Differential equations in momentum space for the three-body problem in the case of pointlike pair interactions
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EI Theory
Differential Equations in Momentum Space for the Three-Body Problem in the Case of Pointlike Pair Interactions F. M. Pen’kov1)* and W. Sandhas2) Received January 29, 2013
Abstract—Corrects schemes for solving equations of three-body dynamics for systems governed by a zerorange two-body interaction are considered. Correlations between spectral features of a three-boson system are obtained. The results are compared with the results obtained by calculating the spectra and scattering lengths in the system of three helium atoms with realistic two-body interaction potentials. DOI: 10.1134/S1063778814040085
Dedicated to Professor Vladimir B. Belyaev on the occasion of the 80th anniversary of his birth
where aki = L(ki , kj ) =
1. INTRODUCTION For physical systems governed by a two-body interaction that satisfies the condition r0 1 (κr0 1), (1) a0 a0 is the where r0 is the range of two-body forces, √ two-body scattering length, and κ = −mε (here, m is the particle mass and ε is the binding energy of two particles), use is frequently made of the interaction model featuring zero-range forces: r0 → 0. This model determines the motion of particles beyond the range of two-body forces and admits a substantial analytic simplification. If the condition in (1) is satisfied, it can be used to describe real physical systems. By way of example, we indicate that, as far back as 1956, Skornyakov and Ter-Martirosyan (STM) proposed an integral equation for the spectrum and the amplitudes of scattering in a system of three nucleons [1]. A simplified form of this equation for the amplitude of boson scattering on a bound boson pair is f (ki , kj ; k0 ) = 2 8 + (aki + κ) 3 π
∞ 0
1)
8 (ak + κ)L(ki , kj ) 3 i
(2)
L(ki , k)k2 f (k, kj ; k0 )dk, k2 − k02 − i0
Institute of Nuclear Physics, ul. Ibragimova 1, Almaty, 050032 Republic of Kazakhstan. 2) ¨ Bonn, D-53115 Physikalisches Institut der Universitat Bonn, Federal Republic of Germany. * E-mail: [email protected]
−mZ + ki2 · 3/4,
ki2 + kj2 + ki kj + λ2 1 ln 2 . 2ki kj ki + kj2 − ki kj + λ2
Here, the symbol k is traditionally used for the momenta of the relative motion of a particle and the remaining pair. Their on-shell values of ki = kj = k0 determine the amplitude of boson scattering on a bound boson pair at the total energy equal to Z = 3k02 /4m + ε. The quantity λ2 = −mZ was introduced in order to render the form of the equations presented here more compact. The equation obtained upon recasting Eq. (2) into the homogeneous form ϕ(k, Z) = ∞ ×
2 (ak + κ) 8 4 2 3 k − 3 (mZ + ε) π
(3)
L(k, k )k2 ϕ(k , Z)dk,
0
determines the bound-state wave function ϕ(k, Z) and the spectrum of the three-boson system. Albeit possessing obvious advantages and being widely applicable, the model of zero-range twobody interaction for three-body systems has a serious drawback. The Hamiltonian for such systems is not self-adjoint (see, for example, [2]), and the ¨ Schrodinger equations have square-integrable solutions at any energy. For example, the STM equations provided an excellent desc
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