Jost function formalism with complex potential

The Jost function formalism is extended with use of the complex potential in this paper. We derive the Jost function by taking into account the dual state which is defined by the complex conjugate the complex Hamiltonian. By using the unitarity of the S-matrix which is defined by the Jost function, the optical theorem with the complex potential is also derived. The role of the imaginary part of the complex potential for both the bound states and the scattering states is figured out. The numerical calculation is performed by using the complex Woods-Saxon potential, and some numerical results are demonstrated to confirmed the properties of extended Jost function formalism.


Introduction
The optical model has been used to analyze and successfully reproduced the experimental cross section of the nucleon-nucleus (NA) scattering at higher energy region [1].As is well known, the optical potential is given by the complex potential, and the imaginary part of the optical potential is important for the quantitative reproduction of the experimental data because it has the absorptive effect to the incident nucleon flux on the NA scattering as written in many papers and textbooks.According to the Feshbach projection theory [2], the origin of the complex optical potential is due to the channels coupling at the intermediate states of the NA-scattering.
Nevertheless, the optical potential parameters are given phenomenological so as to reproduce the experimental data [3,4].
The contribution of the optical potential can be interpreted as the mean contribution from the coupling of many channels at higher incident incident energy of NA scattering.At low energy, the basic shape of the cross section is characterized by the individual contribution of shape resonances.
As expected from the R-matrix theory [9,10,11] and results and analysis by continuum PVC [5], the channel coupling effect will play crucial role for the production of the sharp peaks of resonances.The absorptive effect of optical potential becomes smaller.This is the fact which can be expected from the absolute value of the reaction cross section (which is defined by the difference of the total and elastic cross section).
The important role of the imaginary part of the complex potential is not only for the nuclear reaction.The role of the complex potential for the nuclear structure has been discussed.In order to analyze the PVC effect on the single particle levels, the experimentally determined spectroscopic factor has been analyzed by comparing with the theoretically calculated level density [12].It was confirmed that the experimental spectroscopic factors are reasonably well reproduced in both senses; the position of the centroid energies and the fragmentation.This is another aspect of the complex self-energy function (complex optical potential)'s effect.
The Jost function is one of the useful tool to investigate the role of the potential for bound states, resonances and scattering states, because it is possible to calculate all of those states as the poles of the S-matrix on the complex energy/momentum plane directly from the potential.Recently, we extended the Jost function formalism based on the Hartree-Fock-Bogoliubov (HFB) formalism in order to take into account the pairing [13].This extension of the Jost function is a kind of the extension to the multi-channel system because the HFB is also a kind of two channel system in a broad sense.However, the potential in the Jost function framework is still supposed to be real.Therefore, we shall extend the Jost function framework with the complex potential in this paper.

Jost function with the complex potential
When the potential is given by the complex, the hermiticity of the Hamiltonian is broken.As is well-known, the dual Hilbert space which is defined by the complex conjugate of the Hamiltonian  * exists if the hermiticity of the Hamiltonian is broken.In order to derive the Jost function with the complex potential, it is necessary to take into account the dual space.

Derivation of the Jost function
In this section, we shall derive the Jost function with the complex potential.
The Jost functions satisfy the following symmetric properties. and

Bound state
The boundary condition for a bound state   = (  ) is given by From Eqs.( 24)-( 26), we can obtain Therefore we can find the relation between   and  ̃ as The relation between the energy eigen value for the dual state ( ̃) and (  ) is given by It is very easy to rove that the orthogonality of the bound state wave function is given by By dividing the potential into two parts, real and imaginary parts as and applying the two potential formula to the where  0, (±) (, ′; ) is the Green's function which satisfies the following equation Using the spectral representation of  0, (±) (, ′; ) and   (±) (, ′; ) in Eq.( 33), we can obtain where   0 is the bound state wave function However, is required due to Eq.( 29).Therefore, we can find that   for the real potential becomes pure imaginary, and the energy eigen value becomes real number.

