Complex scaling : physics of unbound light nuclei and perspective
aa r X i v : . [ nu c l - t h ] J u l Prog. Theor. Exp. Phys. , 00000 (38 pages)DOI: 10.1093 / ptep/0000000000 Complex scaling : physics of unbound lightnuclei and perspective
Takayuki Myo and Kiyoshi Kat¯o General Education, Faculty of Engineering, Osaka Institute of Technology, Osaka535-8585, Japan ∗ E-mail: [email protected] Research Center for Nuclear Physics (RCNP), Osaka University, Ibaraki, Osaka567-0047, Japan Nuclear Reaction Data Centre, Faculty of Science, Hokkaido University, Sapporo060-0810, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The complex scaling method (CSM) is one of the most powerful methods of describ-ing the resonances with complex energy eigenstates, based on non-Hermitian quantummechanics. We present the basic application of CSM to the properties of the unboundphenomena of light nuclei. In particular, we focus on many-body resonant and non-resonant continuum states observed in unstable nuclei. We also investigate the contin-uum level density (CLD) in the scattering problem in terms of the Green’s function withCSM. We discuss the explicit effects of resonant and non-resonant contributions in CLDand transition strength functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index D10, D11, D13
1. Introduction
The physics of resonance is closely related to non-Hermitian physics, because resonances arecomplex energy states. At the beginning of the history of quantum mechanics, a complexenergy state was introduced by Gamow [1, 2] to explain the α decay phenomena of nuclei,which emit α particles exponentially, e − λt , in terms of time. The real and imaginary partsof the complex energy E = E α − i Γ α / α particle, respectively. He also explained that the decay constant, λ = Γ α / ~ , canbe evaluated by a tunneling effect of the α particle with the energy E α for a potentialbarrier by the repulsive Coulomb force between the α particle and the daughter nucleus.This description of the decay process is based on a time-dependent picture of the wavefunction. On the other hand, Siegert [3] described such a decaying state as a resonant statedescribed by a purely outgoing wave at large distances. The description by Siegert is based ona stationary-state picture, and then the complex energy state can be obtained as a solutionof the time-independent Schr¨odinger equation under the boundary condition for outgoingwaves.Since then, resonances have been extensively investigated in quantum scattering theory[4, 5], where the real and imaginary parts of the complex energy, corresponding to the reso-nance energy and width, respectively, are theoretically calculated as a pole of the scattering c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. atrix ( S -matrix). However, it is difficult for such conventional methods to treat many-bodyresonances and non-resonant continuum states. Here, we refer to states decaying into morethan two-body constituents as “many-body resonances.”Recent experimental developments in the field of unstable nuclear physics, starting fromthe discovery of the neutron halo structure in neutron-rich nuclei such as He and Li,have shown various interesting phenomena related to the unbound states of nuclei [6, 7]. Inunstable nuclei, a few extra nucleons are bound to the system with small binding energies,in comparison with the typical value of 8 MeV per nucleon for stable nuclei, and they canbe easily released by adding small energies from the outside. This, in turn, implies thatthe position of the lowest threshold is very close to the ground state and that the couplingeffect of the continuum states becomes important even in the ground state. Such a propertyof unstable nuclei is quite different from that of stable nuclei [8]. One of the interestingfeatures of unstable nuclei is also their so-called Borromean nature of a three-cluster system,in which no two-body subsystem has the bound states. The Borromean nucleus, which hassuch a feature, has a few bound states and the lowest threshold for a three-body emission, notfor a two-body one. Most of the excited states in unstable nuclei are unbound and they areobserved as “many-body resonances.” A new approach to many-body resonances, therefore,is very desirable in unstable nuclear physics.A significant development in the treatment of resonances from two-body systems to many-body systems has been brought about via the complex scaling method (CSM) [9–13]. Thewave function of a resonant state with complex energy diverges exponentially at large dis-tances, but such singular behaviour disappears by complex scaling for the resonance wavefunction. In CSM, then, annealed resonance wave functions can be described by square-integrable functions in the same way as bound states. Employing an appropriate basis setof square-integrable functions, we can directly obtain many-body resonant states within theeigenvalue problem. As eigenvalue solutions, in addition to resonances, many-body contin-uum states are obtained on rotated continuum lines (so-called “2 θ ” lines) in the complexenergy plane. It is expected, therefore, that CSM can play an important role not only ininvestigations of the problems of many-body resonances but also in the description of thenon-resonant many-body continuum states.The aim of this article is to provide a brief review of CSM in nuclear physics and itsapplications to many-body resonant and continuum states of light unstable nuclei. For thispurpose, we explain the basic properties of CSM following the studies done by the Hokkaidogroup [15–18].In CSM, the many-body Hamiltonian H and wave functions Ψ are transformed by thecomplex scaling with a parameter θ . We take up several properties of the transformed H ( θ )and Ψ( θ ). The complex-scaled wave functions become non-singular and their matrix elementsturn out to be finite, while the original resonance wave functions diverge asymptotically andtheir matrix elements are infinite. The eigenvalues of H ( θ ) are classified into bound, resonant,and continuum states, and these eigenstates are considered to construct a completenessrelation (called an extended completeness relation). Continuum states are often discretizedand obtained on the 2 θ lines. They are, furthermore, classified into various kinds of continuumstates, such as two-body and three-body continuum states starting from different thresholdenergies. Using the eigenvalues and eigenstates, the complex-scaled Green’s function can be onstructed [12, 14], which is very useful in the calculation of various physical quantities forbound and unbound states, especially concerning the strength functions with respect to theexternal forces.Although the energies of continuum states are discretized with a finite number of basisfunctions, we can obtain a continuum spectrum distribution by projection of their discretizedcomplex energies on the 2 θ lines onto the real energy axis. By taking appropriate numbersof basis functions for a given θ parameter, we can confirm that the continuum level density(CLD) is independent of the value of θ . Using CLD, we show that the scattering phase shiftcan be expressed in a spectral representation. More generally, the Lippmann–Schwingerequation is derived in CSM including many-body scattering contributions explicitly. Thisequation in CSM is called the complex-scaled Lippmann–Schwinger (CSLS) equation and isvery promising in the study of many-body reaction cross sections.After explaining these properties of CSM, we show powerful applications to many inter-esting problems in unstable nuclear physics. In our many applications, we employ Gaussianbasis functions [19–21] because of their flexibility, simplicity, and high accuracy in many-body problems. In the next section (Sect. 2), the CSM and the framework of our approachare explained. In Sect. 3, illustrative applications of CSM are shown for two-body systems.Further, we present applications to light nuclei as nuclear many-body systems in Sect. 4,and some results using the CSLS equation. in Sect. 5. Finally, a summary and perspectiveare given in Sect. 6.
2. The complex scaling method and the present framework
We briefly explain the complex scaling method proposed by Aguilar, Balslev, and Combes[9, 10]. They introduced the transformation (dilation) operator U ( θ ) with a scaling parameter θ for the radial coordinates r i and momentum k i of all particles i = 1 , . . . , n (where n is thenumber of particles) as U ( θ ) r i U − ( θ ) = r i e iθ , U ( θ ) k i U − ( θ ) = k i e − iθ , (1)where U ( θ ) U − ( θ ) = 1. The Schr¨odinger equation for the many-body wave function Ψ isgiven as H Ψ( r , . . . , r n ) = E Ψ( r , . . . , r n ) , (2)where the Hamiltonian H subtracting the center-of-mass kinetic energy is defined as H = n X i =1 T i − T c . m . + n X i Schematic eigenvalue distribution of the complex-scaled Hamiltonian H ( θ ) for asingle-channel system. Continuum states are discretized on the 2 θ line, denoted as solidcircles, in the finite basis function method. re negative real values and independent of θ . The complex energies E r = E res r − i Γ r / θ line as shown in Fig. 1, are also independent of θ , but the number N θr depends on θ because of the condition tan − (Γ r / E res r ) < θ for E r = E res r − i Γ r / 2. Thediscretized energies E c ( θ ) of continuum states, which are obtained on the 2 θ lines, are θ dependent and expressed as E c ( θ ) = ǫ rc − iǫ ic = | E c ( θ ) | e − iθ .In calculating the matrix elements of the operators ˆ O , it is necessary to take notice ofthe non-Hermiticity of the complex-scaled Hamiltonian H ( θ ). For an eigensolution Ψ θ ( k )of Eq. (4) with momentum k , we employ its conjugate solution e Ψ θ ( k ) = Ψ θ ( − k ∗ ) in thebiorthogonal state [13, 16, 22, 23]. This choice is necessary for the eigenstates belonging tothe complex energy eigenvalues. With the biorthogonal solution, the matrix elements areexpressed as h e Φ( k ) | ˆ O | Ψ( k ′ ) i = h U ( θ ) e Φ( k ) | U ( θ ) ˆ OU − ( θ ) | U ( θ )Ψ( k ′ ) i = h e Φ θ ( k ) | ˆ O θ | Ψ θ ( k ′ ) i , (10)ˆ O θ = U ( θ ) ˆ OU − ( θ ) . Using the solutions of the eigenvalue problem in Eq. (8), the matrix elements are alsocalculated: h e Ψ θα | ˆ O θ | Ψ θβ i = N X i,j =1 c αi ( θ ) c βj ( θ ) h ¯ u i | ˆ O θ | ¯ u j i . (11)It is