научная статья по теме UNIVERSAL DESCRIPTION OF THE ROTATIONAL-VIBRATIONAL SPECTRUM OF THREE PARTICLES WITH ZERO-RANGE INTERACTIONS Физика

Текст научной статьи на тему «UNIVERSAL DESCRIPTION OF THE ROTATIONAL-VIBRATIONAL SPECTRUM OF THREE PARTICLES WITH ZERO-RANGE INTERACTIONS»

Pis'ma v ZhETF, vol.86, iss. 10, pp.713-717

© 2007 November 25

Universal description of the rotational-vibrational spectrum of three

particles with zero-range interactions

O. I. Kartavtsev, A. V. Malykh Joint Institute for Nuclear Research, 141980 Dubna, Russia Submitted 26 September 2007

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and the third particle of mass mi in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio m/rrii and the total angular momentum L. It is found that the number of vibrational states is determined by the functions Lc{m/mi) and Lt,{m/mi). Explicitly, if the two-body scattering length is positive, the number of states is finite for Lc{m/mi) < L < Lt, (m/rrii), zero for L > Lt,(m/rrii), and infinite for L < Lc{m/mi). If the two-body scattering length is negative, the number of states is zero for L > Lc{m/mi) and infinite for L < Lc{m/mi). For the finite number of vibrational states, all the binding energies are described by the universal function SLN(m/rrii) = £(£,rj), where £ = (N — 1/2)/^/L(L + 1), Tj = y/m/miL(L + 1), and N is the vibrational quantum number. This scaling dependence is in agreement with the numerical calculations for L > 2 and only slightly deviates from those for L = 1,2. The universal description implies that the critical values Lc{m/mi) and Lt,{m/mi) increase as 0.401-\/m/mi and 0.563-\/m/mi, respectively, while the number of vibrational states for L > Lc(m/mi) is within the range N < iVmax « l.ly/L(L + 1) + 1/2.

PACS: 03.65.Ge, 03.75.Ss, 2I.45.-v, 36.90.+f

The universal low-energy few-body dynamics of two-species compounds is of much interest both for atomic and many-body physics. In this respect, the study of the three-body energy spectrum gives insight into the role of triatomic molecules and few-body scattering. The area of applications includes the investigation of multi-component ultra-cold quantum gases, e. g., binary Fermi-Bose [1, 2] and Fermi [3-5] mixtures and of impurities embedded in a quantum gas [6, 7], which are presently under thorough experimental and theoretical study. In addition, one should mention the reactions with negative atomic and molecular ions [8, 9].

The universal isotopic dependence of the three-body energy spectrum was multiply discussed [10-14], nevertheless, the main objective was the description of Efi-mov's spectrum. Recently, the infinite number of the 1+ bound states was predicted [15] for three identical fermions with the resonant p-wave interaction. Concerning the low-energy scattering, one should mention a two-hump structure in the isotopic dependence of the three-body recombination rate of two-component fermions [16 -18] and the two-component model for the three-body recombination near the Feshbach resonance [19].

The main aim of the paper is a comprehensive description of the finite three-body rotational-vibrational spectrum in the zero-range limit of the interaction between different particles. Both qualitative and numerical results are obtained by using the solution of hyper-

radial equations (HREs) [20-22]. The detailed study of the bound state and scattering problems for the total angular momentum L = 1 was presented in [18].

Particle 1 of mass toi and two identical particles 2 and 3 of mass to are described by using the scaled Ja-cobi variables x = (r2 — rj), y = — (mir1 +

mr2)/(m1 + to)] and the corresponding hyper-spherical variables x = p cos a, y = p sina, x = x/x, and y = y/y, where r* is the position vector of the ith particle and p = totoi/(to+toi) and p = to(to+toi)/(toi +2to) are the reduced masses. In the universal low-energy limit, only the s-wave interaction between different particles will be taken into account provided the s-wave interaction is forbidden between two identical fermions and is strongly suppressed between two heavy bosons in the states of L > 0. The two-body interaction is defined by imposing the boundary condition at the zero inter-particle distance, which depends on a single parameter, e. g., the two-body scattering length a [18]. This type of interaction is known in the literature as the zerorange potential [23], the Fermi [24] or Fermi-Huang [25] pseudo-potential, and an equivalent approach is used in the momentum-space representation [26]. The units % = 2p = |a| = 1 are used throughout; thus, the binding energy becomes the universal function depending on the mass ratio to/toi and the rotational-vibrational quantum numbers L and N. In view of the wave-function symmetry under permutation of identical particles, a sum of

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O. I. Kartavtsev, A. V. Malykh

two interactions between different particles is expressed by a single boundary condition at the zero distance between particles 1 and 2,

lim

a—s-ît/2

d_

da

tana

Ia!

