Jost Function for Singular Potentials

Transcription

1 Jost Function for Singuar Potentias S. A. Sofianos, S. A. Rakityansky, and S. E. Massen Physics Department, University of South Africa, P.O.Box 392, Pretoria 0003, South Africa (January 2, 999) An exact method for direct cacuation of the Jost function and Jost soutions for a repusive singuar potentia is presented. Within this method the Schrödinger equation is repaced by an equivaent system of inear first order differentia equations, which after compex rotation, can easiy be soved numericay. The Jost function can be obtained to any desired accuracy for a compex momenta of physica interest, incuding the spectra points corresponding to bound and resonant states. The method can aso be used in the compex anguar momentum pane to cacuate the Regge trajectories. The effectiveness of the method is demonstrated using the Lennard Jones (2,6) potentia. The spectra properties of the reaistic inter atomic 4 He 4 He potentias HFDHE2 and HFD-B of Aziz and coaborators are aso investigated. PACS numbers: Nk, r,.55.Hx, Cf, s I. INTRODUCTION A new method for ocating potentia resonances and Regge trajectories, based on direct cacuation of the Jost function in the compex k pane, has recenty been deveoped 4. Within this method, the bound, resonant, and scattering states can be found by cacuating the Jost soutions and the Jost function on the appropriate domain of the k pane. The bound and resonant state energies, for exampe, can be found by ocating the zeros of the Jost function on the positive imaginary axis and in the fourth quadrant respectivey. At the same time, as a by-product of the Jost function cacuation, one gets the physica wave function that has the correct asymptotic behavior. The method, in the form deveoped in Refs. 4, cannot be directy appied to potentias which are more singuar than /r 2 at the origin. A significant number of practica probems, however, where the Jost function coud be very usefu, invoves such potentias. For exampe, inter-atomic and inter-moecuar forces at short distances are strongy repusive due to the overap of the eectron couds, and thus they are usuay represented by repusive singuar potentias such as the Lennard Jones(2,6) one which has /r 2 behavior. It is, therefore, desirabe to extend the Jost function method for potentias of this kind. It is we-known that attractive singuar potentias do not admit physicay meaningfu soutions with the usua boundary conditions 5. A soutions of the Schrödinger equation with such potentias vanish at the origin and there is no apparent way to determine the arbitrary phase factor between them. In contrast, repusive singuar potentias do not pose any probem regarding mathematica uniqueness or physica interpretation. However, the integration of the Schrödinger equation as we as of the reevant equations for the Jost function, have inherent difficuties resuting from the fact that the singuarity of the potentia makes the r = 0 an irreguar singuar point of the equation. In particuar, the reguar soution cannot be defined by universa boundary conditions independent of the potentia. This drawback, stemming from the extremey strong repusion near the origin, can fortunatey be tacked by the WKB approximation which provides the correct radia behavior of the wave function in the neighborhood of the point r = 0 6. Therefore, starting with the WKB boundary conditions, one can find the reguar soution by integrating the Schrödinger equation from r = 0 to some intermediate point r = r int. Then the equations for the Jost function can be integrated, from r int outwards as for the nonsinguar potentia using the boundary conditions at r = r int which are expressed in terms of the reguar soution and its first derivative. The paper is organized as foows: In Sec. II our formaism is presented and is tested in Sec. III using an exampe known in the iterature; in Sec. IV the method is appied to reaistic interaction between heium atoms. Our concusions are drawn in Sec. V. Permanent address: Department of Theoretica Physics, University of Thessaoniki, Thessaoniki, 54006, Greece