Scattering state and optical theorem
The scattering wave function   (+) (; ) has the asymptotic behavior with the S-matrix   () .The scattering wave function as the dual state of Eq.(37) has the asymptotic behavior with the S-matrix  ̃ () .By concerning the asymptotic behavior of Eqs.( 4) and ( 5), we can find that   (+) (; ) and  ̃ (+) (; ) are defined by and The S-matrix   () and  ̃ () are defined by From Eqs.( 24)-( 26), we can derive the following properties of the S-matrix.
therefore we can obtain The scattering states (continuum) is defined on the real axis of the complex momentum plane.
Therefore the Unitarity of the S-matrix in the complex potential system is given by when  * =  ( i.e.  is real).
The T-matrix is defined by and Using the Green's function, which is written in many textbook as the free particle Green's function, defined by: Also, the Green's theorem leads the another types of equations: It should be noted that  ̃ (+) (; ) =   (+) (; ) when the potential is real ( i.e.   =   * ).
From the limit  → ∞ of Eq.( 72), we can obtain By inserting Eq.( 73) into Eq.(53), the generalized optical theorem is rewritten as when  is real.This is the optical theorem with the complex potential.The left hand side of Eq.( 74
Note that Figs. 3 and 4 are corresponding to the upper-half and lower-half of the complex- plane shown in Fig. 1, respectively.One can see the discontinuity between the first and fourth quadrant across the branch-cut which is defined by the real axis of the complex energy plane in the positive region in both Figs. 3 and 4.
The analytic continuation of  (1) -and  (2) plane is shown in Fig. 5.The first quadrant of Fig. 3 is connected with the fourth quadrant of Fig. 4, and the first quadrant of Fig. 4 is connected with the fourth quadrant of Fig. 3.This is due to the regularity of the Jost function on the complex momentum plane.
In Table .1,we show the numerical results for the single particle levels with the real potential and complex potential obtained by searching the zeros of the Jost function on the complex momentum plane.The imaginary part of the single particle levels Im   obtained by using the complex potential are given by negative values.This is consistent with the approximated formula given by Eq.( 35) since the imaginary part of the potential is given by the negative value in this study.The real part of the single particle levels are slightly shifted to higher energy due to the effect of the imaginary part of the potential.As is written in many textbooks, the generalized optical theorem which includes the absorption as the effect of the imaginary part of the complex potential is rather well known.In order to confirm our derivation of the Jost function, firstly we derived the generalized unitarity of the Smatrix by using the Jost function which is calculated by the complex potential.Using the generalized unitarity of the S-matrix, we derived the generalized optical theorem.In order to confirm the effect of the imaginary part of the complex potential to the bound states, we derived an approximated formula which shows the effect of the imaginary part of the potential to the single particle levels by using the Green's function method.And we confirmed that the numerical results for the single particle levels obtained by using the Jost function are consistent with the derived formula.
The results of Table 1 are not qualitatively consistent with the results of the previous cPVC calculation.According to the discussion and results in [12], the complex potential which is calculated as the self-energy function within the PVC has the effect to shift the single particle levels to lower energy, and provides large fragmentation to the single particle levels far from the Fermi level.
As shown in [5,6,7,8], the PVC self-energy function works very well as the microscopic optical potential also for the description of the scattering cross section.
For the quantitative reproduction of the experimental data of the -scattering, the global optical potential has adopted the complex Woods-Saxon form with the energy dependence.The energy dependence has been given by adjusting the experimental data, but the energy dependence has been given only for the positive energy region, except the Dispersive Optical Potential [14].In order to obtain the proper interpretation of physics from the analysis of the experimental data using the phenomenological optical potential, the optical potential should be available for both the nuclear structure (single particle levels and their fragmentation and so on) and the nuclear reaction.
The Jost function may be a powerful tool to construct such a new type of the optical potential.
According to the Feshbach projection theory [2], the origin of the complex potential is the coupling of channels, and the channel-coupling equation can be reduced to the single channel problem by introducing the complex optical potential.As we showed in this paper, two kinds of the Riemann sheets are defined for two types of the Jost function with the complex potential.On the other hand, the multiple Riemann sheets (more than two, depending on the number of channels) are expected for the channel-coupling equation (The HFB framework is also a kind of the channelcoupling method of two channels in a broad sense, and three types of the Riemann sheets are defined within the HFB framework [13]).Clarifying the relationship between the Riemann surface defined Green's function, we can obtain the Dyson equation   (±) (, ′; ) =  0, (±) (, ′; ) + ∫ ∞ 0 ′′ 0, (±) (, ′′; )(−   (′′))  (±) (′′, ′; )

Fig. 5 .
Fig. 5. (Color online) The analytic continuation between the first and second Riemann sheets.In the left panel, the upper-half of Fig.3 and the lower-half of Fig.4 are shown.In the right panel, the upper-half of Fig.4 and the lower-half of Fig.3 are shown

4 Conclution
In this study, we extended the Jost function formalism based on the complex potential.Since the Jost function is defined as the coefficient function to connect the regular and irregular solutions of the Schrodinger equation, the Jost function can be derived by finding the relation between the regular and irregular solutions by using the Green's theorem with the proper boundary conditions for each solutions.In the system defined by the Hamiltonian  with the complex potential, two kinds of the Jost function are defined, the Jost function for the system defined by  and the Jost function for the dual system defined by  * .In order to make sure our derivation, we derived the symmetric properties of the Jost function and confirmed them by the numerical results represented on the complex energy/momentum plane.

Table 1 .
Single particle levels for the bound neutrons for the real potential and complex potential, respectively.