Multichannel phaseequivalent transformation and supersymmetry
Abstract
Phaseequivalent transformation of local interaction is generalized to the multichannel case. Generally, the transformation does not change the number of the bound states in the system and their energies. However, with a special choice of the parameters, the transformation removes one of the bound states and is equivalent to the multichannel supersymmetry transformation recently suggested in [20]. Using the transformation, it is also possible to add a bound state to the discrete spectrum of the system at a given energy if the angular momentum at least in one of the coupled channels .
I Introduction
Nucleonnucleon, nucleoncluster and clustercluster potentials are an input for various microscopic calculations of nuclear structure and reactions. Unfortunately, the exact form of the potentials describing these interactions is unknown. It is conventionally supposed that the interactions are local, that is, of course, an approximation only. However, the available scattering data and bound states properties can be fitted with approximately the same accuracy by different local potentials. For example, there is a lot of socalled realistic potentials on the market describing scattering and deuteron properties with high accuracy. More, a description of phenomenological data can be achieved with the potentials very different in structure. In particular, mesonexchange potentials like the Nijmegen one [1], are known to have a shortrange repulsive core in a triplet wave. The same highquality description of the nucleonnucleon data is provided by latest versions of Moscow potential [2, 3] that does not have a repulsive core but instead is deeplyattractive in the triplet wave at short distances and supports an additional forbidden state. The possibility of alternative description of various clustercluster and nucleoncluster interactions by means of repulsivecore and deeplyattractive potentials with forbidden states, is also wellknown (see, e.g., the discussion in [2] and references therein).
Principally it is possible to distinguish experimentally between alternative potentials studying their offshell properties in interaction with an additional particle. The simplest probe is the photon, and as it was shown in [4, 5, 6], the protonproton bremsstrahlung reaction at the energy range of 350–400 MeV can be used to discriminate between various nucleonnucleon potentials. However the reaction has not been examined experimentally in this energy range.
Another possibility is to study properties of three and four body systems bound by twobody potentials of interest. From this point of view, it looks like that we do not have at present satisfactory nucleonnucleon, clusternucleon and clustercluster potentials. It is wellknown that none of the realistic potentials provides proper binding of tritium or He. There are successful attempts in generating phenomenological threenucleon interactions tuned to fit the properties of light nuclei [7] (see also [8] and references therein). However, as it was shown in a detailed study of Picklesimer et al [9], the effect of threenucleon forces consistent with realistic twobody ones on the binding energy of the triton is canceled by effects of the virtual excitation of isobars, etc. Hence the trinucleon cannot be satisfactorily described using known realistic twobody potentials supplemented by threebody potentials consistent with them. All calculations within threebody cluster models also fail to reproduce the correct binding energy of threecluster nuclear systems with known local clustercluster and clusternucleon potentials fitted to the corresponding scattering data.
To design a potential consistent with twobody phenomenological data and providing the correct binding of fewbody systems, it seems promising to make use of phaseequivalent transformations depending on a continuous parameter(s). Some attempts in this direction have been performed using nonlocal phaseequivalent transformations. The results of these attempts are encouraging: in Ref. [10] an oversimplified potential providing a satisfactory description of wave scattering date was fitted to reproduce exactly the triton binding energy, while in Ref. [11] realistic – potentials were tuned to reproduce various He properties including the binding energy within the cluster model. The interactions suggested in Ref. [10, 11] are nonlocal. Various applications (see, e.g., [12, 13]) of local phaseequivalent transformations to fewbody problems were restricted to the supersymmetry transformation [14, 15, 16] that removes one of the bound states in a two body system. The supersymmetry transformation does not contain parameters and cannot be used for fine tuning of the interaction of interest.