noted that the complex conjugate is not taken for the expansion coefficient c αi ( θ ), namelythe radial part of e Ψ θα due to the biorthogonal property [22, 24]. In standard quantum mechanics without complex scaling, bound and scattering (contin-uum) states form a complete set that is represented by the completeness relation with realeigenenergies (momenta) of the Hamiltonian H [25] : = N b X b =1 | Ψ b ih Ψ b | + Z ∞ dE | Ψ E ih Ψ E | = N b X b =1 | Ψ b ih Ψ b | + Z ∞−∞ dk | Ψ k ih Ψ k | , (12)where Ψ b and Ψ E (Ψ k ) are bound and continuum states, respectively. The continuum states(Ψ − k , Ψ k ) in the momentum representation belong to the states on the real k axis. Therefore,integration over the real k axis corresponds to that along the rims of the cut of the firstRiemann sheet of the energy plane, as shown in Fig. 2 (a). In the case of a potential problem,the mathematical proof of the completeness relation in Eq. (12) was given by Newton [25]using the Cauchy’s theorem.In CSM, the momentum axis is rotated down by θ , and the poles ( r , r , . . . , r N θr ) ofresonances can enter the semicircle for the Cauchy integration shown in Fig. 2 (b). Then,the resonances are explicitly included in the completeness relation of the complex-scaled ig. 2 Cauchy integral contours in the momentum and energy planes for the completenessrelation; (a) without CSM ( θ = 0) and (b) with CSM ( θ > b , b and r , r are the poles of bound and resonant states, respectively.Hamiltonian H ( θ ) as follows: = N b X b =1 | Ψ θb ih e Ψ θb | + N θr X r =1 | Ψ θr ih e Ψ θr | + Z L Eθ dE | Ψ θE ih e Ψ θE | = N b X b =1 | Ψ θb ih e Ψ θb | + N θr X r =1 | Ψ θr ih e Ψ θr | + Z L kθ dk | Ψ θk ih e Ψ θk | , (13)where Ψ θb and Ψ θr are the complex-scaled bound and resonant states, respectively. The tilde( e ) in the bra states means the biorthogonal states with respect to the ket states, h e Ψ θα | Ψ θα ′ i = δ α,α ′ , as mentioned in the previous subsection. Resonant states with number of N θr areincluded in the semicircle rotated down by θ in the momentum plane. The continuum statesΨ θE and Ψ θk are the solutions obtained on the rotated cut L Eθ (2 θ line) of the Riemann planeand on the rotated momentum axis L kθ , respectively. Hereafter, we refer to the relation inEq. (13) as the extended completeness relation (ECR) [23], which has been proven for single-and coupled-channel systems [26].In the case of eigenstates within a finite number of square-integrable basis states, theintegration for continuum states is approximated by the summation of discretized states as N b X b =1 | Ψ θb ih e Ψ θb | + N θr X r =1 | Ψ θr ih e Ψ θr | + N − N b − N θr X c =1 | Ψ θc ih e Ψ θc | ≈ . (14)Investigation has shown that the reliability of the approximation of the continuum states isconfirmed by using a sufficiently large basis number N in CSM [16, 18]. Application of ECR to the calculations of physical quantities allows us to see contributionsfrom bound, resonant, and continuum states separately. For this purpose, we explain herea spectral expansion of the Green’s function in CSM. The complex-scaled expression of theGreen’s function is G θ ( E ; r , r ′ ) = U ( θ ) G ( E ; r , r ′ ) U − ( θ ) = (cid:28) r (cid:12)(cid:12)(cid:12)(cid:12) E − H ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) r ′ (cid:29) , (15) a) Coupled-channel system (b) Borromean three-body system Fig. 3 The rotated Cauchy integral contours of (a) coupled-channel and (b) Borromeanthree-body systems on the complex energy plane in CSM.where we used the index ” θ ” instead of (+) as the superscript of the outgoing Green’sfunction and dropped the iǫ in the denominator.Applying ECR as given in Eq. (13) to Eq. (15), we have the complex-scaled Green’sfunction (CSGF) G θ ( E ; r , r ′ ) = N b X b =1 | Ψ θb ih e Ψ θb | E − E b + N θr X r =1 | Ψ θr ih e Ψ θr | E − E r + Z L Eθ dE c | Ψ θE c ih e Ψ θE c | E − E c . (16)It can be seen that the resonance terms are separated from the continuum one, whichconstructs a background to the distinguished resonant structure. The continuum term, fur-thermore, may have various structures in cases of coupled-channel and many-body systems.In these systems, various kinds of continuum states without CSM may exist degenerately onthe real positive energy axis. In CSM, they are rotated and distributed to different channelsof the 2 θ lines starting from the corresponding threshold energies as shown in Fig. 3 (a) and(b). The 2 θ line of L θ ( cb ) starts from the threshold energies of E cb on the energy axis inthe case of coupled-channel systems in Fig. 3 (a). In many-body systems, as shown in Fig. 3(b) for a Borromean three-body case, a 2 θ line ( L θ ( cr )) may start from the complex energy( E cr ), sometimes called the resonant threshold, when the subsystem is in resonant stateswith E cr .Thus, the continuum terms of the CSGF for coupled-channel and many-body systems aredivided into those on the different 2 θ line, and expressed as G θ ( E ; r , r ′ ) = N b X b =1 | Ψ θb ih e Ψ θb | E − E b + N θr X r =1 | Ψ θr ih e Ψ θr | E − E r + N cb X cb Z L cbθ dE cb | Ψ θE cb ih e Ψ θE cb | E − E cb (17)and G θ ( E ; r , r ′ ) = N b X b =1 | Ψ θb ih e Ψ θb | E − E b + N θr X r =1 | Ψ θr ih e Ψ θr | E − E r + N cb X cb Z L cbθ dE cb | Ψ θE cb ih e Ψ θE cb | E − E cb + N cr X cr Z L crθ dE cr | Ψ θE cr ih e Ψ θE cr | E − E cr , (18) espectively.When we solve the complex-scaled Schr¨odinger equation in Eq. (4) using basis functionsof a finite number N , we have the CSGF where the integration in the continuum termis replaced by the sum of the discretized continuum solutions given in Eq. (14) : for thesingle-channel case in Eq. (16), G θ ( E ; r , r ′ ) ≈ N b X b =1 | Ψ θb ih e Ψ θb | E − E b + N θr X r =1 | Ψ θr ih e Ψ θr | E − E r + N − N b − N θr X c =1 | Ψ θE c ih e Ψ θE c | E − E c . (19)In the same way, we have the CSGF for coupled-channel and many-body systems by replacingthe integration in Eqs. (17) and (18), respectively, by the sum of the discretized continuumsolutions. The density of states at the energy E is defined for the Hamiltonian H as [27] ρ ( E ) = Tr (cid:2) δ ( E − H ) (cid:3) . (20)From the relation 1 E ± − H = P (cid:20) E − H (cid:21) ∓ iπδ ( E − H ) , (21)where E ± = E ± iǫ , with ǫ being real and positive and the limit ǫ → ρ ( E ) = − π Im Tr (cid:20) E + − H (cid:21) . (22)The continuum level density (CLD) [28–30] is also defined by∆( E ) = ρ ( E ) − ρ ( E ) = − π Im (cid:2) Tr (cid:8) G ( E + ) − G ( E + ) (cid:9)(cid:3) , (23)where the Green’s functions are G ( E + ) = 1 / ( E + − H ) and G ( E + ) = 1 / ( E + − H ) for theHamiltonian H and free Hamiltonian H .This CLD describes the density of energy levels which result from the interaction with afinite range, and is related to the scattering S -matrix [27]:∆( E ) = 12 π Im ddE ln { det S ( E ) } . (24)The scattering S -matrix for a single-channel system is expressed as S ( E ) = e iδ ( E ) , where δ ( E ) is the scattering phase shift. Then, in the single-channel two-body system, we have∆( E ) = 1 π dδdE and δ ( E ) = π Z E −∞ ∆( E ′ ) dE ′ . (25)Applying CSM to the calculations of Green’s functions in CLD, we obtain∆( E ) ≈ ∆ Nθ ( E ) = − π Im N b X b =1 E − E b + iǫ + N θr X r =1 E − E res r + i Γ r / N − N b − N r X c =1 E − ǫ rc + iǫ ic − N X k =1 E − ǫ rk + iǫ ik , (26)where ǫ rk − iǫ ik ( k = 1 , . . . , N ) are the energy eigenvalues of the free Hamiltonian H . Theapproximated CLD, ∆ Nθ ( E ), may have a θ dependence, because a finite number N of basis tates are used for diagonalization of the complex-scaled Hamiltonian H ( θ ). In the calcula-tion, we adopt a sufficiently large number of N to keep numerical accuracy and to make the θ dependence negligible in the solutions. Thus, we calculate the phase shift ( δ ( E ) ≈ δ Nθ ( E ))from ∆ Nθ ( E ): δ Nθ ( E ) = Z E −∞ dE ′ N b X b =1 π δ ( E − E b ) + N θr X r =1 Γ r / E ′ − E resr ) + Γ r / N − N b − N r X c =1 ǫ ic ( E − ǫ rc ) + ( ǫ ic ) − N X k =1 ǫ ik ( E − ǫ rk ) + ( ǫ ik ) . (27)By performing the integration for every term, we obtain the spectral decomposition of thephase shift: δ Nθ ( E ) = N b π + N θr X n =1 δ r + N − N b − N θr X c =1 δ c − N X k =1 δ k , (28)where cot δ r = E resr − E Γ r / , cot δ c = ǫ rc − Eǫ ic , cot δ k = ǫ rk − Eǫ ik . (29) We often investigate the properties of a nucleus by seeing the response to a perturbation ofexternal forces. The excitation strength from the initial state to the final state with energy E is expressed in terms of the so-called response function R λ ( E ) by using the relation ofEq. (21) as S λ ( E ) = X α h e Ψ | ˆ O † λ | Ψ α ih e Ψ α | ˆ O λ | Ψ i δ ( E − E α ) = − π Im R λ ( E ) ,R λ ( E ) = Z d r d r ′ e Ψ ∗ ( r ) ˆ O † λ G ( E ; r , r ′ ) ˆ O λ Ψ ( r ′ ) , (30)where E is the energy on the real axis, and Ψ , Ψ α , and ˆ O λ are the initial and final statesand the external force field of rank λ , respectively.In CSM, the response function is expressed as R θλ ( E ) = Z d r d r ′ e Ψ ∗ θ ( r ) ˆ O † θλ G θ ( E ; r , r ′ ) ˆ O θλ Ψ θ ( r ′ ) , (31)for the complex-scaled initial and final states, Ψ θ and Ψ θα , the Hamiltonian H ( θ ) and theexternal field operator ˆ O θλ . From the decomposition of the CSGF given in Eq. (16), we havethe complex-scaled response function in the following decomposed form: R θλ ( E ) = R θλ,B ( E ) + R θλ,R ( E ) + R θλ,C ( E ) , (32) here R θλ,B ( E ) = N b X b =1 h e Ψ θ | ˆ O † θλ | Ψ θb ih e Ψ θb | ˆ O θλ | Ψ θ i E − E b ,R θλ,R ( E ) = N θr X r =1 h e Ψ θ | ˆ O † θλ | Ψ θr ih e Ψ θr | ˆ O θλ | Ψ θ i E − E r ,R θλ,C ( E ) = Z L Eθ dE c ( θ ) h e Ψ θ | ˆ O † θλ | Ψ θE c ih e Ψ θE c | ˆ O θλ | Ψ θ i E − E c ( θ ) . (33)Applying this decomposition of the response function, the complex-scaled strength function, S θλ ( E ) = − π Im R θλ ( E ), is also decomposed as S θλ ( E ) = S θλ,B ( E ) + S θλ,R ( E ) + S θλ,C ( E ) . (34)The matrix elements of the complex-scaled operator given in Eq. (10) are independentof θ [31]. While R θλ,B for bound states are θ independent, the θ dependence of R θλ,R and R θλ,C originates from N θr , L Eθ , and E c ( θ ). However, the final result of the strength function S θλ ( E ) is an observable, which is positive definite and independent of θ , i.e. S θλ ( E ) = S λ ( E ),though each term of S θλ,R ( E ) and S θλ,C ( E ) may have negative numbers. In