Low-lying continuum states of drip-line Oxygen isotopes

Low-lying continuum states of exotic oxygen isotopes are studied, by introducing the Continuum-Coupled Shell Model (CCSM) characterized by an infinite wall placed very far and by an interaction for continuum coupling constructed in a close relation to realistic shell-model interaction. Neutron emission spectra from exotic oxygen isotopes are calculated by the doorway-state approach in heavy-ion multi-nucleon transfer reactions. The results agree with experiment remarkably well, as an evidence that the continuum effects are stronger than $\sim$1 MeV, consistently with the shell evolution in exotic nuclei. The results by this CCSM doorway-state approach are compared with calculations on neutron-scattering resonance peaks made within the CCSM phase-shift approach and also with those obtained in the Gamow shell model, by taking the same interaction. Remarkable similarities in peak energies and certain differences in widths are then obtained.


I. INTRODUCTION
Exotic nuclei far from the β-stability line provide us with new interesting features. A good example is the neutron halo and other continuum-related properties of drip-line nuclei [1]. Another example can be the evolution of shell structure, including the change of magic numbers, due to characteristic properties of the nuclear force [2,3], as being increasingly observed [4,5]. Although there are many bound nuclei between β-stability and drip lines showing the shell evolution, one eventually approaches the drip line, by adding more neutrons (or protons). Low-lying states are then in the continuum even if the ground state is still bound. Thus, the physics of continuum and that of shell evolution should meet. It is of much importance to clarify how the coupling to the continuum affects various structures generated by the nuclear force. We shall study, in this paper, continuum spectra of neutron-rich O isotopes. The O isotopes show unusual shell structure and magic numbers (N =14, 16) [2,[6][7][8], ending up with unbound 0d 3/2 orbit.
We extend the normal shell model (SM) so as to include continuum states as a part of SM basis. By this continuum-coupled shell-model (CCSM) calculation, we can see precisely the location and shape of the peaks of various continuum spectra. Although the use of an infinite wall belongs to standard techniques, the wall is placed at unusually far distance so that the SM ba- * Electronic address: otsuka@phys.s.u-tokyo.ac.jp sis includes many (∼3000) discretized continuum states, both resonant and non-resonant. This allows us to produce spectrum shapes accurately and treat the asymptotic behavior of wave functions appropriately. A finiterange nucleon-nucleon (N N ) force, which is consistent with successful SM interaction for bound states, is introduced for the evaluation of continuum effects, resulting in significant energy shifts from the bound(-state) approximation. Such novel approach leads us to quite interesting results, in agreement with recent experimental data.

II. CONTINUUM-COUPLED SHELL MODEL
As the 0d 3/2 orbit remains unbound in oxygen isotopes, we generate discretized continuum single-particle basis states of d 3/2 , by introducing an infinite wall on top of the Woods-Saxon potential. The single-particle Hamiltonian is then written as, where T denotes kinetic energy and U W S is the Woods-Saxon potential with radius R=1.09A 1/3 fm, diffuseness a=0.67 fm. The depth parameter V 0 will be discussed later. The standard LS term is included [16]. Here, V wall is an infinite wall placed at 3000 fm from the center of the nucleus so as to obtain a sufficiently high density of discretized states. As shown later, the final results do not depend on the position of the infinite wall if the wall is far enough. The i-th d 3/2 state is denoted as |id 3/2 with i=1, 2,· · · ,i max . Six neutrons are assumed, for simplicity, to be in the 0d 5/2 orbit. Namely, the inert 22 O core is assumed in the following CCSM calculations. Thus, the basis states of valence neutrons are constructed from the bound 1s 1/2 state and the discretized continuum id 3/2 (i = 1, 2, ...) states with various sets of occupation numbers.