= o

(i)

The problem under study is conveniently treated by using the expansion of the properly symmetrized wave function,

(2)

n=1

which leads to the hyper-radial equations for the functions fn(p) [18]. Here p denotes permutation of the identical particles 2 and 3, S = 1 and S = — 1 if these particles are bosons and fermions, respectively, Ylm(y) is the spherical function. The action of JP on the angular variables in the limit a 7r/2 is given by PYlm(y) (-l)LYLM(y) and Pa w, where w = arcsin(l + toi/to)_1. The functions <p%(a,p) in the expansion (2) are the solutions of the equation on a hypersphere (at fixed p),

d2 da2

L(L + 1) sin2 a

' 7n (P)

rí(a,p) = 0, (3)

complemented by the boundary conditions ip%(0,p) = 0 and

,. ( 9 a\ l, ^ 2S( — )L L. , [to = ^t^p)

(4)

where a set of discrete eigenvalues 72 (p) plays the role of the effective channel potentials in a system of the hyperradial equations [18]. The functions satisfying Eq. (3) and the zero boundary condition are straightforwardly expressed [27] via the Legendre function

Vnfap) = VsinaQ^i^j(cosa) = 0£)7„(„)(«) . (5)

The functions <f>L,-y(a) are odd functions on both variables 7 and a satisfying the recurrent relations sin a: <j>L+I,7(a) = (7 - L - l)cosa 0£,7(a) - (7 + + i(a), which follow from those for the

Legendre functions. It is convenient to write 4>l,7(a) = ain(cota) sin 7a: + -Bl,7(cot a) cos 7a, where ai^(x) and bi^(x) are simple polynomials on 7 and x, which are explicitly given for few lowest l by a0^(x) = 1, b0a(x) = 0, -¿1,7(3;) = -x, bln(x) = 7, A2„(x) = 1 - 72 + 3a:2, b2n(x) = s-yx, a3n(x) = 3x(2-y2 - 3 - 5a:2), and B3tJ(x) = j(15x2 + 4 - 72). Substituting (5) into

the boundary condition (4)

if7(*/2) =

a=7r/2

transcendental equation for 72(p),

and using the identity one comes to the

pj^i <t>l,j(n/2) = (f>l+i,7(7T/2)

2 Si-

sin 2u)

-<t>l,-f(«j) . (6)

The attractive lowest effective potential determined by 7f (p) plays the dominant role for the binding-energy and low-energy-scattering calculations, while the effective potentials in the upper channels for n > 2 contain the repulsive term 72(p)/p2 and are of minor importance. Thus, a fairly good description will be obtained by using the one-channel approximation for the total wave function (2) where the first-channel radial function satisfies the equation [18]

' d2 7i (p) ~ 1/4 dp2 p2

•E

/1 (p) = 0 .

(7)

Note that the diagonal coupling term is omitted in Eq. (7), which does not affect the final conclusions and leads to the calculation of a lower bound for the exact three-body energy. Our calculations [18] shows that the one-channel approximation provides better than few percent overall accuracy of the binding energy.

The most discussed feature [10-13] of the three-body system under consideration is the infinite number of the bound states for small L and large m/mi (more precisely, for the finite interaction radius ro the number of states unrestrictedly increases with increasing \a\/r0). As the effective potential in (7) is approximately given by (72(0) — 1/4)/p2 at small p, the number of vibrational states is finite (infinite) if 7J (0) > 0 (7i(0) < 0)- According to Eq. (6), 7^(0) decreases with increasing m/mi and becomes zero at the critical value (m/mi)C£. Thus, one can define the step-like function Lc(m/m 1), which increases by unity at the points (to/toi)C£, so that the number of vibrational states is infinite for L < Lc(m/mi) and finite for L > Lc(m/m 1). Solving Eq. (6) at 71 ^ 0 and p 0, one obtains the exact values (to/toi)C£, which approximately equal 13.6069657, 38.6301583, 75.9944943, 125.764635, and 187.958355 for L = 1 — 5. Originally, the dependence Lc(m/mi) was discussed in [10].

Analyzing the eigenvalue equation (6) one concludes that for a > 0 and S(—)L = the effective potential exceeds the threshold energy E = —1, 7f(p)/p2 > —1, therefore, the bound states only exist if either two identical particles are bosons and L is even or two identical particles are fermions and L is odd. Furthermore, one obtains the trivial answer if a < 0 and L > Lc(m/m 1), for which 71 (p) > 0 and there are no three-body bound states.

a

l

The mass-ratio dependence of the binding energies ££at(»w/toi) for L > Lc(m/mi) and a > 0 is determined numerically by seeking the square-integrable solutions to Eq. (7). Mostly, the properties of the energy spectrum are similar to those for L = 1, which were carefully discussed in [18]. For given L, there is the critical value of to/toi at which the first bound state arise, in other words, there are no three-body bound states for L > Lb(m/mi), where the step-like function Lj(to/toi) undergoes unity jumps at those critical values. Furthermore, all the bound states arise at some values of m/mi being the narrow resonances just below them. For the mass ratio near these values, the binding energies and resonance positions depend linearly and the resonance widths depend quadratically on the mass-ratio excess. Exactly at these values one obtains the threshold bound states, whose wave functions are square-integrable with a power fall-off at large dis

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