2 II. THEORY A. Basic equations and definitions There are three different types of physica probems associated with the Schrödinger equation (2µ/ h 2 =), 2 r +k 2 (+)/r 2 u (k, r) =V(r)u (k, r), () namey, bound, scattering, and resonant state probems. They differ in the boundary conditions imposed on the wave function at arge distances. Aternativey, a soution can be prescribed by the boundary conditions at the origin. In the case of reguar potentias obeying the condition im r 0 r2 V (r) =0, (2) the soution φ (k, r) which vanishes near r = 0 exacty ike the Riccati Besse function, im φ (k, r)/j (kr) =, (3) r 0 is caed the reguar soution. Since a physica soutions are reguar at the origin, they differ from φ (k, r) onybya normaization constant. Therefore, if the function φ (k, r) can be cacuated at a rea and compex momenta k, one can have, in principe, a soutions of physica interest in a most genera form. For exampe, the cacuation of the scattering soutions on the rea k-axis is simpy a matter of finding the proper normaization for φ (k, r) because the reguar soution has the correct behavior at arge r for any k>0. In contrast, to find the bound and resonant states where the k is compex, one must ensure that the function φ (k, r) has the proper physica asymptotic behavior which exist ony at certain points on the k pane. These spectra points can be found by many different ways, but perhaps the most convenient way to find them is by ocating the zeros of the Jost function. For any compex k, the reguar soution at arge distances can be expressed as a inear combination of the Riccati Hanke functions h (±) (kr), φ (k, r) h (+) r (kr)f 2 (k )+h ( ) (kr)f (k), (4) where the r-independent but momentum dependent coefficient f (k) is the Jost function. From the asymptotic form (4) it is cear that the zeros of f (k) on the positive imaginary axis of the compex k pane correspond to bound states whie those in the fourth quadrant to resonances. In order to find f (k), we ook for the reguar soution on the whoe interva 0, ) in the form φ (k, r) = h (+) (kr)f (+) (k, r)+h ( ) (kr)f ( ) (k, r), (5) 2 where the new unknown functions F (±) (k, r) are subjected to the additiona condition h (+) (kr) r F (+) (k, r)+h ( ) (kr) r F ( ) (k, r) =0. (6) Eq. () is then transformed into an equivaent system of first order equations r F (±) (k, r) =± h( ) (kr) 2ik V (r) h (+) (kr)f (+) (k, r)+h ( ) (kr)f ( ) (k, r). (7) In Refs.,4 it was shown that at arge distances F ( ) (k, r) coincides with the Jost function, im F ( ) r (k, r) =f (k), (8) but this imit ony exists when Im kr 0. (9) If r is rea, the condition (9) is ony satisfied for bound and scattering states but not for resonances. To cacuate f (k) we, therefore, make a compex rotation of the coordinate in Eqs. (7), in the first quadrant r = x exp(iθ), x 0, 0 θ< π 2, (0) with a sufficienty arge θ (see Refs. 4 for more detais). Such a rotation is ony possibe if the potentia is an anaytic function of r and tends to zero when x for the chosen ange θ. 2

3 B. Boundary conditions In the case of reguar potentias the boundary conditions for Eqs. (7) are very simpe, F (±) (k, 0) =. () They foow immediatey from (3), (5), and the fact that 2 h (+) (kr)+h ( ) (kr) = j (kr). Going over to singuar potentias, Eq. (3) does not hod anymore. Due to the extremey strong repusion, the reguar soution vanishes much faster than j (kr)whenr 0. In fact, it vanishes exponentiay 7 and therefore the conditions () must be modified accordingy. In order to find the exact behavior of the reguar soution near the origin we appy the famiiar semi-cassica WKB method. Though the strong repusion makes things rather compicated, it has the advantage that the criterion of the appicabiity of the WKB approximation is satisfied when r 0. Indeed, the WKB method works we when the oca waveength λ varies sowy, i.e. dλ/dr. (2) It can be shown 5 that this derivative is given by dλ/dr = dv (r) 2 k 2 V (r) 3/2. (3) dr Assuming that V (r) approaches its singuarity near r = 0 monotonicay, we can find an r min that for a r<r min the momentum in (3) is negigibe, i.e. we may write dλ/dr dv(r) r 0 2 V(r) 3/2. (4) dr When r 0, the right hand side of Eq. (4) for usua singuar potentias tends to zero. For exampe, if the condition (2) is aways satisfied for n>2, V (r) r 0 g/r n, dλ/dr nr 2 n r 0 2 g 0, if n>2. Therefore, assuming that the necessary condition (2) is fufied and choosing a sma enough r min, we can express the reguar soution on the interva 0,r min using the WKB approximation (see, for exampe, Ref. 8), viz. φ (k, r) = a exp i p(ρ)dρ, r 0,r min, (5) p(r) r where the cassica momentum p(r) is defined by p(r) k 2 V (r) ( + 2 )2 /r 2 (6) and the upper imit a in the integra is an arbitrary vaue a>r min. Usuay a is taken to be the inner turning point 8, but it is obvious from Eq. (5) that an additiona integration from a to the turning point can ony change the overa normaization of the soution which is not our concern at the moment. Thus, Eq. (5) together with the derivative r φ (k, r) = dv(r) dr ( 2 + ) 2 r 3 2 4p(r) 5/2 i p(r) exp i a r p(ρ)dρ, r (0,r min (7) 3