A local phaseequivalent transformation which preserves the number of the bound states and depends on a continuous parameter, exists and is wellknown in the inverse scattering theory [17]. Recently the effect of this transformation on the properties of three and four nucleon systems was studied in Ref. [18] using as an example a semirealistic MalflietTjon interaction [19]. It was shown in Ref. [18] that a slight phaseequivalent modification of interaction is enough to reproduce the trinucleon binding energy and to improve simultaneously the description of fournucleon binding. However the local transformation was developed for a singlechannel case only and cannot be applied without some approximations to realistic interactions that mix triplet and partial waves. Another drawback of the transformation is that it involves a bound state wave function and thus cannot be used to modify and interactions and the interaction in all ‘nondeuteron’ partial waves.
Recently Sparenberg and Baye [20] suggested a multichannel supersymmetry transformation. We use some ideas of Ref. [20] to derive in what follows a multichannel phaseequivalent transformation which depends on continuous parameters. The transformation can be treated as a generalization both of the singlechannel phaseequivalent transformation [17] and of the multichannel supersymmetry transformation of Ref. [20]. Generally, the transformation does not change the number of the bound states in the system and their energies. However, with a special choice of the parameters, the transformation removes one of the bound states and becomes equivalent to the multichannel supersymmetry transformation suggested in [20]. If the angular momenta in all coupled channels are less than 2, a parameterdependent family of local interactions phaseequivalent to the given initial one can be constructed by means of the transformation even in the case when the system does not have a bound state. If the angular momentum at least in one of the coupled channels , the transformation can be used to add a bound state to the discrete spectrum of the system at a given energy . Having a bound state, one can construct a family of phaseequivalent potentials and afterwards remove the bound state by the supersymmetry version of the transformation. Thus, the suggested transformation can be used in a multichannel case to produce phaseequivalent interactions without any restriction on the structure of the discrete spectrum of the system. In particular, the transformation can be applied to the realistic interaction in all partial waves.
Ii General form of local multichannel phaseequivalent transformation
Multichannel scattering and bound states we describe by Schrödinger equation
(1) 
where indexes and label channels, is the energy, the Hamiltonian
(2) 
is the reduced mass, and stands for the angular momentum in the channel . We suppose that the potential (i) is Hermitian and (ii) at large distances it tends asymptotically to a diagonal constant matrix,
(3) 
where is a threshold energy in the channel . We suppose that and if .
Boundary conditions for the wave functions are
(4)  
(5) 
Except for the discussion in section III.3, we suppose that there is at least one bound state in the system at the energy . The corresponding wave function, , is supposed to be normalized,
(6) 
where denotes the complex conjugation. Of course, fits more severe boundary condition at than (5):
(7) 
We define the transformed potential as
(8) 
where
(9) 
and , and are arbitrary real parameters.
The main result of this paper can be formulated as the following statement.

The wave function
(10) fits inhomogeneous multichannel Schrödinger equation
(11) where the Hamiltonian
(12) and the quasiWronskian
(13)
We use prime to denote derivatives: .
To prove the statement, one can verify Eq. (11) by the direct calculation of using the definitions (8)–(10), (12) and (13) and other formulas given above as well as the fact that the interaction is Hermitian, . The calculation is lengthy but straightforward.
It is clear from (10) and (7) that the suggested transformation is phaseequivalent at any energy ; all the bound states supported by the initial potential are preserved by the transformation since the wave functions for the corresponding energies (including ) fit both boundary conditions (4) and (7). However, the denominator in the last term in (10) should be nonzero at any distance , and therefore one should be accurate in assigning values to arbitrary parameters , and . This requirement can be easily satisfied in a wide and continuous range of parameter values.