coupled-channeland many-body systems, the continuum-state term of the CSGF is decomposed further intomany kinds of continuum states on the different 2 θ lines. Owing to such a decompositionof the final-state contributions, we can investigate which state largely contributes to theformation of the structures observed in the strength function. This is a prominent featureof CSM and is applicable to many-body unbound states such as the Borromean three-bodysystems, as shown in Sect. 4. We proposed a method of describing many-body scattering using CSM, referred to asthe complex-scaled solutions of the Lippmann–Schwinger equation (CSLS) [32–35]. In theCSLS method, the scattering states are described by combining the formal solutions of theLippmann-Schwinger equation with the CSGF defined by Eq. (15), which automaticallysatisfies the correct boundary conditions using the complex-scaled eigenstates.The formal solution of the Lippmann–Schwinger equation is expressed asΨ (+) = Φ + 1 E + − H V Φ , (35)where Φ is a solution of an asymptotic Hamiltonian H . The total Hamiltonian is also givenas H = H + V .In the CSLS method, we utilize the CSGF given in Eq. (15), which is related to thenon-scaled Green’s function G ( E, r , r ′ ) with outgoing boundary conditions as1 E + − H = G ( E, r , r ′ ) = U − ( θ ) G θ ( E, r , r ′ ) U ( θ ) . (36)Using the eigenstates Ψ θν of the complex-scaled Hamiltonian H ( θ ) with state index ν andtheir biorthogonal states e Ψ θν , we rewrite the Green’s function in Eq. (36): G ( E, r , r ′ ) = X ν Z U − ( θ ) | Ψ θν i E − E θν h e Ψ θν | U ( θ ) . (37) hus we have the outgoing scattering state as | Ψ (+) i = | Φ i + X ν Z U − ( θ ) | Ψ θν i E − E θν h e Ψ θν | U ( θ ) V | Φ i . (38)The scattering amplitude f ( k ) of a two-particle collision with energy E = ~ k µ iscalculated from the following expression [36, 37]: f ( k ) = − µ ~ k h Φ ( k ) | V | Ψ (+) ( k ) i = f Born ( k ) + f sc ( k ) , (39)where f Born ( k ) = − µ ~ k h Φ ( k ) | V | Φ ( k ) i f sc ( k ) = − µ ~ k X ν Z h Φ ( k ) | V | U − ( θ ) | Ψ θν i E − E θν h e Ψ θν | U ( θ ) V | Φ ( k ) i (40)The first term is the Born term and the second is a spectrum decomposition of the scatteringamplitude for a short-range interaction V . In the case of a long-range interaction such asthe Coulomb potential, it should be included in the Hamiltonian H and Ψ is expressed byusing the asymptotic Coulomb wave functions.In the coupled-channel case of two-body initial and final systems, the transition amplitudefrom α to β channels is f βα ( k β , k α ) = − √ µ α µ β ~ k α k β h Φ ( k α ) | V | Ψ (+) β ( k β ) i = f Born βα ( k β , k α ) + f sc βα ( k β , k α ) , (41)and f Born βα ( k β , k α ) = − √ µ α µ β ~ k α k β h Φ ( k α ) | V | Ψ ( k β ) i f sc βα ( k β , k α ) = − √ µ α µ β ~ k α k β X ν Z h Φ ( k α ) | V | U − ( θ ) | Ψ θν i E − E θν h e Ψ θν | U ( θ ) V | Φ ( k β ) i , (42)where V is a coupling interaction. 3. Illustrative application to simple two-body systems In this section we explain the characteristics of the CSM approach by showing the applicationof the present framework explained in Sect. 2 to simple two-body systems. We demonstrate the calculation of resonances using CSM. For this purpose, we employ thepotential model [23, 24, 31, 38] with the simple Hamiltonian given as H = − ∇ + V ( r ) , V ( r ) = − e − . r + 4 e − . r . (43)In this model, we calculate the J π = 0 + and 1 − states and also the dipole strength functionin CSM. The energy spectra are shown in Fig. 4; panel (a) shows the several energy levelswith the potential shape and (b) is the energy eigenvalue distribution in the complex energy lane. There is one bound state in each state and we obtain many resonances including theones with large decay widths in CSM.In Fig. 5 we show the wave function of the 0 +3 state, the resonance energy of which is locatedat the top of the potential barrier ( E r = 1 . 63 MeV, Γ = 0 . 246 MeV), as shown in Fig. 4 a).The left panel of Fig. 5 shows the wave function without CSM, i.e. obtained directly underthe divergent boundary condition at the asymptotic region. The wave function is complexand its divergent behavior is clearly confirmed. The right panel of Fig. 5 shows the wavefunction transformed in CSM with a scaling angle of 10 ◦ . The damping behavior of the wavefunction is confirmed around 10 fm in CSM.We discuss the matrix elements of resonances using the wave functions obtained in CSM.In Fig. 6 we show the calculation of the matrix element of r for the 0 +3 resonant state intwo different ways. One is that the resonance wave function without CSM is used and theconvergence factor is introduced in the numerical integration of the matrix element [39–42]. -4-3-2-1 0 1 2 3 0 1 2 3 4 5 6 7 8 90 +1 +2 +3 -1 -2 -3 (a) V (r) [ M e V ] r [fm] + - I m ( e n e r gy ) [ M e V ] Re(energy) [MeV]-15-10-5 0 -2 -1 0 1 2 3 40 + -2 -1 0 1 2 3 4 (b) - Fig. 4 a) Potential shape and low-lying energy levels for 0 +1 and 1 − states. b) Energyeigenvalues in the complex energy plane. -0.10-0.050.000.050.100.15 0 10 20 30 40 50 θ =0 ° w a v e f un c ti on [f m - / ] r [fm] RealImaginary -0.4-0.20.00.20.40.60.81.01.21.4 0 2 4 6 8 10 12 14 16 θ =10 ° w a v e f un c ti on [f m - / ] r [fm] RealImaginary Fig. 5 Wave functions of the 0 +3 resonant state in the schematic potential model; left:without CSM, and right: with CSM. r 〉 [f m ] α -1/2 [fm] 0 5 10 15 20 25 30 35 40 45RealImaginary θ c θ [deg.] Fig. 6 Matrix element of r for 0 +3 in the schematic potential model. Left: Convergencefactor method with the parameter α [39]. Right: Complex scaling method with the scalingangle θ [31]. The arrow indicates the critical angle θ c for the 0 +3 resonance.In general, the resonance wave functions diverge exponentially at an asymptotic region andthe matrix elements of resonances are difficult to calculate directly in a usual integration.This property has been discussed as being evaluated by introducing a convergence factorsuch as e − αr and taking continuation to the limit of α → h e Φ | b O | Φ i = lim α → Z e Φ ∗ ( r ) b O Φ( r ) e − αr d r , (44)where b O is an arbitrary operator acting on the resonance wave function Φ.In CSM, the above matrix element is transformed using the complex-scaled wave functionΦ θ = U ( θ )Φ and the corresponding complex-scaled operator b O θ = U ( θ ) b OU − ( θ ) as h e Φ | b O | Φ i = h e Φ θ | b O θ | Φ θ i . (45)We show the matrix element of b O = r for the 0 +3 resonance in a schematic potential casein Fig. 6 , which becomes a complex number. We display two kinds of calculation of thesquared radius of the 0 +3 resonance, with α dependence in the convergence factor method[39] and θ dependence in CSM. In CSM, we can see that both real and imaginary values of thesquared radius become independent of the scaling parameter θ after passing the critical angle θ c = 12 tan − (Γ / E r ). In practical calculations, we use a finite number of basis functions andthen have to take a θ value rather larger than θ c . We can see good agreement between theresults of CSM and the convergence factor method for both real and imaginary parts. InCSM, when we take a sufficient value of θ , we can discuss the properties of resonances fromtheir matrix elements.In CSM, we obtain the CSGF using the complex-scaled wave functions not only of res-onances but also non-resonant continuum states. This Green’s function is utilized in thecalculation of various kinds of strength functions, and also the few-body scattering prob-lems. We take the example of the calculation of the dipole strength function from 0 +1 to thescattering states of 1 − in CSM, where the transition operator is given as b O = rY (ˆ r ) / √ π [23]. We discuss the effects of resonances and non-resonant continuum states of 1 − stateson the dipole strength. For 1 − states, when the scaling angle θ is 3 ◦ we obtain one boundstate ( E = − E = 1 . − i . × − MeV ), and the residual (cid:9) I m ( E n e r gy ) [ M e V ] (cid:9) Re(Energy) [MeV] r o t a t e d - c on ti nuu m li n e ( θ = ° ) r o t a t e d - c on ti nuu m li n e ( θ = ° ) θ c =0.12 ° )2nd res. ( θ c =6.78 ° ) Fig. 7 Energy eigenvalues of the resonances of the first and second 1 − states in thecomplex energy plane with two rotated continuum lines at θ = 3 ◦ and 10 ◦ . s t r e ng t h f un c ti on [f m ] Energy [MeV] -0.50.00.51.01.52.02.5 0 1 2 3 4 ( × -4 ) res. + conti.1st res.conti. ( θ =3 ° )exact 0 1 2 3 4 5res. + conti.1st res.2nd res.conti. ( θ =10 ° )exact Fig. 8 Dipole transition strength in the schematic potential model from the 0 + groundstate to the 1 − states with CSM using θ = 3 ◦ (left) and 10 ◦ (right). The dashed and dash-dotted lines are the contributions from the first and second resonances, respectively. Thedotted lines are the continuum contributions, and the solid lines are the sum of all theterms. The open circles are the exact calculation by solving the scattering problem for 1 − states.non-resonant continuum states. From Fig. 7, the second resonance is not obtained for small θ . When θ is 10 ◦ , one new resonance ( E = 2 . − i . 