4 can be used as boundary conditions for the reguar soution of the Schrödinger equation at any point in the interva (0,r min. To obtain the corresponding boundary conditions for the functions F (±) (k, r), we need to express them in terms of φ (k, r) and r φ (k, r). For this we can use Eq. (5) together with reation r φ (k, r) = F (+) (k, r) r h (+) (kr)+f ( ) (k, r) r h ( ) (kr), (8) 2 which foows from (6). From (5) and (8) we find that F (±) (k, r) =± i φ (k, r) r h ( ) (kr) h ( ) (kr) r φ (k, r) k (9) which is vaid for any r 0, ). Therefore Eqs. (9) taken at some point r<r min with φ (k, r) and r φ (k, r) given by (5) and (7), provide us the boundary conditions, required in Eqs. (7), for singuar potentias. It can easiy be checked (by using j (kr) for the reguar soution near r = 0) that Eq. (9) gives the correct boundary conditions for reguar potentias as we, Aternativey to impose the boundary conditions on the functions F (±) (k, r) near the origin, one can simpy sove the Schrödinger equation from a sma r up to some intermediate point b where, using (9), the F (±) (k, b) canbe obtained and propagated further on by integrating equations (7). C. Integration path The use of more compicated boundary conditions at r = 0 does not change the condition (9) for the existence of the imit (8). Indeed, in deriving this condition we used ony the behavior of the potentia and the Riccati Hanke functions at arge distances,4. Therefore, the Jost function for a singuar potentia can aso be cacuated by evauating the function F ( ) (k, r) at a arge r. When we are deaing with resonances, i.e. working in the fourth quadrant of the k pane, we need to integrate Eqs. (7) aong the turned ray (0). As can be seen from the WKB boundary conditions (5), the use of a compex r near the origin, makes φ (k, r) osciatory from the outset. Athough this does not formay cause any probem, in numerica cacuations such osciations may reduce the accuracy. To avoid this we sove Eqs. (7) from a sma r min to some intermediate point b aong the rea axis and then perform the compex rotation, r = b + x exp(iθ), x 0, ), 0 θ< π 2, (20) as is shown in Fig.. Therefore, on the interva r min,b we can use Eqs. (7) as they are, whie beyond the point r = b these equations are transformed to x F (±) (k, b + xe iθ )= ± eiθ h ( ) (kb + kxe iθ ) 2ik + h ( ) (kb + kxe iθ )F ( ) (k, b + xe iθ ) V (b + xe iθ ) h (+) (kb + kxe iθ )F (+) (k, b + xe iθ ) Though the compex transformation (20) is different from (0), the proof of the existence of the imit (8) given in the Appendix A.2 of Ref. 4 remains appicabe here. Indeed, that proof was based on the fact that for Im kr > 0 the Riccati Hanke function h (+) (kr) decays exponentiay at arge r, and thus the derivative r F ( ) (k, r) vanishes there and the function F ( ) (k, r) becomes a constant. Under the transformation (20) the asymptotic behavior of the Riccati Hanke function, h (+) (kr) i exp i(kr π/2), (22) r has ony an additiona r independent phase factor exp(ikb) which does not affect the proof. From the above, it is cear that we can identify the Jost function f (k) as the vaue of F ( ) (k, b + xe iθ )ata sufficienty arge x beyond which this function is practicay constant. In the bound and scattering state domain,. (2) 4

