1. (macfeds3.htm    5-11-98)

    	Borromean states are states of three neutral particles
 which are bound by two-body interaction potentials 
V{ij}(r{ij}), even though none of 
the two-body subsystems are bound.  Efimov states are a special
 case with the additional property that the three-body states 
become unbound as the strength of the two-body interaction increases. 
 In the unusual circumstance where the two-body scattering length 
becomes infinite, Danilov showed that there are an infinite number 
of three-body bound states. No clear examples of Efimov states exist, 
but Borromean states occur in nuclei, atoms and molecules. 
 As an introduction to this fascinating aspect of few-body dynamics 
that is not easily extrapolated from two-body quantum
 theory we present a simple study problem dealing with  Danilov's effect.  

I. Review of effective range theory
	1. Consider a two-body square well in spherical coordinates

  2. V{ij}(r{ij})  = -V0,  r{ij}  < r0

  3. V{ij}(r{ij})  =0,               r{ij}  >  r0

Derive expressions essions for the s-wave phase shift 
d and wave function 
f(r{ij}) for this two-body system.  


  4.  REFERENCES

[1] Danilov. G. S., Zh. Eksp. Teor. Fiz.  40, 498 (1961)
[Sov.  Phys. JETP  13, 349 (1961)]
[2] Efimov, V. N., Yad. Fiz,12, 1080 (1970)
[  Sov. J. Nuc. Phys.  12, 589 (1971)].
[3]  Macek, J., { Zeit. Phys. D},3, 31 (1986).
[4] Federov,  D. V., and Jensen, A. S.  Phys. Rev. Lett., 71, 4103 (1993).
[5] Esry, B. D., Lin, C. D., Greene, Chris H.,  Phys. Rev.A,54, 394 (1996).

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