49 MeV ) is decomposed from thecontinuum states.In Fig. 8, we show the dipole strength function from the 0 + ground state into the 1 − states. We compare the strengths obtained in CSM with the exact calculation (open circles),which is obtained by solving the ordinary scattering problem for 1 − states. We calculatethe distribution using the CSGF and the matrix elements of the dipole transition in CSM,which is shown in the solid line and agrees with the exact one. In the figure, a sharp peakis observed just above 1 MeV. The dashed curve denotes the contribution from the firstresonance. We conclude that the sharp peak originates from the first resonance. The left R e ( δ ( E θ ) / π ) Energy |E θ | [MeV] θ= 0°0.12°7°18°25°31°35°38°41°43° Fig. 9 Scattering phase shift of 1 − states as functions of the energy of rotated continuumstates, E θ changing the scaling angle θ .panel of Fig. 8 is the results with a small θ of 3 ◦ , and shows that the continuum contribution(dotted curve) forms a second peak around 2 MeV. The right panel of Fig. 8 shows thecase in which a large θ of 10 ◦ is used to describe the second resonance. It is found that thecontribution of the second resonance (dash-dotted curve) is a major component in makinga peak around 2 MeV, whereas the contribution of the residual continuum states (dottedcurve) is very small. From these results, it is concluded that two two major peaks of dipolestrength come from the contributions of two 1 − resonances.We now discuss the properties of the rotated continuum states in CSM. For this purpose,we calculate the scattering phase shift of 1 − states on the rotated continuum line withenergy of E θ = E e − iθ by increasing the scaling angle θ . It is interesting to see the effectof resonances on the phase shift. For the phase shift δ ℓ ( E θ ) with the partial wave ℓ on therotated continuum line, the resonances appearing in a wedge region between the real energyaxis and the rotated continuum line are treated as bound states as shown in Fig. 7. From thisproperty, the difference between the real part of δ ℓ ( E θ = 0) and that of δ ℓ ( | E θ | → ∞ ) goes to (cid:0) N b + N θr (cid:1) π , where N b and N θr are the numbers of bound and resonant states obtained withthe scaling angle θ , respectively. Rittby et al. called this property the generalized Levinsontheorem [43].In Fig. 9, we show the phase shifts for 1 − states ( ℓ = 1) with increasing θ , in which the realpart of the phase shifts is presented. We can see one bound state from Fig. 4 (b), N b = 1, andthe phase shift starts from π at E = 0 and goes to zero at E = ∞ when θ = 0. At E = 1 . θ increases and passes beyond θ c = 0 . ◦ of itsresonance value, because the resonance is decomposed from the rotated continuum states.Furthermore, a shift by a step of π is clearly seen at a large energy for every θ value, whichis chosen to be an appropriate angle between resonance positions. Thus, the asymptotic ehavior of δ ℓ ( E θ ) at large energy, certainly corresponds to the number of resonances N θr with the scaling angle θ . The continuum level density (CLD) is an important quantity in the description of quantumscattering phenomena, since it is connected with the scattering S -matrix [27, 44, 45]. TheCLD is defined as the variation of the level densities due to the interaction, and the scatteringphase shift ( S -matrix) is derived from CLD. In the analysis of the CLD, we can discuss therelation between resonance solutions and the scattering phase shifts. In this section we give anapplication of CLD in CSM, which provides the decomposition of the scattering phase shiftsinto resonance and non-resonant continuum contributions [46]. We also show the applicationof CLD to a realistic α + n system.In Ref. [48] we proposed a method to calculate CLD using CSM in terms of the discretizedenergy eigenstates. In this method, the smoothing procedure for each eigenenergy is auto-matically performed using the scaling angle θ as shown in Fig. 10 without other artificialparameters such as the Strutinsky method [29, 49]. The concept of the method is based onthe extended completeness relation [23], originally proposed by Berggren [22], for bound,resonance, and continuum states in CSM. The Green’s functions can be expressed usingECR in terms of discretized eigenvalues in CSM expanded with a finite number of basisfunctions. Because the complex-scaled Hamiltonians H θ and H θ have complex eigenvalues,singularities such as a δ -function are replaced with Lorentzian functions as shown in Fig. 10.We show an example of the level density in the schematic potential given in Eq. (43). Thelevel density ρ θN ( E ) for the 0 + state is shown in Fig. 11 using the basis number N by chang-ing θ , where we take N = 30 to get converging results. It is found that ρ θN ( E ) is smoothedby increasing θ . It has been shown that the CLD denoted by ∆ θN ( E ) = ρ θN ( E ) − ρ θ ,N ( E ) isobtained independent of the scaling angle θ in CSM [48, 50]. I m ( E n e r gy ) ρ θ N ( E n e r gy ) Re(Energy) Fig. 10 Schematic energy eigenvalue dis-tribution with complex-scaled Hamiltonianon a complex energy plane (black circles),and the contributions of each eigenstate tothe level density (solid lines). ρ N θ ( E ) [ M e V - ] Energy[MeV] θ=5 ° θ=10 ° θ=15 ° θ=20 ° Fig. 11 Level density ρ θN ( E ) with respectto the complex-scaled Hamiltonian H θ inthe schematic potential model. Several θ cases are shown. ( r a d . / ) Energy (MeV) J =0 + Nr=0NNN r=1 N r=2 NN r=3 NN r=4 N Energy (MeV) Fig. 12 Phase shifts of the schematic potential for the 0 + state and the subtraction ofthe number of N θr of resonance terms one by one.First, we show the example of the calculation of phase shifts using CLD in the schematicpotential. Using the property of CLD, we calculate the phase shifts by integration from E = 0 as δ N ( E ) = N b π + π Z E ∆ θN ( E ′ ) dE ′ . (46)Here N b is the number of bound states. In Fig. 12 we show the phase shifts of the 0 + statewith N b = 1. In order to see the explicit effect of resonances on the phase shifts, we calculatethe phase shifts subtracting the resonance term from δ N ( E ) as δ NN θr ( E ) = δ N ( E ) − N θr X r =1 Z E dE ′ Γ r / E ′ − E res r ) + Γ r / . (47)Here N θr is the number of resonances, which depends on the scaling angle θ . The resonanceenergy and decay width are given as E res r and Γ r , respectively. In Fig. 12 the results are shownfor N θr = 0 , , , , and 4. It is shown that the phase shifts go downward with increasing N θr , in a step of π at higher energies from the Levinson theorem. It is found that the effectsof the first and second resonances are remarkable on the structures of the phase shifts withenergy up to 2 MeV, but the third and fourth resonances having larger decay widths, do nothave notable effects.Next, we show the realistic example of the unbound He nucleus as a two-body α + n clustersystem. For the interaction between the α particle and n , we use the so-called microscopicKKNN interaction [51], which consists of the central and LS terms and is commonly usedin the calculation of He isotopes with the α core plus valence-neutron model in the nextsection.Using the Gaussian basis states, we calculate the energy eigenvalues of the complex-scaledHamiltonian of He with θ = 35 ◦ , and the results for the three states of 3 / − , 1 / − , and1 / + are shown in the bottom panels of Fig. 13. It is found that the 3 / − and 1 / − stateshave one resonance pole of the p -wave neutron, corresponding to the observed resonances of He. One resonance pole of the α – n system is obtained, ( E res r , Γ r ) = (0 . , . 59) MeV for p / and (2 . , . 82) MeV for p / , which agree well with the experimental data [16, 52, 53].The energy difference of two states comes from the LS force in the α + n interaction. The1 / + state has no resonance of the s -wave neutron.In addition to resonances, the discretized non-resonant continuum solutions are obtainedalong the 2 θ line as shown in Fig. 13. Several continuum solutions are off the 2 θ line. Thisis because that the couplings between resonance and non-resonant continuum states are notaccurately described due to the finite number of basis functions. However, the resonances aresolved with appropriate accuracy and the resulting CLD is obtained from these eigenstatessatisfactorily, although the positions of some continuum solutions are slightly off the 2 θ line.According to ECR for α + n system, we can decompose CLD and the phase shifts intoresonance and non-resonant components, as shown in Fig. 14 together with experimentaldata. We can see good agreement between the theoretical and experimental results. Theresonance component of the phase shift of p / increases rapidly due to the small decaywidth. Although p / has a larger width, the phase shift of p / shows clear resonancebehavior. The continuum components (dotted lines) of phase shifts for both states are verysimilar. This behavior seems due to the same p -wave scattering and a small effect of the LS force on the background states. The present α – n systems show clear resonance behavior. Wealso investigate similar analysis for the α – α two-cluster system [46]. The analysis of CLD inthe three-body system is straightforward, as for the triple- α states in C [18, 47]. δ [ d e g . ] - - + -0.6-0.4-0.20.00.20.40.60.81.01.2 ∆ θ N [ M e V - ] -5-4-3-2-10 0 2 4 6 I m ( E n e r gy ) [ M e V ] Re(Energy) [MeV] Re(Energy) [MeV] Re(Energy) [MeV] Fig. 13 Various properties of the α + n system with J π = 3 / − , / − , / + states. Upper:phase shifts, middle: continuum level density, and lower: energy eigenvalue distributions ina complex energy plane. The scaling angle θ is 35 ◦ . .51.0 d . / ) p d . / ) p (a) (b) ( r a d . / Energy (MeV) p ( r a d . / Energy (MeV) p Energy (MeV) ReE (MeV) 420 0 2 4 6 8 10 M e V ) ReE (MeV) Energy (MeV) (a') (b') p I m E ( M e V p (a') (b') p p Fig. 14 Decomposition of the scattering phase shifts of the α – n system for (a) p / and(b) p / . The dashed and dotted lines represent the contributions of the resonance andcontinuum terms, respectively. The solid lines display the total scattering phase shifts. Theexperimental data [54] are shown with open circles. . Many-body resonances and non-resonant continuum states We apply CSM to the resonances phenomena observed in unstable nuclei, in particular many-body resonances beyond the two-body case, which are located above the threshold energy ofthe many-body decay. We take the case of the structures of neutron-rich He isotopes and theirmirror proton-rich nuclei, most states of which are unbound due to the weak binding propertyof valence nucleons to the α particle. In the present nuclear model, we treat the α particleas a frozen core nucleus and solve the motions of valence neutrons/protons surrounding the α core for neutron-rich He isotopes/proton-rich cases. We explain the nuclear model describing He isotopes and their mirror nuclei on the basisof the cluster model assuming an α core. We use the so-called cluster orbital shell model(COSM) [55]. In COSM, the motion of valence nucleons around a core nucleus is solved asshown in Fig. 15, which has an analogy with the single-particle motion of the shell model.In addition, COSM has the merit of the cluster model to obtain the relative wave functionof valence nucleons with respect to the core nucleus precisely. This COSM is extendable totreat excitations of the core nucleus in terms of a multi-configuration representation [56, 57].The Hamiltonian for COSM consisting of the α particle with a mass number A c = 4 and N v valence