The CCSM is proposed to treat correlations due to nuclear forces in continuum states in a manner consistent with the SM for bound states. Realistic SM interactions are given usually in the form of two-body matrix elements (TBME's). In order to calculate matrix elements of the Hamiltonian for the continuum basis states discussed above, the SM interaction is extended to an analytic function,V , depending on variables of two interacting nucleons. As the realistic SM interaction contains many effects, from free-nucleon scattering to in-medium polarization corrections, it is virtually impossible to ex-pressV in a simple functional form for all single-particle orbits. On the other hand, the continuum coupling is important for few selected orbits in many cases. For instance, in the present case, neutron active orbits are 1s 1/2 and 0d 3/2 , and d 3/2 is extended to continuum basis states. We introduce a simpleV as a modeling of a realistic interaction within this small model space. Continuum id 3/2 states differ among themselves in their radial wave functions, and are mixed due to the coupling by nuclear force, producing correlated eigenstates even in the continuum. The coupling occurs primarily through the inner (i.e., resonance-like) part of such wave functions, because in the outer part nucleons are far from each other. The inner part resembles, more or less, bound 0d 3/2 , apart from the overall amplitude. The interaction with a neutron in 1s 1/2 is important too. Therefore, it is the minimal condition thatV should reproduce relevant TBME's of a realistic SM interaction involving bound (i.e., Harmonic Oscillator (HO)) 1s 1/2 and 0d 3/2 orbits. In this work,V is fixed by this minimal condition.
As a simple yet general form ofV , a superposition of two Gaussian functions with spin dependences is taken: where r is the inter-nucleon distance, and σ implies spin of a nucleon. Here, d 1,2 =1.4, 0.7 fm are prefixed for simplicity. The other parameters are fixed later. The CCSM Hamiltonian is then written as whereǫ j and n j are, respectively, the single-particle energy (SPE) and the occupation number operator, and j denotes single-particle states including discretized continuum ones.

III. CONSTRUCTION OF CCSM HAMILTONIAN
We define the SM Hamiltonian to be used for fixingV . The SDPF-M Hamiltonian [17], a revision of USD [18], is taken. The SDPF-M Hamiltonian has been shown to be successful for describing bound sd-shell nuclei including those with a large number of neutrons. We begin with the fine tuning of the SDPF-M so that it can reproduce one neutron separation energy (S n ) of 23 O exactly, by the following minor change of the monopole interaction, δ 1s 1/2 0d 5/2 |V |1s 1/2 0d 5/2 T =1 = −0.03 MeV. (4) Note that this change is less than 5% of the original value. While this fine-tuned Hamiltonian is for the full SM diagonalization, we introduce a simpler SM Hamiltonian to be used in the filling configuration scheme. The filling configuration means that valence nucleons are put into the lowest possible orbit one by one, and the diagonalization is carried out within such configurations. This truncation makes sense if the valence orbits are well separated. For oxygen isotopes with N ≥ 14, this is the case as represented by N =14 and 16 gaps ∼4 MeV [17]. The SPE's and TBME's appropriate for the filling scheme are determined so that the results of the fully mixed SM calculations are reproduced by the calculations in the filling scheme, as far as low-lying energy levels of 23−26 O relative to the 22 O ground state are concerned. Note that the active model space is (1s 1/2 -0d 3/2 ) for the filling scheme. The resultant Hamiltonian is referred to as modified SDPF-M.
Next, the parameters ofV are determined from the modified SDPF-M Hamiltonian. Relevant TBME's are calculated for each term ofV by using HO singleparticle wave functions. The parameters ofV are optimized by the χ 2 fit to the corresponding TBME's of the modified SDPF-M, under the condition |V (r)| < 400 MeV. One can uniquely determine the four parameters, a i , g i (i = 1, 2), by four constraints. If we do this, however, the parameters appear to be unnaturally large, while the final SM result turned out not to change much from the above one. We therefore choose the parameters fixed by the the χ 2 fit mentioned above. The obtained values are g 1 =39.9 MeV, g 2 =-409 MeV, a 1 =0.38 and a 2 =0.09. Their validity will be examined from a different viewpoint later. The Woods-Saxon depth parameter of H 0 in eq. (1) is determined so that when the HO wave function is used for 0d 3/2 , the expectation value of H 0 becomes equal to the 0d 3/2 SPE of the modified SDPF-M Hamiltonian. The same Woods-Saxon potential is used in all CCSM calculations in this paper as well as the other parameters in the Hamiltonian in eq. (3). The CCSM Hamiltonian is thus fixed.