5 where Imk 0, one can choose any rotation ange θ aowed by the potentia, incuding θ = 0. In the resonance domain, however, where k = k exp( iϕ), ϕ > 0, the rotation ange θ must be greater or equa to ϕ. If the condition θ ϕ is fufied, the vaue of the imit (8) does not depend on the choice of θ. This provides us with a reiabe way to check the stabiity and accuracy of the cacuations by comparing the resuts for f (k) obtained with two different vaues of θ. From Eq. (22) it is cear that the anguar momentum appears ony in the phase factor of the asymptotic behavior of the Riccati Hanke functions and hence of the reguar soution. Therefore, the use of any compex cannot change the domain of the k pane where the imit (8) exists. This means that the Jost function can be cacuated, for any compex anguar momentum, using the same equations. Moreover, when ooking for the Regge poes in the pane, the compex rotation is not necessary because these poes correspond to rea energies. Locating Regge poes as zeros of the Jost function in the compex pane is easier than by cacuating them via the S matrix using three integration paths (in the r pane) as suggested in Ref. 9. D. Jost soutions By storing the vaues of F (±) (k, r) on the integration grid one can aso obtain the reguar soution in the form (5) on the interva r min,r max. It is noted that the use of the Riccati Hanke functions in (5) guarantees the correct (in fact exact) asymptotic behavior of the wave function. The reguar soution thus obtained consists of two terms: 2 h(+) (kr)f (+) 2 h( ) (kr)f ( ) (k, r) r i 2 exp +i(kr π/2) f (k ), (k, r) r + i 2 exp i(kr π/2) f (k). Asymptoticay they behave ike e ±ikr and thus at ong distances they are proportiona to the commony used Jost soutions f (±) (k, r) for which f (±) (k, r) h (±) r (kr). (23) In practice, the Jost soutions can be cacuated, via (5), by integrating Eqs. (7) inwards from a sufficienty arge r max with the boundary conditions F (+) (k, r max ) 2 F ( ) =, for f (+) (k, r max ) 0 (k, r), F (+) (k, r max ) 0 F ( ) =, for f ( ) (k, r max ) 2 (k, r), which obviousy compy with the definition (23). The advantage of such an approach is that at arge r a the osciations of f (±) (k, r) are described exacty by the Riccati Hanke functions whie the functions F (±) (k, r) are smooth. III. LENNARD JONES POTENTIAL In order to evauate the accuracy and efficiency of our method we appy it to the Lennard Jones potentia (d ) 2 ( ) 6 d V (r) =D 2. (24) r r 5

6 which is we known in atomic and moecuar physics. Combined with a rotationa barrier, this potentia supports narrow as we as broad resonant states (see, for exampe, Ref. 8). To ocate them, any method empoyed must be pushed to the extreme, thus exhibiting its advantages and drawbacks. To be abe to compare our resuts with other cacuations, we chose the parameters in (24) to be the same as those used in Refs. 8,0, namey, d =3.56 Åand with D varying from 5 cm to 60 cm.thechoiced=60cm together with the conversion factor h 2 /2µ = cm Å 2 (which was used for a vaues of D) approximatey represents the interaction between the Ar atom and the H 2 moecue 8. In Tabes I and II the energies and widths of the first resonant states in the partia wave = 8 are presented for different vaues of D. The resuts obtained with three other methods described in Refs. 8,0 are aso given. The digits shown there are stabe under changes of the rotation ange and thus they indicate the accuracy achieved. The third coumn of these tabes, contains the resuts obtained in Ref. 8 using a Compex Rotation (CR) method which in some aspects is simiar to ours. The authors of that reference perform the rotation directy in the Schrödinger equation and integrate it from r = 0 outwards and from a arge r max inwards. At the origin they use the WKB boundary conditions and at r max they start from the Siegert spherica wave. In other words, the wave function is cacuated using physica boundary conditions. In such an approach a resonance corresponds to a compex energy which matches the inward and outward integration. As indicated in Ref. 8, this method fais for broad resonances due to instabiity in the outward integration. In the fourth coumn the resuts obtained in Ref. 8 using the Quantum Time Deay (QTD) method are cited. This method is expected to be reiabe for narrow resonances but its appicabiity to broad states is questionabe. Finay, in the ast coumn of Tabe I and II we give the resuts obtained in Ref. 0 using the Finite Range Scattering Wave (FRSW) method. The main idea of this method is based on the fact that whie the scattering wave function cannot be expanded propery by a finite number of square integrabe functions on an infinite range, it is possibe to do so for a finite range. The test cacuations show that our method works we, especiay for narrow resonances. Broad resonances can aso be ocated. In contrast to the CR method of Ref. 8, which was unstabe for broad resonances corresponding to D<35 cm, we succeeded even in the case of D =5cm which generates an extremey broad state (its width is greater than the resonance energy by a factor of 2). Our resuts for sma vaues of D, reproduce we the curve depicted in Fig. 3 of Ref. 8 which was produced semi-cassicay. The greater stabiity of the Jost function method as compared to the CR method of Ref. 8 can be attributed to the use of the ansatz (5) for the reguar soution. The Riccati Hanke functions, expicity extracted there, describe correcty a osciations at arge distances with the remaining functions F (±) being smooth. Another reason for this stabiity is the use of the deformed integration path shown on Fig., which enabes us to avoid fast osciations at short distances. IV. AZIZ POTENTIALS The mode potentia considered in the previous section, though of typica form for inter-moecuar interactions, does not describe any rea physica system. To give a more practica exampe, we appy our formaism to study the interaction between two 4 He atoms. This interaction is of interest in the Bose Einstein condensation and super-fuidity of heium at extremey ow temperatures. It is known that two heium atoms form a dimmer moecue with binding energy of mk, but, to the best of our knowedge, the possibiity of forming dimmer resonances has not been investigated yet. The search for a reaistic 4 He 4 He potentia is a ong-standing probem in moecuar physics. The eariest successfu potentia of the Lennard Jones(2-6) form was fitted just to reproduce the second viria coefficient. Later on some other characteristics of heium gas, such as viscosity, were incuded into the fitting (for a more detai review see Refs.,2). Nowadays, the potentias suggested by Aziz and co-workers are considered as be reaistic. Therefore, in this section, we appy our method using two versions of these potentias, namey, the HFDHE2 and the HFD-B 2 potentias. They can both be described using the same anaytica form V (r) =ε Aexp( αζ βζ 2 ) { exp (B/ζ ) 2, if ζ B, F (ζ) =, if ζ>b, ζ=r/r m, ( C6 ζ 6 + C 8 ζ 8 + C 0 ζ 0 ) F (ζ), (25) 6

