 1) with l = 0 this is the equation first derived by Danilov [22], which has an imaginary solution for ν.…”

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confidence: 93%

“…This provides an independent confirmation of results of Griesshammer for the power counting in these systems [9]. That work solved the Skorniakov-Ter-Martirosian equation [15] in momentum space, whereas here I work in coordinate-space following the approach developed by Efimov for attractive three-body systems [16] and recently extended by Gasaneo and Macek to cases with nonzero angular momentum [17]. I also comment on differences between the counting for derivative interactions in systems with strong long-range forces compared with the pure short-range case.…”

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confidence: 73%

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Renormalization-group analysis of repulsive three-body systems

Birse

2005

J. Phys. A: Math. Gen.

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A coordinate space approach, based on that used by Efimov, is applied to three-body systems with contact interactions between pairs of particles. In systems with nonzero orbital angular momentum or with asymmetric spatial wave functions, the hyperradial equation contains a repulsive 1/r 2 potential. The resulting wave functions are used in a renormalisation group analysis. This confirms Griesshammer's power counting for short-range three-body forces in these systems. The only exceptions are ones like the 4 S channel for three nucleons, where any derivatives needed in the interaction are found to be already counted by the scaling with the cut-off.Effective field theories (EFT's) are now being widely applied to few-nucleon systems, see Refs. [1,2]. The starting point is usually an organisation of the terms in an effective potential according to naive dimensional analysis (NDA), as originally suggested by Weinberg [3]. This classifies terms according to number of powers of low-energy scales they contain. In some cases, most notably S-wave nucleon-nucleon scattering, the leading-order (LO) terms turn out to be unnaturally large. This is a consequence of low-energy bound or virtual states. It means that the LO terms need to be iterated to all orders in solving the Schrödinger or Lippmann-Schwinger equation [4,5,6].However it is now clear that NDA is not valid in all systems. There can be nonperturbative effects associated with strong long-range potentials that significantly change the power counting for short-range interactions. This was first noted in the context of attractive three-body systems (such as three bosons, or the 2 S channel for three nucleons) [7,8], where the leading three-body forces must be promoted to LO. More recently, failures of NDA have been observed in repulsive three-body systems [9], and for nucleon-nucleon scattering in spin-triplet waves [10,11]. In the first example, short-range three-body forces in most low partial waves are demoted to higher orders than naively expected; in the second, short-range forces are promoted to lower orders in P -and D-waves. In both cases, the terms in potentials scale with noninteger anomalous dimensions, and so the standard classification of terms as LO, next-to-leading order (NLO) etc. is no longer convenient.In this note I examine repulsive three-body systems using the renormalisation group (RG) approach developed in Refs. [12,13,14]. This provides an independent confirmation of results of Griesshammer for the power counting in these systems [9]. That work solved the Skorniakov-Ter-Martirosian equation [15] in momentum space, whereas here I work in coordinate-space following the approach developed by Efimov for attractive three-body systems [16] and recently extended by Gasaneo and Macek to cases with nonzero angular momentum [17]. I also comment on differences between the counting for derivative interactions in systems with strong long-range forces compared with the pure short-range case. This corrects the counting in Ref. [9] for the leading three-body fo...

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confidence: 93%

supporting

confidence: 73%

Renormalization-group analysis of repulsive three-body systems

Birse

2005

J. Phys. A: Math. Gen.

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“…Then the real space rotation R (ij) in Eq. (29) is (39) and the corresponding operator has matrix elements ℓ, m z |R (14)…”

