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note
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One can equivalently consider a dimer wavefunction of a well defined angular moment component m=0,±1 along the quantization axis z, by replacing (3/4π) 1/2 rγ /r in Eq. 7 by Y1m (θ,).
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If one assumes αres =0, taking u (k) =1/ (k2 Vs) for low k, as done in Ref., one gets a dimer for Vs <0 (contrarily to the realistic case) and one reaches a contradiction in Eq. 9 |N| 2 <0.
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If one assumes αres =0, taking u (k) =1/ (k2 Vs) for low k, as done in Ref., one gets a dimer for Vs <0 (contrarily to the realistic case) and one reaches a contradiction in Eq. 9 |N| 2 <0.
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note
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In the resonant limit | Vs | →+, on the side αres Vs <0 of the resonance, there is a so-called quasibound state in the two-body problem, which was studied experimentally in Ref.: From Eq. 4 and using the fact that αres >0, one finds that the scattering amplitude f (k) has a complex pole with a real part k00 1/ (αres | Vs |) 1/2 much larger than its imaginary part - (k00) 2 / (2 αres). Physically this is related to the existence of a centrifugal barrier when the two particles approach with non-zero relative momentum, here l=1. This quasibound state does not play an important role in the present work: in the exceptional case where we consider a negative rather than positive scattering volume (see end of Sec. 5), the total energy E in the three-body problem is still assumed to be much smaller than 2 (k00) 2 /m so that the relative momentum krel appearing in the three-body problem, as defined in Eq. 39, cannot reach the value k00.
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This inequality was obtained by a different technique in Ref. and was generalized to higher partial wave channels in PLRAAN 1050-2947 10.1103/PhysRevA.73.012701
-
This inequality was obtained by a different technique in Ref. and was generalized to higher partial wave channels in L. Pricoupenko, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.73.012701 73, 012701 (2006).
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(2006)
Phys. Rev. A
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Pricoupenko, L.1
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55
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0007125811
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This inequality can also be obtained using the generalization of the Smorodinskii formula for the effective range to the case of a resonant p -wave interaction, as done in TMFZAL 0564-6162
-
This inequality can also be obtained using the generalization of the Smorodinskii formula for the effective range to the case of a resonant p -wave interaction, as done in V. D. Mur and V. S. Popov, Teor. Mat. Fiz. TMFZAL 0564-6162 27, 204 (1976).
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(1976)
Teor. Mat. Fiz.
, vol.27
, pp. 204
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Mur, V.D.1
Popov, V.S.2
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56
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33845202266
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RMPHAT 0034-6861 10.1103/RevModPhys.78.1311
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T. Köhler, K. Goral, and P. Julienne, Rev. Mod. Phys. RMPHAT 0034-6861 10.1103/RevModPhys.78.1311 78, 1311 (2006).
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Rev. Mod. Phys.
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Köhler, T.1
Goral, K.2
Julienne, P.3
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58
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42149095859
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This reasoning extends to an arbitrary, everywhere finite envelope function f (k) such that χ (k) =kf (k).
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This reasoning extends to an arbitrary, everywhere finite envelope function f (k) such that χ (k) =kf (k).
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59
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42149112120
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note
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We used the identity r1,..., rN | a k1 † a kN † | 0 = (N!) -1/2 σ SN (σ) exp [i (k1 rσ (1) ++ kN rσ (N))].
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60
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42149155927
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note
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Checking the consistency of these two ways of calculating β is actually useful: contour integration in the space of complex k was used to derive Eq. 9 from Eq. 8, and it is not straightforward that the contribution of the contour integral on an infinite radius half circle is zero, since | f (iq) | actually rapidly diverges as q→ for our two-channel model. This check reduces to proving that I (r) = d3 k (2π) 3 e- k2 b2 /2 eikr / (k2 + q dim 2) ∼ e- qdim r e q dim 2 b2 /2 / (4πr) for r→. This can be done by using the analytic expression for I (r) or by noting that I (r) is the convolution of a Gaussian with a Yukawa function.
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61
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11444254122
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PRLTAO 0031-9007 10.1103/PhysRevLett.93.143201
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D. S. Petrov, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.93. 143201 93, 143201 (2004).
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Phys. Rev. Lett.
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Petrov, D.S.1
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62
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42149190521
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note
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In the second term, an integral of the form I= 0 Kmax d K′ v (K′) P [1/ (K′2 - K dim 2)] appears, with Kmax being the numerical momentum cut-off, which may lead to numerical difficulties since the integrand diverges in K= Kdim. We rewrite it using v (K′) = [v (K′) -v (Kdim)] +v (Kdim). Using 0 Kmax d K′ P [1/ (K′2 - K dim 2)] =-ln [(Kmax + Kdim) / (Kmax - Kdim)] / (2 Kdim), we are left with the numerical evaluation of 0 Kmax d K′ [v (K′) -v (Kdim)] / (K′2 - K dim 2).
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65
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0033474619
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EPJDF6 1434-6060 10.1007/s100530050586
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M. Holzmann and Y. Castin, Eur. Phys. J. D EPJDF6 1434-6060 10.1007/s100530050586 7, 425 (1999).