IV. CONTINUUM STATES IN 24 O
We move to the structure of 24 O. As the filling configuration is taken, the ground state is composed of two neutrons in the 1s 1/2 orbit. This orbit is well bound with S n =2.7 MeV for 23 O [19], and hence is represented by a HO wave function. The excited state of the angular momentum J (=1,2) lies in the continuum, and is given by a superposition of basis states, where id 3/2 denotes the i-th discretized continuum state introduced earlier. In the present CCSM calculation, i max for id 3/2 corresponds to excitation energy of ∼50 MeV above threshold. After the diagonalization, we obtain the k-th eigenstate (eigen energy E k ) of the angular momentum J, where c (J,k) i denotes mixing amplitude. We discuss a reaction leading to 24 O continuum states through doorway state created by the sudden removal of a proton and a neutron from 26 F. We assume that after the removal of the proton, there is no proton in the sd shell in the doorway state. For 26 F (N =17), six neutrons are assumed to remain in 0d 5/2 of the 22 O core before and after the reaction, because of a large gap (∼ 4 MeV) between 0d 5/2 and 1s 1/2 around N =14 [17]. The remaining three neutrons are assumed to be in the 1s 1/2 -0d 3/2 space in the initial state. After the removal of one of the three neutrons, there are configurations: (a) 1s 2 1/2 , (b) 1s 1 1/2 0d 1 3/2 , and (c) 0d 2 3/2 . For (c), the initial state should be of either configurations 0d 3 3/2 or 1s 1 1/2 0d 2 3/2 , which are excluded in the filling scheme and should be indeed negligible because of a large 1s 1/2 -0d 3/2 neutron gap in nuclei being studied [2,17]. We do not discuss the configuration (a), because this is nothing but the bound ground state of 24 O. Thus, only the configuration (b) is considered for the doorway state. Since 26 F is bound, 1s 1/2 and 0d 3/2 orbits of the doorway state are expressed by the HO wave functions. The 1s 1/2 wave function keeps this feature after the reaction, because it is well bound as confirmed experimentally by Kanungo et al. [8]. However, the 0d 3/2 is not bound in 24 O, and the neutron 1s 1 1/2 0d 1 3/2 HO wave function becomes the initial state (doorway state) for the following time evolution. After the reaction, this two-neutron system of 24 O is described by the CCSM Hamiltonian in eq. (3). Its eigenstate with the energy E k is |J + k in eq. (8). The 1s 1 1/2 0d 1 3/2 doorway state is expanded by |J + k 's with amplitudes and its overlap probability is given by, b) 24 Prob. (arb. unit) Namely, with this probability, the doorway state evolves in time after the reaction as the state |J + k . The |J + k state represents a freely moving neutron at far distance with the kinetic energy E k -E 0 with E 0 being the 23 O ground-state energy. This is nothing but the emission of d 3/2 neutron at the energy E k -E 0 in the channel J + . Although the reaction phase space should be considered for the neutron emission cross section and it varies in principle as a function of E k , this change is rather minor and is neglected, because of high bombarding energy. Thus, the probability p (J) k is considered to represent the spectrum of d 3/2 neutron emitted in the channel J + except for the overall absolute magnitude. We do not discuss the absolute magnitude of the cross section. Figure 1 shows p is defined for a discrete value of E k originally, we smear it out by Gaussian function under the usual condition that the value of p (J) k is equal to the integral of this Gaussian. The width of the Gaussian is varied in accordance with the interval of discrete energies E k 's. Figures 1 (a,b) show the values of p (J) k for J π =1 and 2, as functions of the energy of emitted neutron. The solid curve shows the present CCSM calculation. This curve should be compared to the experimentally observed spectrum, except for the overall magnitude.
Regarding the comparison to experiments, we first mention the absence of bound excited states in 24   sistently with experiments [20,21]. Peak energies of various continuum spectra are depicted in Fig. 2. The CCSM results for 24 O agree very well to the experimental values reported recently by Hoffman et al. [7]. Note that the present calculation is done without adjustment to these experimental neutron spectra and also that it has been done before the experiment [22].
