Three two-component fermions with contact interactions: correct formulation and energy spectrum

Abstract

Properties of two identical particles of mass m and a distinct particle of mass m 1 in the universal low-energy limit of zero-range two-body interaction are studied in different sectors of total angular momentum L and parity P . For the unambiguous formulation of the problem in the interval619 and µ c (1 − ) ≈ 13.607, µ r (2 + ) ≈ 32.948 and µ c (2 + ) ≈ 38.630, etc.) in each L P sector an additional parameter b determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. Within the framework of this formulation, dependence of the bound-state energies on m/m 1 and b in the sector of angular momentum and parity L P is calculated for L ≤ 5 and analysed with the aid of a simple model. A number of the bound states for each L P sector is analysed and presented in the form of "phase diagrams" in the plane of two parameters m/m 1 and b.

Minlos-Faddeev regularization of zero-range interactions in the three-body problem

“…As is well established, the zero total angular momentum and positive parity P (L P = 0 + ) are those quantum numbers, for which the regularization is certainly required, therefore, namely this case will be considered in this note. One should mention that the regularization is also needed for even L ≥ 2 and positive parity if the mass ratio m/m 1 of the two-component system is sufficiently large [5][6][7][8]. Nonetheless, description of the regularized problem for L ≥ 2 is analogous to the case of L = 0 and will not be given in this work.…”

“…To proceed, one introduces the hyper-radius ρ and hyper-angular variables on a hyper-sphere {Ω} = {α, x, ŷ}, which are defined by x = ρ cos α, y = ρ sin α and x = x/x, and ŷ = y/y. As in [5,[8][9][10], the total wave function is expanded in a set of functions Φ n (α, x, ŷ; ρ), which are the solutions of an auxiliary eigenvalue problem on a hyper-sphere (for fixed ρ),…”

“…In the eigenvalue equations (2.5) and (2.6), whose different branches determine an infinite set of eigenvalues γ 2 n (ρ), the regularization gives rise to an additional term in the well-known result [5,8,11].…”

“…By means of the expansion (2.2), the original problem is formulated as a set of coupled hyper-radial equations [8][9][10] for the channel functions f n (ρ),…”

“…Here the notations γ ≡ γ 1 (0) and q ≡ dγ 2 1 dρ ρ=0 are introduced for brevity. A number of aspects concerning the Schrödinger equation with inverse square singularity, as in V sing (ρ), were multiply discussed in literature, e. g., in [8,[13][14][15]. Following [8,10], the essential features of the quantum problem for the singular potential of the form V sing (ρ) (2.8) are briefly summarized below.…”