nucleons [58–61] is H = N v X i =0 t i − T c . m . + N v X i =1 V αNi + N v X i Set of spatial coordinates in COSM consisting of a core nucleus and valencenucleons N . n COSM, the total wave function Ψ JT COSM with total mass number A = A c + N v , spin J ,and isospin T is represented by the superposition of the different configurations Φ JTc asΨ JT COSM = X c C JTc Φ JTc ( A ) , Φ JTc ( A ) = N v Y i =1 a † κ i | i , (50)where the vacuum | i represents the α particle with spin J = 0 and isospin T = 0. Thecreation operator a † κ is for the single-particle state of a valence nucleon above the α corewith the (0 s ) closed configuration of a harmonic oscillator wave function. The quantumnumber κ is a set of { n, ℓ, j, t z } , where the index n represents the different radial componentand ℓ is the orbital angular momentum of a valence nucleon. The z component of theisospin of each nucleon (proton or neutron) is given as t z . The index c represents the setof κ i as c = { κ , . . . , κ N v } for all valence nucleons, which determines the configuration ofthe COSM wave function. The expansion coefficients { C JTc } in Eq. (50) are determined bydiagonalization of the Hamiltonian matrix with complex scaling.The coordinate representation of the single-particle state a † κ is given as ψ κ ( r ) as functionsof the relative coordinate r between the center of mass of the α and a valence nucleon [55].An illustration of the coordinate set is shown in Fig. 15 up to the N v = 4 case. In COSM,we expand the radial part of ψ κ ( r ) using the Gaussian basis functions for each orbit [16, 21]as ψ κ ( r ) = N ℓj X k =1 d kκ φ kℓjt z ( r , b kℓj ) , (51) φ kℓjt z ( r , b kℓj ) = N r ℓ e − ( r/b kℓj ) / [ Y ℓ (ˆ r ) , χ σ / ] j χ τt z , (52) h ψ κ | ψ κ ′ i = δ κ,κ ′ , (53)where the index k is to distinguish the range parameter b kℓj of the Gaussian functions with thenumber N ℓj , which is determined to converge the physical solutions. The length parameters b nℓj are typically chosen in a geometric progression [16, 19, 20, 60]. The normalization factorsof the basis are given by N . The coefficients { d kκ } in Eq. (51) are determined in the Gram–Schmidt orthonormalization, and the basis states ψ κ are orthogonal to each other in Eq. (53).The same technique using Gaussian bases as a single–particle basis is used in the tensor-optimized shell model [56, 62–65] and tensor-optimized antisymmetrized molecular dynamics[66, 67] for the analysis of light nuclei with bare nuclear forces.The antisymmetrization between the α core and a valence nucleon is described in theorthogonality condition model [16], where the single particle states ψ κ are orthogonal to the0 s state in the α core.We apply CSM to the many-body COSM wave function. In CSM, all of the relative coor-dinates r i between α and a valence nucleon are complex-scaled into r i e iθ for i = 1 , . . . , N v with a common scaling angle θ . The Hamiltonian in Eq. (49) becomes the complex-scaledHamiltonian H θ COSM , and the complex-scaled Schr¨odinger equation is written as H θ Ψ JT, θ COSM = E θJT Ψ JT, θ COSM , (54)Ψ JT, θ COSM = X c C JT, θc Φ JTc ( A ) , (55) H θ = U ( θ ) H U − ( θ ) . (56) he expansion coefficients C JT, θc depend on θ and are obtained from the eigenvalue problemof H θ with the COSM basis functions using the complex-scaled Hamiltonian matrix elements; h e Φ JTc ( A ) | H θ | Φ JTc ′ ( A ) i = h e Φ JTc ( A ) − θ | H | Φ JTc ′ ( A ) − θ i , (57)where the biorthogonal relation is taken in the wave functions as explained in Eq. (10).The inverse scaling of the basis states with − θ is treated in the Gaussian basis functionstransforming r/b kℓj to r/ ( b kℓj e iθ ) in Eq (52).The energy eigenvalues E θJT are obtained on a complex energy plane for each spin J and isospin T . We employ a finite number of basis states, which bring the discretizedrepresentation of the continuum states as well as the resonances in CSM.After solving the eigenvalue problem with COSM, we can categorize each eigenstate Ψ ν into bound, resonance and non-resonant continuum states, which constructs the extendedcompleteness relation (ECR). We explain the case of He with isospin T = 1 consisting ofthe α core and two valence neutrons. The various states of He consist of the three-bodyECR [22, 68] as = X ν | Ψ ν ih e Ψ ν | (58)= { Three-body bound states of He } + { Three-body resonances of He } + { Two-body continuum states consisting of He ( ∗ ) + n } + { Three-body continuum states consisting of α + n + n } (59)where { Ψ ν , e Ψ ν } consist of a set of biorthogonal bases with a state ν . Note that the Henucleus, a subsystem of He, has resonances but no bound states. The above relation is usedto calculate the CSGF in Eq. (15) and the transition strength into the three-body unboundstates of He such as the E E We discuss the spectroscopy of He isotopes and their mirror nuclei with COSM. In theHamiltonian of COSM, two kinds of the interactions between core– N and N – N are necessary.In the present study, the α - n interaction V αn is given by the microscopic KKNN potential[51, 52], also used in the previous section. We use the effective Minnesota central potential[70] as the nuclear part of V NN in addition to the Coulomb interaction.We explain the model space of COSM for He isotopes [56, 60, 61]. For the single-particlestates, we take the angular momenta ℓ ≤ V NN slightly to reproduce the two-neutron separation energy of He (0 + ) from experiment of 0.975 MeV, which is small for theneutron-halo structure in the ground state.We show the energy levels of He isotopes and their mirror nuclei with COSM in Fig. 16,measured from the energy of the α particle [71–73]. The small numbers near the levelsrepresent the decay widths of the states. One can see good agreement for the energy positionbetween theory and experiment up to the five-body case of He and C with isospin T = 2,in which only the 0 + states are shown. It is found that the order of energy levels are the samebetween proton-rich and neutron-rich sides. Theory also gives many predictions of the excited He He He E thr ( He+ N v n ) Γ =3.19 Expt Theor He Γ =5.84 0.0482.771.802.29 Γ =6.340.1323.785.88 Γ =8.76 E - E t h r ( H e + N v n ) [ M e V ] E thr ( He+ N v p ) Γ=6.49 Expt Theor Li Be B E - E t h r ( H e + N v p ) [ M e V ] Γ=9.86 Γ=11.21 Γ=6.64 C Fig. 16 Energy levels of He isotopes (left) and their mirror nuclei (right) measured fromthe threshold energy of the α particle emission in units of MeV. The black and gray linesindicate the theoretical and experimental values, respectively. The small numbers are thedecay widths in theory. The 0 + states are shown in theory for He and C. Li Be B E x [ M e V ] C He He He He Fig. 17 Excitation energy spectra of mirror nuclei of A = 5 , , 7, and 8 in units of MeV.states. In Fig. 17, we compare the excitation energy spectra of proton-rich and neutron-rich sides to examine the mirror symmetry in nuclei. It is found that the good symmetryis confirmed between the corresponding nuclei. The differences in excitation energies forindividual levels are less than 1 MeV [72].The matter and charge radii of the ground states of He and He provide good informationon the valence neutron distribution. These values obtained in COSM reproduce the recentexperiments, as shown in Table 1. From the results, the COSM wave functions can explainthe special extensions of neutrons in the halo and skin structures of He isotopes. + resonances and continuum states in He For resonance and non-resonant continuum states in CSM, the energy eigenvalue distributionof He (2 + ) is shown in Fig.18. In this figure, we can identify the locations of various kinds ofunbound state: the three-body resonances of He (2 +1 and 2 +2 ), and two kinds of continuumstates of He(3 / − ,1 / − )+ n and α + n + n . able 1 Matter ( R m ) and charge ( R ch ) radii of He and He obtained with COSM inunits of fm. Experimental values are taken from a[74], b[75], c[76], d[77], and e[78].COSM Experiments He R m a b c R ch d8 He R m a b c R ch d e Re(E)Im(E)three-body continuum resonance bound θ =0) θ two-body continuumcore+n+n [core+n] * res +n[core+n] res +n -4.0-3.5-3.0-2.5-2.0-1.5-1.0-0.50.0 0 0.5 1 1.5 2 2.5 3 3.5 4 I m ( E ) [ M e V ] Re(E) [MeV] He(2 + ) θ =30 ° +1 +2 Fig. 18 Left: Energy eigenvalue distribution of core+ n + n in CSM. Right: Energy eigen-values of the He(2 + ) states with θ = 30 ◦ in the complex energy plane measured from the α + n + n threshold energy [60]. The three schematic lines with dots indicate the continuumstates of α + n + n and He(3 / − ,1 / − )+ n , from left to right, respectively.Corresponding to the 2 + energies in Fig. 18, we calculate the E He excited from the 0 + ground state to the three-body scattering states of α + n + n . Itis interesting to investigate the contributions of resonances and non-resonant continuumstates. Before showing the results of the E +1 , resonances, the squaredvalues of which are 2 . 78 + i . 43 for 2 +1 and − . i . +2 in units of e fm . Transitionstrengths for resonances are obtained as complex numbers as well as the energy eigenvalues.When the resonance pole is very close to the real energy axis as for 2 +1 with Γ = 0 . 