We next evaluate the magnitude of the continuumcoupling effect. For this purpose, we first show the results of a normal SM calculation performed by still usinĝ V in eq. (2) but replacing the discretized continuum basis states, id 3/2 , with the 0d 3/2 HO wave function. In other words, this calculation is the same as the CCSM calculation except that there is no continuum coupling, and will be referred to as bound approximation. Vertical bars in Figure 1 stand for such results. Figure 1 (a,b) indicates that J=1 and 2 continuum peaks appear, respectively, 1.4 and 1.2 MeV below the corresponding "bound approx" levels. This is purely due to the continuum coupling, as the interaction is the same between the CCSM and the bound approximation. One thus finds that the continuum coupling lowers energies by more than 1 MeV. We emphasize that a good agreement to experiments is owing to this size of the continuum coupling effect. We note also that the J π =1 + and 2 + states remain unbound after including such significant effects of continuum coupling.
The spacing between J π =1 + and 2 + peaks (levels) becomes smaller in the CCSM than in the bound approximation. This difference is consistent with experimental data as seen in Fig. 2 [23]. The two peaks with J π = 1 + and 2 + are split due to the residual interactionV in eq. (2), which works attractively for J π =2 + but repulsively for J π =1 + . The weaker splitting in the CCSM result is due to a weaker effect ofV for the wave functions of CCSM as compared to HO wave functions. Namely, in the former, even inner part is more spread than the HO wave function, making the effect ofV smaller. Note that the repulsive effect for J π =1 + disappears if we use a zerorange interaction because of the exclusion of the state of total spin, S=1. The penetrability should be different between J π =2 + and J π =1 + because of their splitting, giving smaller width to J π =2 + than to J π =1 + . Thus, the widths also reflect properties of the residual interaction. This effect is not spherical and hence may not be taken into account by renormalizing the potential barrier height. Figure 1 (a,b) includes the results obtained by settinĝ V = 0, i.e., no residual interaction. This is referred to as "no int." in the figure. In this case, the neutron spectrum becomes identical to the neutron emission from the doorway state formed by the 0d 3/2 HO wave function on top of the 22 O core. This is nothing but the pure single-particle picture and the J π dependence disappears, as is clear in Fig. 1. Figure 2 exhibits also the levels of the fully mixed sdshell calculation by SDPF-M. First of all, these levels are pretty high in energy. We can confirm that these levels are nearly identical to the corresponding levels of the bound approximation withV of Eq. (2) in the filling scheme, reinforcing the validity of the usage ofV in the present case. Figure 2 includes the levels of the normal sd-shell calculation by usd-b interaction [24], where continuum effects seem to be included into the effective interaction to a certain extent.

V. WAVE FUNCTIONS OF CONTINUUM STATES IN 24 O
We next discuss the density distribution of two valence neutrons with one in continuum d 3/2 orbits. The density ρ(r) is defined in general as the expectation value of the operator ρ(r) ≡ψ † (r)ψ(r) = m,n u * m (r)u n (r)a † m a n , where u m (r) is the spatial wave function of the singleparticle state with quantum number m, and a † (a) denotes usual creation (annihilation) operator. We consider hereafter the density with the angular integration : This makes sense because one neutron occupies always one of the discretized d 3/2 orbits, while the other valence neutron is in the s orbit. This quantity is expressed as . Solid lines indicate densities (arbitrary unit) of the corresponding single-particle "resonance" with phase shift of π/2 with the same energy.
Clearly, ρ kd represents the contribution from discretized continuum d 3/2 orbits to the kth discrete eigenstate. Figure 3 (a) and (b) show, respectively, r 2 ρ kd (r), for the appropriate eigenstates denoted by k which are close to 1 + and 2 + peaks in Figs. 1 (a) and (b). Both show enlarged densities in and near the oxygen nucleus, while the enlargement differs between (a) and (b). The first node of the radial wave function is somewhat further away for 2 + than for 1 + . These density distributions are compared to the one calculated by harmonic oscillator 0d 3/2 . The large differences between the CCSM and HO results are seen, as expected. In addition, one sees notable differences between 1 + and 2 + , which are consequences of the different configuration mixing in continuum due to the nuclear force.