7 but with different choices of the parameters (see Tabe III). The ony principa difference in the functiona form between them is the absence of the Gaussian term (β = 0) in the HFDHE2 potentia. Formay, the HFDHE2 and HFD-B are reguar potentias since the presence of the cut-off function F (ζ) in (25) makes them finite at r =0, V(r) r 0 εa. (26) However, the product εa is very arge ( 0 6 ) as compared with the vaues of the potentia in the attractive region. This causes numerica instabiities when one tries to sove the Schrödinger equation using methods designed for reguar potentias. To avoid this difficuty, we notice that ike in the case of singuar potentias the fast growth of the repusion near the origin aows the use of the WKB boundary conditions near r = 0. Indeed, the derivative of the potentia in the vicinity of this point, dv dr αεa, (27) r 0 r m is of the same order of magnitude as V, which makes the derivative of the oca waveength (4) very sma because of the arge A, dλ/dr r 0 α 2r m 2µ h 2εA, (28) where the conversion factor h 2 /2µ =2.2 KÅ 2, corresponding to the choice of the units in Tabe III, shoud be used. With the parameters given, formua (28) gives and for the potentias HFDHE2 and HFD-B respectivey. These vaues of dλ/dr are sma enough to compy with (2) and aow the use of WKB boundary conditions. We can, therefore, appy the method described in the preceding sections, to the potentias HFDHE2 and HFD-B as if they were singuar potentias. To begin with, we tested the abiity of our method to dea with this kind of potentias by cacuating the dimmer binding energy. The resuts of these cacuations are given in Tabe IV where, for comparison, we aso cite the binding energies obtained in severa earier works. It is seen that the potentias HFDHE2 and HFD-B support a dimmer bound state at energies which differ by a factor of 2. A question then arises whether these potentias generate aso quite different distribution of resonances which woud resut in different on and off the energy she characteristics of the scattering ampitude. To study this we ocated severa zeros of the Jost function in the momentum as we as in the pane (Regge poes) for both potentias. Due to the absence of a potentia barrier there are no resonances in the S wave (at east with a reasonaby sma width). They appear, however, at higher partia waves, starting from =. The energies and widths of severa such resonances are given in Tabe V. They are the owest resonant states in each partia wave as they beong to the same Regge trajectory which starts from the ground state. The trajectories for the potentias HFDHE2 and HFD-B are practicay indistinguishabe and are shown in Fig. 2 by a singe curve. Few points of this curve which correspond to resonances, are aso given in Tabe VI. It is seen that, to a practica purposes, the position of the Regge poes are the same. As can be seen in Tabe V, in each partia wave the potentia generates a broad resonance which covers the whoe ow energy region. This, together with the fact that the bound state poe of the ampitude is very cose to k =0, impies that the cross section at energies 0 K( 0 3 ev) must be quite arge. V. CONCLUSIONS We presented an exact method for cacuating the Jost soutions and the Jost function for singuar potentias, for rea or compex momenta of physica interest. We demonstrated in the exampes considered, the suggested method is sufficienty stabe and effective not ony in the case of true singuar potentias but aso when a potentia has strong, though finite, repusion at short distances. 7