Iii Particular cases of the phaseequivalent transformation
iii.1 Homogeneous Schrödinger equation
Of course, we are mostly interested in phaseequivalent transformations that results in homogeneous Schrödinger equation
(14) 
instead of the inhomogeneous Schrödinger equation (11). To derive the transformation leading to Eq. (14), we can fix the parameters , , and in such a way that the r.h.s. of Eq. (11) will take zero value. The choice brings us to the equivalent (contrary to phaseequivalent) transformation that is of no interest. Thus we should search for the parameters that fit the equation
(15) 
Two obvious solutions of Eq. (15) are and . Various other solutions of Eq. (15) can be found for particular potentials . However, the nonzero finite solutions of Eq. (15) are energydependent. With the solutions of Eq. (15) we can obtain energydependent potentials phaseequivalent to the initial energyindependent potential . It may be interesting for some applications, but we shall not discuss the energydependent transformation and shall concentrate our attention on the solutions and .
The case presents a generalization of the singlechannel phaseequivalent transformation of Ref. [17]. For the bound state at the energy , the wave function obtained by means of the transformation is of the form:
(16) 
The wave function (16) is not normalized. The normalization constant can be easily calculated. The normalized bound state wave function is
(17) 
It is interesting that the components of the bound state wave function in all channels are modified by the transformation sinchronically: all the components are multiplied by the same multiplier . Nevertheless the relative weight of the components in the norm of the total multichannel wave function can be changed by the transformation.
Now let us discuss the case . The transformed wave function in this case is of the form:
(18) 
If , the functions and are orthogonal:
(19) 
With the help of (19) and (6), we can rewrite (18) as
(20) 
It is seen from (20) that the case is identical (up to the redefinition of the parameter ) to the case if . It is clear, however, that after the redefinition of the parameter , the potential obtained with becomes identical to the potential corresponding to the case . Hence the the case appears to be identical to the case at any energy including . To demonstrate this explicitly, let us examine the wave function in the case . Substituting by in (18) we obtain:
(21)  
(22) 
Replacing by and normalizing the wave function (22), we obtain the expression (17).
iii.2 Supersymmetry
Let us discuss a particular choice of parameters: , , and . The wave function in this case is
(23)  
(24) 
Equation (23) can be used at any energy while Eq. (24) is applicable only if . In the case , the wave function can be rewritten in a simpler form as
(25) 
Equation (23) is just the Eq. (4) of Ref. [20]. In Ref. [20], Sparenberg and Baye suggested a multichannel supersymmetry transformation. Thus equations (24)–(25) describe the multichannel supersymmetry transformation, or, in other words, the multichannel supersymmetry transformation is a particular case of the phaseequivalent multichannel transformation discussed in this paper that corresponds to the particular choice of the parameters. Let us discuss how it works.
It is clear from Eq. (25) that as . Hence at the energy , the wave function does not match the required boundary condition (4) at . At the same time, fits the boundary condition (7) at . Therefore it is impossible to construct another solution of the Schrödinger equation (14) consistent with both boundary conditions at the energy . As a result the phaseequivalent transformation removes the bound state at . At the same time, it is clear from (24) that for all energies , zero in the denominator arising in the limit is canceled by the zero in the numerator and the wave function (24) matches the boundary conditions at the origin and at the infinity both at once. So, the transformation in this case removes the bound state at but not any of the other bound states, while the matrix at any energy is unchanged.
Of course, the supersymmetry transformation can be also formulated in the case . It is interesting that the bound state in this case is removed by a different mechanism. Suppose and is arbitrary. The wave function at any energy in this case may be written as
(26) 
It is seen that at the wave function .
We used the boundary condition (4) to construct the supersymmetry transformation: the bound state is removed because for some particular parameter values the wave function diverges at the origin and appears to be inconsistent with (4). One can suppose that it is also possible to use the boundary condition at to remove the bound state and to construct another supersymmetry transformation. It is not so. Let us discuss the case of , and arbitrary . As is seen from (18), in this case, thus the bound state is removed. However the transformation is no more phaseequivalent. Really, at energies the last term in (18) does not vanish at and provides an additional phase shift, or, in other words, it modifies the matrix.
An example of an application of the multichannel supersymmetry transformation to the Moscow potential can be found in Ref. [20].
iii.3 Inverse supersymmetry
A transformation that adds a bound state to the discrete spectrum of the system and leaves unchanged the matrix and the energies of all bound states supported by the initial Hamiltonian, we shall refer to as inverse supersymmetry transformation.