We next discuss how the CCSM wave functions can be compared to scattering states, particularly, resonance ones. For this purpose, we consider a linear combination of the incoming and outgoing states of d 3/2 with the phase shift π/2. As only asymptotic behavior is relevant here, wave functions of a free particle are considered. The energy is fixed to be the same as that of the emitted neutron from the peaks of Figs. 1 (a) and (b). This phase shift corresponds apparently to the single-particle resonance. We simply assume it without dealing with the potential. The resultant densities are shown also in Figs. 3 (a) and (b) as S.P. "resonance". Note that the S.P. "resonance" density has no normalization factor, and only its shape matters here. The asymptotic behavior of the density of the S.P. "resonance" is very close to that of CCSM densities at larger r as seen in Fig. 3 (a). For the 2 + state in 24 O, the S.P. "resonance" density shows some deviation from the CCSM density (see Fig. 3 (b)). It becomes basically identical to that of CCSM density at even larger distance as shown in Fig. 4, while this slower convergence for the 2 + state is of a certain interest. Thus, the CCSM calculation always gives the correct wave length for the emitted neutron asymptotically, and furthermore the wave function at the peak has the same phase shift to what we can expect from single-particle resonance at its energy.

VI. RESONANCE IN NEUTRON SCATTERING
In this section, we examine the similarity between the spectrum of the present approach and that of the neutron resonance on 22 O nucleus. We can calculate the phase shift of the neutron scattering for the potential of eq. (1) without the wall. Figure 5 (a) shows the (tangent of) phase shift thus calculated, which exhibits a clear pattern of the resonance. Figure 6 (a) shows two spectra. One is of them (dashed line) is the spectrum of a neutron emitted from the HO d 3/2 doorway state built on the 22 O core. This is a straightforward application of the where k is the usual notation of the momentum and J (l) denotes the total (orbital) angular momentum of the channel. In this paper, l=2, as only d wave is considered for continuum. The similarity is remarkable between the two results, while the vertical scale is adjusted. In the resonance state, the incoming particle with the resonance energy is considered to stay inside the nucleus for the resonance life time. If this meta-stable state inside the nucleus has a wave function somewhat similar to the HO wave function, the outgoing particle follow a spectrum resembling that of the HO doorway state emission. The similarity between the meta-stable wave function and the HO wave function is expected naturally because they reflect attractive mean effects of the nucleus, but these wavefunction cannot be identical either. The meta-stable one should be more diffuse.
It is then of interest to see the cases of 24 O. Since the 1 + and 2 + states in 24 O are not of single-particle nature, we introduce effective phase shift first. The wave functions of two valence neutrons can be rewritten, by The effective phase shifts are obtained from asymptotic behavior of the wave function of the state |d 3/2;J,k . Figure 5 (b) shows the (tangent of) effective phase shift thus obtained. The effective phase shift behaves quite similarly to the phase shift of single-particle state, while it makes sense only in a particular channel, which is specified by J in the present study. The actual wave function of a particle in continuum corresponds to a certain superposition of wave functions of different channels, and therefore there is no unique phase shift describing the scattering. We can still use the effective phase shift for a particular channel very conveniently. The "resonance" energy can be shifted due to the correlation by the interaction in eq. (2). Figure 6 (b) shows four spectra. Two of them (dashed line) are those shown in Fig.1. The other two (solid line) are the cross sections calculated by eq. (14) with the effective phase shifts. The spectra obtained by the two methods appear to be similar to each other, while certain differences are visible also. We stress that these two methods are for different processes and certain differences are rather natural. Table I summarizes peak energies and widths of the doorway-state emission spectra (shown as "CCSM") and the phase-shift spectra (shown as "phase shift"), in comparison to experiments.