Section: B Rotational Invariancementioning

confidence: 99%

Absence of a four-body Efimov effect in the2+2fermionic problem

Endo

Castin

2015

Phys. Rev. A

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In the free three-dimensional space, we consider a pair of identical ↑ fermions of some species or in some internal state, and a pair of identical ↓ fermions of another species or in another state. There is a resonant s-wave interaction (that is of zero range and infinite scattering length) between fermions in different pairs, and no interaction within the same pair. We study whether this 2 + 2 fermionic system can exhibit (as the 3+1 fermionic system) a four-body Efimov effect in the absence of three-body Efimov effect, that is the mass ratio α between ↑ and ↓ fermions and its inverse are both smaller than 13.6069. . . . For this purpose, we investigate scale invariant zero-energy solutions of the four-body Schrödinger equation, that is positively homogeneous functions of the coordinates of degree s − 7/2, where s is a generalized Efimov exponent that becomes purely imaginary in the presence of a four-body Efimov effect. Using rotational invariance in momentum space, it is found that the allowed values of s are such that M (s) has a zero eigenvalue; here the operator M (s), that depends on the total angular momentum ℓ, acts on functions of two real variables (the cosine of the angle between two wave vectors and the logarithm of the ratio of their moduli), and we write it explicitly in terms of an integral matrix kernel. We have performed a spectral analysis of M (s), analytical and for an arbitrary imaginary s for the continuous spectrum, numerical and limited to s = 0 and ℓ ≤ 12 for the discrete spectrum. We conclude that no eigenvalue of M (0) crosses zero over the mass ratio interval α ∈ [1, 13.6069 . . .], even if, in the parity sector (−1) ℓ , the continuous spectrum of M (s) has everywhere a zero lower border. As a consequence, there is no possibility of a four-body Efimov effect for the 2+2 fermions.We also enunciated a conjecture for the fourth virial coefficient of the unitary spin-1/2 Fermi gas, inspired from the known analytical form of the third cluster coefficient and involving the integral over the imaginary s-axis of s times the logarithmic derivative of the determinant of M (s) summed over all angular momenta. The conjectured value is in contradiction with the experimental results. PACS numbers: 67.85.-d, 21.45.-v, 34.50.-s R dxdx ′ 1 −1 dudu ′ mz,m ′ z (−1) ℓ+1 x, u, ℓ, m z |[M (ℓ) (iS)] −1 |x ′ , u ′ , ℓ, m ′ z × x ′ , u ′ , ℓ, m ′ z | d dS M (ℓ) (iS)|x, u, ℓ, m z (B21)

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“…However, aside for some particular potentials, there are no analytic solutions for the eigenvalue problem defined by Eq. (4) with the boundary conditions (6)- (7) or (10)- (11).…”

Section: Methods For Obtaining Sturmiansmentioning

confidence: 99%

“…parameter and using other parameter appearing in the equation as eigenvalue [1,2]. These solutions for a particular set of boundary conditions, are referred to as Sturmian functions [1,[3][4][5][7][8][9] The Schrödinger Coulomb Sturmian functions (CSFs) for a two-body system are of particular interest in atomic physics. These functions are solutions of the two-body Schrödinger equation for a Coulomb potential, where the energy is fixed and the charge is considered as the eigenvalue.…”

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confidence: 99%

L² discretization of Sturmian wave functions for Coulomb‐like potentials

Frapiccini

González

Randazzo

et al. 2006

Int J of Quantum Chemistry

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ABSTRACT:In this work we introduce a method to construct Sturmian functions for general interaction potentials in two-body problems. We expand these Sturmians on a finite L 2 space, using N Laguerre basis functions to obtain a discrete set of eigenvalues for positive and negative energies. Orthogonality and closure relations are thus rewritten for these expansions; completeness is achieved through increasing the basis size. We apply the method to the Coulomb and Herman and Skillman potential. We study the behavior of the functions obtained and their convergence for an overall range of energies. The Sturmian functions are applied to solve the Schrödinger equation for an active electron in a He-like system.

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Hyperspherical adiabatic eigenvalues for zero-range potentials

Cited by 26 publications

References 29 publications

Renormalization-group analysis of repulsive three-body systems

Renormalization-group analysis of repulsive three-body systems

Absence of a four-body Efimov effect in the2+2fermionic problem

L² discretization of Sturmian wave functions for Coulomb‐like potentials

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Hyperspherical adiabatic eigenvalues for zero-range potentials

Cited by 26 publications

References 29 publications

Renormalization-group analysis of repulsive three-body systems

Renormalization-group analysis of repulsive three-body systems

Absence of a four-body Efimov effect in the2+2fermionic problem

L2 discretization of Sturmian wave functions for Coulomb‐like potentials

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L² discretization of Sturmian wave functions for Coulomb‐like potentials