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(1999)
Eur. Phys. J. D
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, pp. 425
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Holzmann, M.1
Castin, Y.2
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66
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42149150911
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-
note
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We get Rα ψ † Rβ ψ 0 = Mαβ / L3, with Mαβ = i=1 3 k i,α 0 k i,β 0. From Wick's theorem the expectation value 0 in Eq. 81 is equal to 3 L-9 (Mαα Mββ - M αβ 2). One may then check that (α,β) Mαα Mββ - M αβ 2 =3 (K0 × k0) 2, where the sum is taken over the three values (x,y), (x,z), and (y,z) of (α,β).
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67
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42149098630
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note
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One sees that the quantity t appearing in the definition of C0 and C2 in Appendix is small and can be neglected; explicit expressions without relying on special functions can be obtained; they can be further simplified by performing a high v expansion, where the parameter v appears in Appendix, and by keeping the leading term 2/ (3v). Since the next term in the large v expansion is O (1/ v3), this further approximation of keeping the leading term 1/v only is already good at the minimal possible value of v,v=2, where it introduces an extra error of ∼5%.
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68
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42149188728
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One can show the following result, for a fixed value of αres: if there exist constants A and B such that, for all Vs above some value, one has | F (K) |
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One can show the following result, for a fixed value of αres: if there exist constants A and B such that, for all Vs above some value, one has | F (K) |
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69
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2842515744
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PHRVAO 0031-899X 10.1103/PhysRev.124.1866
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U. Fano, Phys. Rev. PHRVAO 0031-899X 10.1103/PhysRev.124.1866 124, 1866 (1961).
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Phys. Rev.
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Fano, U.1
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72
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33847136470
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PLRAAN 1050-2947 10.1103/PhysRevA.75.020702
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G. Smirne, R. M. Godun, D. Cassettari, V. Boyer, C. J. Foot, T. Volz, N. Syassen, S. Dürr, G. Rempe, M. D. Lee, K. Goral, and T. Koehler, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.75.020702 75, 020702 (R) (2007).
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(2007)
Phys. Rev. A
, vol.75
, pp. 020702
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-
Smirne, G.1
Godun, R.M.2
Cassettari, D.3
Boyer, V.4
Foot, C.J.5
Volz, T.6
Syassen, N.7
Dürr, S.8
Rempe, G.9
Lee, M.D.10
Goral, K.11
Koehler, T.12
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73
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34547526100
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PLRAAN 1050-2947 10.1103/PhysRevA.76.012720
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D. M. Lee, T. Koehler, and P. S. Julienne, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.76.012720 76, 012720 (2007).
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(2007)
Phys. Rev. A
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, pp. 012720
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Lee, D.M.1
Koehler, T.2
Julienne, P.S.3
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74
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0036655468
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PLRAAN 1050-2947 10.1103/PhysRevA.66.012705
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E. Nielsen, H. Suno, and B. D. Esry, Phys. Rev. A PLRAAN 1050-2947 10.1103/PhysRevA.66.012705 66, 012705 (2002).
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(2002)
Phys. Rev. A
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Nielsen, E.1
Suno, H.2
Esry, B.D.3
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75
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27744515758
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PRLTAO 0031-9007 10.1103/PhysRevLett.94.213201
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J. P. D'Incao and B. D. Esry, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.94.213201 94, 213201 (2005).
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Phys. Rev. Lett.
, vol.94
, pp. 213201
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D'Incao, J.P.1
Esry, B.D.2
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76
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33749624515
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PRLTAO 0031-9007 10.1103/PhysRevLett.97.150401
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F. Werner and Y. Castin, Phys. Rev. Lett. PRLTAO 0031-9007 10.1103/PhysRevLett.97.150401 97, 150401 (2006).
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Phys. Rev. Lett.
, vol.97
, pp. 150401
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Werner, F.1
Castin, Y.2
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77
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42149162749
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note
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This arbitrary normalization is 0 + dK K2 | b (K) | 2 =1.
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78
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42149099862
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note
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We shall also assume that the first order derivative of E (K) with respect to K is uniformly bounded as a function of K and Ktrim.
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79
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42149175866
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note
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From the low- K expansion of the matrix elements in the second line of M (K,k), we get BL=2 (K) =O (1/ K2 + K trim 2), so that K | BL=2 (K) | is uniformly bounded in K and Ktrim, similarly to what happens in the even sector. The crossed terms in BL=0 BL=2 give at most a logarithmic divergence in the normalization integrals.
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80
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42149137821
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note
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We use here the convention that, in the two-body calculation of the dimer wave function (r), β= p closed 1/2 ez.
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81
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42149116817
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note
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The infinite Vs bg pure single channel model is equivalent to an infinite Vs two channel model with no open channel interaction and with Λ→+, since both models have the same scattering amplitude in these limits.
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82
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42149177675
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note
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More precisely, if one assumes that v0 (K) diverges as 1/ Kγ for low K, γ<3 to ensure convergence in the integral, one sees that the integral in the limit K→0 either diverges as 1/ Kγ-1, for γ>1, or does not diverge, for γ<1. Since α th even >0, this leads to a contradiction in the integral equation α th even v0 (K) /4= I0† [v0] (K), the left-hand side diverging more rapidly than the right-hand side.
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