The width of the doorway-state emission spectrum shows larger widths than the corresponding phase-shift spectrum (see also Fig. 5). The difference here is considered to stem from the difference between the doorway state (HO wave function) of the knockout reaction and the inner wave function of the phase-shift approaches. In the former, radial wave functions are more confined inside the nucleus, making the mixing with various continuum states stronger and the width larger. Table I indicates that the difference of the width is largest for the 2 + state in 24 O. This fact is consistent with the above argument as the 2 + state has the lowest peak among the peaks shown in Table I, and its inner wave function should be most different from HO wave function.

VII. LOW-LYING STATES IN 25 O AND 26 O
We now discuss the continuum properties of 25 O and 26 O. As for 25 O, there are three neutrons in the present model space. We diagonalize the Hamiltonian in eq. (3), using the basis, |(1s 1/2 ) 2 ⊗ id 3/2 ; 3/2 + (i = 1, · · · , i max ). Namely, one neutron is placed in the continuum d 3/2 space, for simplicity. We discuss the reaction of single proton knockout from 26 F, namely, 9 Be( 26 F, 25 O)X [7]. Here we assume, by a similar argument to the 24 O case, that the neutron emission probability is proportional to p (J) k calculated in this configuration space. The resultant peak energy is included in Fig. 2, exhibiting a good agreement to experiment [7]. The spectrum of the emitted neutron is shown in Fig. 1(c). The density distribution of this continuum d 3/2 state is shown in Fig. 3.
The ground and the first 2 + states of 26 O are calculated within the basis states: |(1s 1/2 ) 2 ⊗ id 2 3/2 ; J + (i = 1, · · · , i max ). These states turn out to be unbound in agreement with experiment [10,11]. The resultant peak energies are included in Fig. 2. The 0 + -2 + spacing in the CCSM result is smaller than in the normal SM, which may indicate a weakening of the pairing gap. Note that the density dependence of the pairing interaction is not considered here. The results obtained by the usdb interaction [24] are shown too. Note that a part of continuum effects are included in the usdb TBME's by fit.  I: Peak energies and widths of spectra of emitted neutrons. Peak energies are measured from the particle-emission thresholds [MeV]. "CCSM" refers to spectra of emitted neutron from HO doorway states, while "phase shift" to those of neutron scattering calculated in the phase-shift approach. Results of the normal shell model calculation in the full sdshell and the bound-state approximation are shown. GSM results with the same Hamiltonian are exhibited. 23

VIII. CONVERGENCE OF CCSM WALL POSITION
We shall comment on the convergence of the results with respect to the position of the infinite wall, L. In Fig. 7, the peak energies discussed already for oxygen isotopes are plotted as a function of L. It is seen that the results do not change so much if the value of L is taken to be sufficiently large. It should be noted that the results with even usual values of L ∼ 50 fm are not stable. Note that the value of L is fixed to 3000 fm in this paper.

IX. GSM CALCULATION BY THE SAME HAMILTONIAN
The present continuum properties can be studied by other methods, e.g., Gamow Shell Model (GSM) [26,27] and Coupled Channel calculation [28]. The shell model in continuum (CSM) [29] is another approach where a non-hermitian energy-dependent Hamiltonian is used, by evaluating continuum coupling through one-body potential.
We here investigate the relation of the GSM [26,27] to the CCSM using the present Hamiltonian, Eq. (3).
The GSM is one of the extensions of the normal shell model to continuum, by introducing the so-called Gamow basis. The Gamov basis is generated by analytically continued Schrödinger equation where µ is the reduced mass between a neutron and 22 O core in this study. V l (k, k ′ ) is the Fourier-Bessel transform of spherical Woods-Saxon potential which is the same as in Eq. (1) except for the wall. TBME's of the effective interaction in Eq. (2) are calculated following the prescription in [27]. Diagonalizing the Hamiltonian matrix which is complex and symmetric, we obtain the eigenstate whose imaginary part is interpreted as the width of the resonance state. Note that the GSM provides us with resonance properties. This calculation produces practically the same peak energies but smaller (by 5%∼30%) widths (see Table I) as compared to the CCSM calculations with the same Hamiltonian. The CCSM phase-shift calculations produce results in between or closer to CCSM doorway-state-emission calculation.