8 The method is based on simpe differentia equations of the first order, which can be easiy soved numericay. Thus, the spectrum generated by any given potentia can be thoroughy investigated. At the same time, physica wave function can be obtained having the correct asymptotic behavior. When the potentia has a Couomb tai one can simpy repace the Riccati Hanke functions in the reevant equations by their Couomb anaogous, H (±) (η, kr) F (η, kr) ig (η, kr) 2. In the case of a non-centra potentia the Jost function as we as the differentia equations assume a matrix form with somewhat more compicated, but sti tractabe boundary conditions at r = 0 4. The method is aso appicabe when the anguar momentum is compex. This enabes us to ocate Regge trajectories as we. This coud be usefu, for exampe, in moecuar scattering probems where the partia wave series in many cases converges sowy 7. This sow convergence can be overcome by aowing the anguar momentum to become compex vaued which aows the use of the Watson transformation. However, such a procedure requires the knowedge of the positions of the Regge poes. ACKNOWLEDGEMENTS Financia support from the University of South Africa, the Foundation for Research Deveopment (FRD) of South Africa, and the Joint Institute for Nucear Research (JINR), Dubna, is greaty appreciated. 8

9 S. A Rakityansky, S. A. Sofianos, and K. Amos, Nuovo Cimento B, 363 (996). 2 S. A. Sofianos and S. A. Rakityansky, J. Phys. A: Math. Gen. 30, 3725 (997). 3 S. A. Sofianos, S. A. Rakityansky, and G. P. Vermaak, J. Phys. G: Nuc. Part. Phys. 23, 69 (997). 4 S. A Rakityansky and S. A. Sofianos, J. Phys. A: Math. Gen. 3, 549 (998). 5 W. M. Frank and D. J. Land, Rev. Mod. Phys. 43, 36 (97). 6 L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 965). 7 R. G. Newton, Scattering Theory of Waves and Partices, 2nd ed. (Springer, New York, 982). 8 J. N. L. Connor and A. D. Smith, J. Chem. Phys., 78, 66 (983). 9 C. V. Sukumar and J. N. Bardsey, J. Phys. B 8, 568 (975). 0 H. W. Jang and J. C. Light, J. Chem. Phys. 99, 057 (993). R. A. Aziz, V. P. Nain, J. S. Carey, W. L. Tayor, G. T. McConvie, J. Chem. Phys., 70, 4330 (979). 2 R. A. Aziz et a., Mo. Phys., 6, 487 (987). 3 E. A. Koganova, A. K. Motoviov, and S. A. Sofianos, J. Phys. B: At. Mo. Opt. Phys., 3, 279 (998). 4 T. Corneius and W. Göcke, J. Chem. Phys., 85, 3906 (986). 5 S. Nakaichi-Maeda and T. K. Lim, Phys. Rev. A, 28, 692 (983). 6 Y. H. Uang and W. C. Stwaey, J. Chem. Phys. 76, 5069 (982). 7 J. N. L. Connor, J. Chem. Soc. Faraday Trans. 86, 627 (990). 9

10 Ref. This work CR 8 QTD 8 FRSW 0 D (cm ) E res (cm ) E res (cm ) E res (cm ) E res (cm ) TABLE I. Energies of the owest resonances, in the = 8 partia wave, for the Lennard Jones potentia with different D. Ref. This work CR 8 QTD 8 FRSW 0 D (cm ) Γ res (cm ) Γ res (cm ) Γ res (cm ) Γ res (cm ) TABLE II. Widths of the owest resonances, in the = 8 partia wave, for the Lennard Jones potentia with different D. parameter HFDHE2 HFD-B ε (K) r m (Å) A α β C C C B TABLE III. Parameters of the two versions of the Aziz 4 He 4 He potentia. 0

11 4 He- 4 He binding energy (mk) Ref. HFDHE2 HFD-B This work TABLE IV. Binding energies of 4 He 2 di-atomic moecue for the two versions of the Aziz potentia. HFDHE2 HFD-B E (K) Γ(K) E(K) Γ(K) TABLE V. Energies and widths of the owest resonant states generated by the two versions of the Aziz potentia in severa partia waves. HFDHE2 HFD-B E (K) E (K) i i i i i i i i i i.584 TABLE VI. Regge poes corresponding to resonances generated by the two versions of the Aziz potentia.

12 Im r b Re r FIG.. Deformed contour for integration of the differentia equations Im ` 0 ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp s s s s s s p Re ` FIG. 2. Regge trajectory for the HFD-B potentia. Fied circes indicate bound and resonant states. 2