Let us suppose that there is no bound state at the energy . By we now denote the wave function at energy that matches the boundary condition (7) at infinity but diverges at the origin as (see, e.g.^{1}^{1}1The divergence of the wave functions at the origin is derived in Ref. [21] for the singlechannel case only. However the derivation of the rule of Ref. [21] can be easily generalized to the multichannel case, at least for the potentials that do not diverge in the origin., [21]) where is the angular momentum in the channel .
Having , we can use our transformation to obtain the homogeneous Schrödinger equation (14) in the case . The transformed wave function is given by (18). It is seen from (18) that does not diverge in the origin and matches the boundary conditions both at the origin and at infinity, at any energy . For , the transformed wave function is given by (21). It is clear that at the origin is proportional to where . Hence, matches the boundary condition (4) if and is not consistent with (4) if . Therefore our transformation with irregular at the origin, is the inverse supersymmetry transformation in the case . In the case the transformation appears to be a phaseequivalent transformation that does not make use of the bound state and can be applied to the system that does not support a bound state. If the transformation is applied to the free Hamiltonian with in the or partial wave, it produces a nonzero ‘transparent’ potential that provides phase shift at any energy . The multichannel version of the transformation couples and partial waves to produce a twochannel ‘transparent’ interaction that provides the matrix of the form .
It is interesting that the inverse supersymmetry transformation is not unique: we have three parameters , and that provide a family of inverse supersymmetry partner potentials. Contrary to it, the supersymmetry transformation is unique; however, it can be used in combination with the phaseequivalent transformation to construct a family of potentials phaseequivalent to the initial one but not supporting one of the bound state.
The multichannel inverse supersymmetry transformation is discussed in more detail in a very recent paper of Leeb et al [22]. One can find in this paper examples of the applications of the transformation to realistic potentials. This transformation is discussed in Ref. [23], too; in particular the authors of Ref. [23] also conclude that it is possible to create a new bound state by means of the phaseequivalent transformation only in the case .
Iv Conclusions
We derived a multichannel phaseequivalent transformation that can be used without restrictions on the structure of the discrete spectrum of the system in various scattering problems like scattering, nucleoncluster or clustercluster scattering. The multichannel supersymmetry and inverse supersymmetry transformations appear to be particular cases of the suggested general phaseequivalent transformation corresponding to particular choices of the parameter values. The inverse supersymmetry transformation is possible if only the orbital angular momentum at least in one of the coupled channels. It is interesting to note that from the point of view of the system, this means that a deep attractive potential supporting an additional forbidden state like Moscow potential, can be constructed by the inverse supersymmetry transformation of the realistic mesonexchange potential with repulsive core only due to the wave admixture in the deuteron wave function.
With the help of the suggested transformation, one can construct a family of phaseequivalent potentials depending on continuous parameters. Such families may be very useful for fine tuning of the interaction aimed to fit not only twobody observables but also three and fewbody ones. If the system has at least one bound state, the phaseequivalent potential family is constructed using directly formulas (8) and (9). One can construct phaseequivalent single or multichannel potential families also in the case when there are no bound states in the system: if all channel orbital angular momenta , one can apply directly the transformation with the irregular function ; if at least one of the channel orbital angular momenta , one can produce a bound state using inverse supersymmetry at the first stage and remove the bound state at the last stage with the help of supersymmetry version of the transformation. So, one can, for example, construct a family of phaseequivalent potentials for any combination of coupled partial waves in the system.
We hope that the suggested transformation will be useful in
various fewbody applications.
Acknowledgements. We are thankful to A. I. Mazur, A. Mondragon, V. N. Pomerantsev, D. L. Pursey, Yu. F. Smirnov, J. P. Vary, T. A. Weber, and B. N. Zakhar’ev for stimulating discussions. The work was supported in part by the State Program “Universities of Russia”, project No 992306 and by the Competitive Center at St. Petersburg State University.