132 MeV,its matrix elements show large real parts. On the other hand, the 2 +2 resonance has a largedecay width of 3.78 MeV and a large imaginary part of the transition strength.The obtained E +1 , reso-nances, two-body continuum states of He(3 / − ,1 / − )+ n , and three-body continuum onesof α + n + n . From Fig. 19, we can see that the 2 +1 resonance gives the main contributionshowing a sharp peak around the resonance energy of 0.81 MeV with a small decay width.On the other hand, the contribution from the 2 +2 resonance is very small due to the largedecay width. The components associated with two- and three-body continua are smaller thanthose of 2 +1 . However, in the continuum transitions the two-body continuum component of He(3/2 - )+n2 +1 +2 d B ( E ) / d E [ e f m / M e V ] Energy [MeV]Total He( - )+n He( - )+n He+n+n Fig. 19 E He from the ground state to the 2 + scattering state in CSM,measured from the α + n + n threshold energy. He(3 / − )+ n shows a peak at around 1 MeV, just above the two-body threshold energy ofthis channel. This component mainly contributes to making shoulder structure in the totalstrength at around 1.5 MeV measured from the α + n + n threshold energy.The reason why the two-body component of He(3 / − )+ n makes a peak at around 1MeV is as follows: In this two-body component, the last neutron is in the continuum stateand we have confirmed that this neutron has a similar structure to the plane wave [68].Hence the peak at around 1 MeV is interpreted as a two-body threshold effect reflectingthe spatially extended structure of neutrons in the initial neutron-halo state. A similarpeak structure can be observed in the case of E − states, which aredominated by the He(3 / − )+ n channel and no three-body resonances. The effect of two-body continuum components of He(1/2 − )+ n is very small. It is found that the three-bodycontinuum component of α + n + n also contributes to making a shoulder at around 1.5 MeVin the total transition strength.In the calculation, it is found that the E +1 state, but not of the 2 +2 state. It might be hard to observeexperimentally the existence of the 2 +2 state from the E He(3/2 − )+ n and α + n + n channels contribute to makinga shoulder-like structure. We compare the structures of proton- and neutron-rich nuclei from the viewpoint of mirrorsymmetry and focus on two nuclei, He and C, having four neutrons/protons above the α particle, respectively. In the two nuclei, only the ground state of He is a bound state as seenin Fig.16. It is interesting to investigate the effect of the Coulomb interaction on the mirrorsymmetry of these nuclei. In general, the structures of resonances and weakly bound statesare influenced by the open channels of the particle emissions. In unstable nuclei, the mirrorsymmetry is considered to be closely related to the coupling between the open channels andthe continuum states. We have also performed a similar analysis on the comparison of Heand Be with two valence nucleons [79]. able 2 Squared amplitudes ( C JTc ) of the ground states of He and C.Configuration He (0 +1 ) C (0 +1 )( p / ) . − . i ( p / ) ( p / ) . 057 + 0 . i ( p / ) (1 s / ) . 010 + 0 . i ( p / ) ( d / ) . 007 + 0 . i ( p / ) ( d / ) . 037 + 0 . i Table 3 Squared amplitudes ( C JTc ) of the 0 +2 states of He and C.Configuration He (0 +2 ) C (0 +2 )( p / ) . − . i . 044 + 0 . i ( p / ) ( p / ) . − . i . − . i ( p / ) (1 s / ) − . − . i − . 001 + 0 . i ( p / ) ( d / ) . 018 + 0 . i . 020 + 0 . i ( p / ) ( d / ) . 002 + 0 . i . 002 + 0 . i We first discuss the configuration mixing of the single-particle states in He and C, theresults of which are shown in Table 2 for the ground states of He and C [72]. We show thedominant configurations and the corresponding squared amplitudes ( C JTc ) defined in Eq.(50). The states of He and C show similar values for mixing, and the ( p / ) configurationdominates their wave functions.We can see the configuration mixing of the 0 +2 resonances of He and C in Table 3 [72].It is found that the ( p / ) ( p / ) configuration dominates the wave functions of He and C with large squared amplitudes of about 0.93–0.97 in the real parts. These states can beregarded as the two-particle two-hole excitations from the ground states. The magnitudesof the dominant configuration in 0 +2 are larger than those of the ground states of He and C commonly. This mainly comes from the reduction of the coupling strengths betweendifferent configurations for valence nucleons [79]. When the 0 +2 resonances of He and Care spatially extended, their wave functions are widely distributed and the amplitudes ofvalence nucleons easily penetrate from the interaction range to the outside. From this effect,the couplings between the different configurations are reduced and the single configurationof ( p / ) ( p / ) largely survives in the 0 +2 states.We next discuss the effect of the Coulomb interaction on the spatial motion of valenceprotons in C in comparison with the valence neutrons in He. This is related to the mirrorsymmetry in two nuclei. In this study we compare the various radii for He and C to inves-tigate the motion of valence nucleons. Note that the radius of Gamow resonances generallybecomes a complex number, because the states have the complex amplitudes belonging tothe complex energies. In the numerical results for resonances, their radii show the imaginaryparts, which are smaller than the real values. A similar tendency is confirmed in the squaredamplitudes shown in Tables 2 and 3. From this property, we use the real part of the complexradius to discuss the spatial property of resonances. The operator forms of the various radii able 4 Radius of matter ( R m ), proton ( R p ), and neutron ( R n ) of the 0 +1 , states in Heand C in units of fm. He(0 +1 ) C(0 +1 ) He(0 +2 ) C(0 +2 ) R m . − . i i i R p . − . i i i R n . − . i i i R m R p R n [fm] +18 He C R m R p R n [fm] +28 C He Fig. 20 Real parts of matter ( R m ), proton ( R p ), and neutron ( R n ) radii of He and C in units of fm. Left: Ground states. Right: 0 +2 states. The circles with error bars areexperimental values of the matter radius of He [74–76].in COSM are given in Ref. [55]. The radii of matter ( R m ), proton ( R p ), and neutron ( R n )have the relation of AR = ZR + N R with proton number Z and neutron number N .We show the results of the radii for He and C in Table 4 for the matter, proton, andneutron parts. For the 0 +1 states, the matter radius of C is larger than the value of He byabout 12% in the real part. For 0 +2 states, all the radii become complex numbers because ofthe property of resonances. Their imaginary parts are shown to be large in He(0 +2 ), but stillsmaller than the real parts. The values in C have small imaginary parts and the bound-state property can survive for this state. In two nuclei, various radii for the 0 +2 states becomelarger than those for their ground 0 +1 states. This trend represents the spatial extension ofvalence nucleons occurring in the 0 +2 states.It is found that the matter radius of C is smaller than the value of He for the 0 +2 states.In particular, the proton radius of C is found to be smaller than the neutron radius of He from a mirror relation. This is an opposite relation to that for the ground 0 +1 statesof He and C. To make clear this difference, we plot the real parts of the matter, proton,and neutron radii of 0 +1 . in Fig. 20. For He(0 +2 ), the large matter radius is due to the largeneutron radius. For C(0 +2 ), the large matter radius comes from the large proton radius,which is smaller than the neutron radius of He.We conclude that the spatial relation between He and C depends on the states, andthis property is understood from the Coulomb interaction. The Coulomb interaction actsrepulsively and shifts the energy of C upward from the case of He as seen in Fig. 16. In theground state of C, this repulsion increases the distances between α and a valence proton andbetween valence protons. In addition, the Coulomb interaction makes the barrier above the article threshold in C, and the 0 +2 resonance is influenced by this barrier, which preventsthe four valence protons in C from extending spatially. In He, there is no Coulomb barrierfor valence neutrons and the neutrons in the resonances can be extended to a large distance.This effect of the Coulomb interaction makes the radius of C(0 +2 ) smaller than the value of He (0 +2 ). 5. Complex-scaled Lippmann–Schwinger equation In unstable nuclei, separation energies of valence nucleons are much smaller than for stablenuclei by one order. Due to this weakly bound property, unstable nuclei are broken up in alow-excitation energy. Utilizing this property, the breakup reactions become important toolsexperimentally to explore the exotic properties of unstable nuclei [7]. Theoretically, a reliabledescription of the many-body scattering states of weakly bound systems is required.Recently, several theoretical methods have been proposed to solve many-body scatteringproblems [80, 81]. We have also developed a method to describe the many-body scatteringstates in CSM, called the “complex-scaled solutions of the Lippmann–Schwinger equation”(CSLS equation) [32–35]. In the CSLS equation, scattering states are described in the formalsolutions of the Lippmann–Schwinger equation using the CSGF. This CSGF given in Eq.(15) is automatically imposed to satisfy the correct boundary conditions of many-bodyscattering states using the complex-scaled eigenstates. We have shown that the CSGF givesthe precise CLD of the two-body scattering states [48, 50] and works consistently to obtainthe three-body CLD [18]. We describe the many-body scattering states of unstable nucleiin the CSLS equation, and obtain the physical quantities as functions of the energies of thesubsystems in a many-body states. This gives useful information on the internal correlationsof unstable nuclei. This approach contributes to clarifying the exotic properties of weaklycoupled systems, such as the neutron motion in a halo.In this paper we explain the following two topics related to the three-body scattering statesin the CSLS equation: (i) the Coulomb breakup reactions of two-neutron halo nuclei, Heand Li; (ii) the elastic scattering and radiative capture reaction of the α + d system.We comment on the continuum discretized coupled channel (CDCC) approach for breakupreactions [82]. In CDCC, the reaction process between the projectile and target nuclei isdynamically solved in a coupled-channel problem. In CDCC, the continuum states of theprojectile nucleus are discretized in a two-body breakup case. For three-body breakup casessuch as the two-neutron halo nuclei, there is an extension of CDCC by applying ECR andthe CSGF to the projectile nucleus [83]. Coulomb breakup reactions are utilized to understand the excitation properties of two-neutron halo nuclei such as He and Li [84–89], and in particular the dipole responsesof the halo structure. In two-neutron halo nuclei, no binary subsystems have bound statesin a core+ n + n picture, which is called the Borromean property. We theoretically describetwo-neutron halo nuclei in the core+ n + n three-body model and investigate their Coulombbreakup reactions. We describe here the three-body scattering states of halo nuclei by usingthe CSLS equation. In this review, we focus on the Coulomb breakup reactions of He and Li. For Li, we consider the configuration mixing of the Li core nucleus with particle–hole xcitations, which is essential to reproduce the halo properties of Li with a large s -wavemixing of two neutrons [56].In the Coulomb breakup reactions of two-neutron halo nuclei, the asymptotic Hamiltonian H is given as H = h core + X i =1 t i − T c . m . , (60)where h core is the internal Hamiltonian of the core nucleus of the α particle for He and Lifor Li. The kinetic energy operators for each particle and for the center-of-mass part are t i and T c . m . , respectively. The solution of H is given asΦ ( k , K ) = Φ coregs ⊗ φ ( k , K ) , (61)where k and K indicate the asymptotic momenta with a three-body Jacobi coordinate. Inthe asymptotic region, the core nucleus becomes the ground state Φ coregs . For He, the groundstate of the α core is the (0 s ) closed configuration [32, 33]. For Li, the ground-state wavefunction of Li is obtained in the tensor-optimized shell model (TOSM) [35, 56, 57]. Theasymptotic wave function φ for the relative motion part in the core+ n + n three-body systemis given as φ ( k , K ) = 1(2 π ) e i k · r + i K · R , (62)where the coordinates r and R are the conjugates to the momenta k and K , respectively.We construct the Green’s function given in Eq. (15) and three-body ECR in Eq. (59) todescribe the scattering states of two-neutron halo nuclei. In a similar manner to COSM,we obtain the eigenstates { Ψ θν } with the index ν and their eigenvalues { E θν } by solving thecomplex-scaled Schr¨odinger equation, H θ Ψ θν = E θν Ψ θν . (63)For the core+ n + n three-body system, the total Hamiltonian H is given as H = h core + X i =1 t i − T c . m . + X i =1 V core – n ( r i ) + V n – n , (64)where V core – n and V n – n are the interactions for core- n and n - n , respectively. The coordinatebetween the core nucleus and the i th neutron is given by r i . We remove the Pauli forbiddenstates occupied by the nucleons in the core nucleus from the core– n relative motion. For Hewe adopt the same Hamiltonian used in COSM and for Li we adopt the same Hamilto-nian defined in Refs.[56, 57]. We transform the total Hamiltonian H in CSM and solve thecomplex-scaled eigenvalue problem with the Hamiltonian H θ in Eq. (63). He and Li. We show the Coulomb breakupcross sections as functions of the excitation energies of He and Li. The target nucleus isPb and the incident energies of the He and Li projectiles are 240 and 70 MeV/nucleon,respectively. The cross sections are evaluated using the E d σ / d E [ m b / M e V ] Energy [MeV]CSLSExp. (a) d σ / d E [ b / M e V ] Energy [MeV]CSLSExp. (b) Fig. 21 Coulomb breakup cross sections measured from the three-body threshold energiesof core+ n + n . Panels (a) and (b) indicate the results for He and Li, respectively. Theexperimental data for He and Li are taken from Refs. [84] and [89], respectively, shownas open squares with error bars.equivalent photon method [90], d σd k d K = 16 π ~ c N ( E γ ) d B ( E d k d K , (65)where N ( E γ ) is the virtual photon number with the photon energy E γ that the projectilenucleus absorbs. The E d B ( E d k d K = 12 J + 1 (cid:12)(cid:12)(cid:12)(cid:10) Ψ ( − ) ( k , K ) || b O ( E || Ψ (cid:11)(cid:12)(cid:12)(cid:12) , (66)where the operator b O ( E 1) represents the E and J , respectively.By integrating the differential cross sections in Eq. (65), we obtain the cross section asfunctions of the scattering energy E [32, 33, 35], dσdE = Z Z d k d K d σd k d K δ (cid:18) E − ~ k µ − ~ K M (cid:19) , (67)where µ and M are the reduced masses for the corresponding momenta.In Fig. 21, we show the breakup cross sections of He and Li measured from the thresholdenergies of core+ n + n with the experimental data [84, 89]. For He, low-energy enhancementare confirmed around 1 MeV and the cross section decreases gradually with the excitationenergy. The result reproduces the observed cross section [84], especially at low excitationenergy below 2 MeV. It has been demonstrated that this cross section is dominated by the He(3/2 − )+ n decay channel [16, 68]. For Li, the results show good agreement with theexperimental data [89] in the whole energy region. This cross section also shows a low-energyenhancement around 0.3 MeV and rapidly decreases as the energy increases.For He and Li, it has been commonly confirmed that the low-energy enhancementsin the cross sections are affected by the strong final-state interactions (FSIs) in the dipoleexcited states [33, 35]. This result means that the Coulomb breakup cross sections are muchinfluenced by the final three-body scattering states. From this property, information on theground-state structure of halo nuclei is difficult to obtain due to the strong FSI. We further d σ / d E α - n [ a r b . un i t s ] E α -n [MeV] He(3/2 - ) CSLSExp. (a) d σ / d E n - n [ a r b . un i t s ] E n-n [MeV]CSLSExp. (b) Fig. 22 Invariant mass spectra of the Coulomb breakup cross section for He with arbi-trary units. Panels (a) and (b) are the strengths as functions of the energies of the α – n and n – n binary subsystems, respectively. The experimental data is shown by the open squares[84]. The arrow in panel (a) shows the He(3/2 − ) resonance energy.investigate the invariant mass spectra of the binary subsystems to clarify the mechanismsof the Coulomb breakup reactions in the next section.Using the three-body ECR of two-neutron halo nuclei given in Eq. (59), we can decomposethe breakup cross sections into the components of three-body resonances and two-bodyand three-body continuum states; the latter two kinds of continuum states indicate thesequential breakup process and direct breakup, respectively. We have discussed the amountsof different breakup processes with respect to the total cross sections [61, 68, 69]. Note thatthis decomposition does not correspond to the experimental observables directly. He. We calculate the invariantmass spectra for binary subsystems, which is the cross section as functions of the relativeenergy of the binary subsystems such as core– n and n – n . This quantity provides the infor-mation on the correlations of the subsystems in the reaction. Using Eq. (65), we define theinvariant mass spectra with the strength distribution obtained in the CSLS equation as dσdε = Z Z d k d K d σd k d K δ (cid:18) ε − ~ k µ (cid:19) , (68)where ε is the relative energy of the binary subsystem with momentum k .In Fig. 22, we show the invariant mass spectra of He as compared with the experimentaldata [84]. Panels (a) and (b) are the spectra of the α – n and n – n subsystems, respectively.Two spectra show good agreement with the experimental data, which indicates the reliabilityof the CSLS equation. For the α - n system in Fig. 22 (a), the peak position of the strengthagrees with the resonance energy of 0.74 MeV in He(3/2 − ). This result shows the dominanceof the sequential breakup process of He into α + n + n via the He+ n channel. For the n - n system in Fig. 22 (b), low-energy enhancement is confirmed near the zero energy, whichoriginates from the n – n virtual state in the attractive S -wave channel. α and deuteron in the α + p + n three-body model The deuteron, denoted as d , is a weakly bound system with an energy of 2.2 MeV andeasily excited inside the nucleus. In Li, the threshold energies of the α + d and α + p + n ystems are very close, with excitation energies of 1.47 and 3.70 MeV, respectively. Thisproperty suggests that the α + d and α + p + n structures coexist in the low excitation energyregion. In the scatterings associated with Li, it is important to describe the three-bodyscattering states with the α + p + n configurations. We describe the scattering states of Liin the CSLS equation, and investigate the α + d elastic scattering and the radiative capturereaction of H( α , γ ) Li. In both reactions, we examine the dynamical effects of α + p + n three-body structures. We investigate the effects of deuteron breakup in Li and the rearrangementto the He+ p and Li+ n channels on the above reactions [34].The scattering states of Li are described in the α + p + n three-body model with the CSLSequation. We define the asymptotic Hamiltonian H for the α + d system as H = h d + T rel + V Coul ( R ) , (69)where h d is the internal Hamiltonian of the deuteron with the Argonne V8 ′ realistic N – N potential [91]. The kinetic energy and the Coulomb interaction between α and d are givenas T rel and V Coul , respectively. The relative coordinate between α and d is denoted as R .The solution of H is expressed asΦ ℓJ π ( K , r , R ) = h χ + d ( r ) ⊗ φ ℓ ( K , R ) i J π , (70)where ℓ is the relative orbital angular momentum for α (0 + ) and d (1 + ), and J π is thetotal spin and parity. The relative momentum is given as K . The deuteron wave function is χ + d ( r ). The asymptotic relative wave function φ l for the α + d system is defined as φ ℓ ( K , R ) = (2 ℓ + 1) i ℓ F ℓ ( η, KR ) KR X m Y ℓm ( ˆ K ) Y ∗ ℓm ( ˆ R ) , (71)where F ℓ and η are the regular Coulomb wave function and the Sommerfeld parameter,respectively. Using Eq. (70), we obtain the scattering states of the α + d system Ψ (+) ℓJ π ( K ) inthe CSLS equation using Eq. (35). We solve the α + p + n three-body model with Gaussianexpansion and complex scaling, and prepare the set of eigenstates { Ψ θν } . Using these eigen-states, we describe the α + p + n three-body components in the Green’s function in the CSLSequation. α + d scatterings. In Fig. 23, we show the α + d elastic phaseshifts for the relative D -wave scatterings with total spin J . The resulting phase shifts repro-duce the trend of the experimental data for the J = 1 + , 2 + , and 3 + states. This agreementrepresents the reliability of the CSLS scheme in three-body description of Li. Three kindsof the phase shifts show resonance behavior, and in CSM the resonance energies measuredfrom the α + d threshold and the decay widths ( E r , Γ) are obtained as (4.12, 3.60) for 1 + ,(2.83, 0.91) for 2 + , and (0.72, 0.19) for 3 + in units of MeV.We investigate the effect of a three-body configuration on the α + d scatterings. We focuson two kinds of effects of deuteron breakup and the rearrangement to the He+ p and Li+ n channels. For this purpose, we show two kinds of calculations in addition to that shown inFig. 