References
 [1] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen, and J. J. de Swart, Phys. Rev. C 49, 2950 (1994).
 [2] V. I. Kukulin and V. N. Pomerantsev, Progr. Theor. Phys. 88, 159 (1992).
 [3] V. I. Kukulin, V. N. Pomerantsev, A. Faessler, A. J. Buchmann, and E. M. Tursunov, Phys. Rev. C57, 535 (1998).
 [4] V. G. Neudatchin, N. A. Khokhlov, A. M. Shirokov, and V. A. Knyr, Yad. Fiz. 60, 1086 (1997) [Phys. At. Nucl. 60, 971 (1997)].
 [5] A. M. Shirokov, in: Proc. XIth Int. Workshop on Quantum Field Theory and High Energy Phys. (ed. B. B. Levtchenko), p. 397 (Moscow, 1997).
 [6] N. A. Khokhlov, V. A. Knyr, V. G. Neudatchin, and A. M. Shirokov, Nucl. Phys. A 629, 218 (1998); Phys. Rev. C 62, 054003 (2000).
 [7] B. S. Pudliner, V. R. Pandharipande, J. Carlson, S. C. Pieper, and R. B. Wiringa, Phys. Rev. C 56, 1720 (1997).
 [8] R. B. Wiringa, Nucl. Phys. A 631, 70c (1998)
 [9] A. Picklesimer, R. A. Rice, and R. Brandenburg. Phys. Rev. Lett. 68, 1484 (1992); Phys. Rev. C 45, 547 (1992); 2045 (1992); 2624 (1992); 46, 1178 (1992).
 [10] A. M. Shirokov, Yu. F. Smirnov, and S. A. Zaytsev, Revista Mex. Fis. 40, Supl. 1, 74 (1994).
 [11] Yu. A. Lurie and A. M. Shirokov, Izv. RAN, Ser. Fiz. 61, 2121 (1997) [Bul. Rus. Acad. Sci., Phys. Ser. 61, 1665 (1997)].
 [12] E. Garrido, D. V. Fedorov, and A. S. Jensen, Nucl. Phys. A 650, 247 (1999).
 [13] H. Fiedeldey, S. A. Sofianos, A. Papastylianos, K. Amos, and L. J. Allen, Phys. Rev. C 42, 411 (1990).
 [14] A. A. Andrianov, N. V. Borisov, and M. V. Ioffe, Phys. Lett. A 105, 19 (1984).
 [15] C. V. Sukumar, J. Phys. A 18, 2937 (1985).
 [16] D. Baye, Phys. Rev. Lett. 58, 2738 (1987).
 [17] R. G. Newton, Scattering theory of waves and particles, 2nd. ed. (SpringerVerlag, New York, 1982).
 [18] A. I. Mazur, A. M. Shirokov, T. A. Weber, J. P. Vary, and D. L. Pursey, in: Proc. XIV Int. Workshop on High Energy Physics and Quantum Field Theory (QFTHEP’99) (eds. B.B.Levchenko and V.I.Savrin), p. 531 (Moscow, MSUPress, 2000).
 [19] R. A. Malfliet and T. A. Tjon, Nucl. Phys. A 127, 161 (1969); Ann. Phys. (N.Y.) 61, 425 (1970).
 [20] J. M. Sparenberg and D. Baye, Phys. Rev. Lett. 79, 3802 (1997).
 [21] L. D. Landau and E. M. Lifshitz, Quantum mechanics: nonrelativistic theory (Pergamon Press, New York, 1977).
 [22] H. Leeb, S. A. Sofianos, J.M. Sparenberg, D. Baye, nuclth/0008054 (2000).
 [23] B. N. Zakhar’ev and V. M. Chabanov, Phys. Part. Nucl. 30, 111 (1999).