23. One is the calculation in which only the elastic channel of α + d is considered withoutthe effects of the deuteron breakup and rearrangement, and is named “Elastic”. The otheris the calculation in which only the deuteron breakup is included in the breakup effect, andis named “Breakup”. From these calculations, we estimate the effects of deuteron breakupand the rearrangement channels on the α + d scattering. P ha s e s h i ft δ [ deg ] D D D E [MeV] α -d Fig. 23 Elastic D -wave phase shifts D J with spin J for the α + d scatterings as func-tions of the α + d relative energies. The red,green, and blue lines show the calculationsfor D , D , and D scattering states, cor-responding to the 3 + , 2 + , and 1 + states,respectively. Experimental results [92, 93]are shown as open circles in the same colorsas the lines. P ha s e s h i ft δ [ deg ] E [MeV] α -d Fig. 24 The effects of deuteron breakupand rearrangement on the α + d phase shifts.The solid lines are the full calculations andare as shown in Fig. 23. The dotted anddashed lines are the Elastic and Breakupcalculations, respectively. The three dottedlines for the elastic calculation give almostidentical solutions.For the Elastic case, the p + n wave function in Li is kept as the deuteron during the scat-tering. We solve only the relative motion between α and d with the coordinate R and obtainthe set of new eigenstates { Ψ θν, el } of Li. For the Breakup case, we allow excitations of the p - n relative motion. First, we prepare the eigenstates of the p + n system with the Gaussianexpansion, which involves the ground and excited states. Next, we solve the coupled-channelproblem on the relative motion between α and the set of the p + n states, and obtain anotherset of eigenstates { Ψ θν, br } for Li . Using the different sets of eigenstates { Ψ θν } , we obtaintwo different ECRs for the Elastic and Breakup calculations of Li.We show the results in Fig. 24. In the Elastic case with dotted lines, the deuteron is keptin the ground state and all the phase shifts exhibit no structure. In the Breakup case withdashed lines, the calculations show resonance behaviors for each state. These results indicatethat the deuteron breakup is very important in α + d scattering, as suggested in Ref. [82].Note that the positions of the resonance energies are higher than in the full calculations. Thisdifference indicates the rearrangement effect, which shifts the resonance energies down byabout 0.5 MeV. In the present analysis the deuteron breakup is found to have a significantrole on the resonance description of Li in the α + d scattering, while the rearrangementeffects give a small contribution to the phase shift. Li. We now discuss the α + p + n three-bodyeffect on the radiative capture reactions of Li. We calculate the radiative capture crosssection of H( α, γ ) Li, σ cap , in the relation σ dis ( E ) = 2(2 J gs + 1)(2 J α + 1)(2 J d + 1) k γ K σ cap ( E ) , (72) C ap t u r e c r o ss s e c t i on σ c ap [ nb ] Calc.Robertson81Kiener91E [MeV] α -d α +p+n threshold Fig. 25 The radiative capture crosssection of H( α, γ ) Li is shown with redlines. The open and solid circles with errorbars are the experimental data from Refs.[94] and [95], respectively. C ap t u r e c r o ss s e c t i on σ c ap [ nb ] FullElasticBreakupE [MeV] α -d Fig. 26 The effects of deuteron breakupand rearrangement on the radiative capturecross section of Li. The red solid line isthe same as in Fig. 25. The blue dottedand green dashed lines are the results of theElastic and Breakup cases, respectively.where J gs , J α and J d are the spins of the Li ground state (1 + ), α , and deuteron, respectively.The wave number of the emitted photon is k γ = E γ / ~ c . The photodisintegration cross section σ dis is obtained using σ dis ( E ) = 4 π (cid:18) E γ ~ c (cid:19) Z d K d B ( E d K δ (cid:18) E − ~ K M + ε d (cid:19) , (73)where M is the reduced mass for the momentum K and ε d is the binding energy of thedeuteron. The photon energy is defined as E γ = E + ε gs , and ε gs is the binding energy of the Li ground state with respect to the α + p + n threshold energy. In this calculation, we considerthe contribution of the dominant E E d B ( E d K = 12 J gs + 1 (cid:12)(cid:12)(cid:12) h Ψ ( − ) ℓJ π ( K ) || b O ( E || Ψ i (cid:12)(cid:12)(cid:12) , (74)where the operator O ( E 2) represents the E is the wave function of the Li ground state.In Fig. 25, we show the radiative capture cross section of H( α, γ ) Li. The calculationshows a good agreement with experiment below the α + p + n threshold energy. Above theenergy of E α - d = 3 MeV, the present result underestimates three of the experimental datapoints. One possibility to improve this underestimation is to include higher-order transitions,such as the M α + p + n three-body effect on the cross section. For this purpose, weperform the same analysis as for the α + d scattering. We show the results in Fig. 26.Similar to the phase shift case, the Elastic result shows a structureless distribution. Thedeuteron breakup result (Breakup) shows peaks and bumps, the energies of which are the3 +1 , 2 +1 , and 1 +2 resonances, although these energies are slightly higher than the full three-body calculations of Li. This difference is explained via the He+ p and Li+ n rearrangementchannels in a similar situation to the phase shifts. . Summary and perspective We have explained the frameworks of the complex scaling method (CSM) to study many-body resonances and continuum states, and presented applications to many-body nuclearsystems in the recently developed physics of unstable nuclei. CSM has been developed as avery promising method to obtain the resonance energies and decay widths of many-body sys-tems, and here we emphasized that CSM based on the non-Hermitian Hamiltonian providesus with a powerful method to investigate not only the resonance state but also many-bodycontinuum states.The basic idea of CSM is that bound, resonant, and continuum states satisfying the bound-ary condition of the asymptotic outgoing waves with an exp ( ikr ) form can be expressed withsquare-integrable functions ( L class) by a complex scaling (a dilation transformation). Thismeans first that the non-Hermiticity of the complex-scaled Hamiltonians is caused by theboundary condition of the outgoing waves for the state space. This idea indicates that bound,resonant and continuum states are obtained simultaneously by solving an eigenvalue problemusing a set of appropriate basis functions. The obtained eigensolutions describe a completeset, and the Green’s function under the outgoing wave boundary condition is constructedin terms of these eigensolutions in CSM. This Green’s function is a spectrum representa-tion consisting of the bound, resonant, and continuum energy states, and gives a kind ofprojection of the matrix elements associated with complex energy states into an observablequantity defined on the real energy axis. Thus, although the matrix elements of resonant andcontinuum energy states are complex numbers, the quantities projected on the real axis arereal numbers and can be investigated through the decomposition into each resonant stateand different kinds of continuum sates.After explaining the frameworks of CSM, we showed its basic application to the simpletwo-body systems. An important advantage of CSM is a natural description of many-bodysystems, and its application to nuclear many-body systems including unstable nuclei was alsodemonstrated. Most of the states in an unstable nucleus are observed as unbound states dueto the weak binding nature of valence nucleons. CSM provides us with a powerful frameworkto investigate the interesting properties of unstable nuclei. We presented our recent resultsfore many-body systems of up to five bodies.One of the main interests in the many-body resonances is the breakup dynamics associatedwith correlations between constituents. To see such correlations among constituents, we havedeveloped the complex-scaled Lippmann–Schwinger equation. The validity of this methodwas shown for three-body breakup reactions of neutron halo nuclei.Thus, CSM is expected to bring a unified description of the structures and reactions ofnuclei, though there are still many problems to be tackled. Several theories have put forthab-initio descriptions of resonances developed from the bare nucleon–nucleon interaction,such as the few-body method [81, 96] and the no-core shell model approach [97]. Furtherextension of the many-body resonant and continuum states covering a wide range of massnumbers is the current task in this direction. The application of CSM can be extended tohadron and strangeness physics. There are studies of many-body resonances observed inhypernuclei with hyperons such as He [98], and also in mesic nuclei consisting of baryonsand mesons such as ¯ KN N [99, 100]. n CSM, because resonant states are described with square-integrable functions as well asbound states, we can investigate their structures by analyzing the wave functions directly.Various kinds of structures of resonant states are expected, as discussed in bound states, inmany-body nuclear systems. An important subject to investigate in nuclear cluster physicsis multi-cluster structure, such as an α linear-chain structure, in resonant states observedaround the corresponding multi-cluster threshold energies [17]. In atomic and molecularphysics, two kinds of resonances have been discussed, a shape-type resonance and a Feshbach-type one [13].CSM describes resonances with complex energy eigenvalues, the imaginary part of whichrepresents the total decay width. It is important to evaluate the partial decay widths ofmany-body resonances for each decaying channel, which provide useful information on thedecay properties of the many-body resonances, but are not yet available. It is desirable todevelop a method for extracting the partial decay widths of the many-body resonances inCSM. There is a theoretical development of obtaining the partial decay widths by using thecontinuity equation based on the time-dependent Schr¨odinger equation [13, 101].In addition to the resonances, the virtual states are a kind of unbound state which oftenplay an important role in nuclear structure around threshold energies, and are difficult toobtain directly in CSM with θ < π/ 2, different from resonance poles. 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