Since we discretize the integral equation (16), we have to check the convergence of the GSM results as a function of the number of single-particle states belonging to the Gamow basis. We confirmed that converged results can be obtained with 30 basis states.
In the GSM calculation, there is a degree of freedom to choose the deformed contour. We use two different types of the contours as shown in Fig. 8, specified by the points A(0.3 − 0.2i fm −1 ), B(0.5 fm −1 ) and C(0.3 − 0.2ifm −1 ). It has been confirmed that the result does not change much between these two contours. FIG. 9: (color online) Single-particle resonance state for 23 O and a free-particle state with δ l = π/2.

X. METHODOLOGICAL RELATION BETWEEN CCSM AND GSM
We shall look into the methodological relation of the CCSM calculation to the GSM.
The resonance energy can be determined in general by two different ways. One is to determine it as a point where the phase shift goes through π/2, as has been done with CCSM wave functions. The other one is to treat a resonance state as a pole of the S-matrix. The latter definition is equivalent to the situation that the radial wave function of the resonant state has the pure outgoing boundary condition [33]. These two are of course related to each other, but their mathematical relation, which may have been noticed, appears to be not so simple, as discussed below. The asymptotic behavior of the radial wave function is given by the combination of the Hankel functions R l (r) → Ah where k= √ 2µE/ . The coefficients can be expressed as A=e 2iδ l (k) /2 = S l (k)/2 and B=1/2. One then obtains at large r limit, At the resonance state defined by δ l = π/2, R l (r) is represented as a standing wave, but not an outgoing wave as in the case of complex approaches. The asymptotic wave functions discussed in Figs. 3 and 4 correspond to this situation. Figure 9 shows the relation between the wave function of the single-particle resonance of 23 O shown in Fig.5 (a) and that of the free particle of the phase shift δ l = π/2. If the phase shift δ l sharply increases by about π in the vicinity of k = k 0 (k, k 0 ∈ R), going through π/2 at k 0 , the energy dependence may be approximated by It then follows that near k = k 0 , where E 0 and Γ correspond to the peak energy and the width of the peak, respectively. Combining with eq. (14), this leads us to the Breit-Wigner formula, with Another usual approach of the resonance phenomenon is the S-matrix [31]. The S-matrix takes, for real k's, the form When this S-matrix is analytically continued to the complex momentum-plane, S l has a pole at k = k 0 − ik 1 in the lower half of the complex k-plane. We then obtain E 0 and Γ, as discussed in [34], from the pole of the S-matrix, Thus, the pole of S-matrix corresponds to the resonance state, only if this pole in the complex k-plane is close enough to the positive real axis and well isolated from other poles. If one applies the width for the 1 + state of 24 O shown in Table I, the estimated shift of the peak energy turn out to be about 0.004 MeV, which is small enough. The peak energies and widths obtained by the CCSM doorway-state-emission and phase-shift calculations can be compared with GSM results in Table I for the same Hamiltonian. At first glance, the peak energies are in a good agreement among the three calculations. The width becomes smaller in going from the CCSM doorwaystate-emission to the CCSM phase-shift, and to the GSM calculations in general. The difference may be partly due to the coupling/mixing between basis states evaluated differently among the three, and also due to the different boundary conditions as discussed.
We comment on the relation to Ref. [36], where the microscopically derived two-body interaction is used for GSM calculation. The oxygen isotopes were studied. As well known, the direct use of microscopic interaction into the shell model calculation does not work for bound states. The same feature is seen in continuum calculations in Ref. [36]. This problem has been avoided in the present work, focusing on the continuum coupling effect. We note that the present work has been started in 2002 [37], and has been refined in 2008 [22]. The present work does not owe to results of [36], as it has been completed before [36].

XI. SUMMARY
In summary, we introduced the Continuum-Coupled Shell Model (CCSM). The CCSM Hamiltonian is designed to include correlations due to a two-body nuclear force in the continuum, in a manner consistent with the realistic SM interaction for bound states. Spectra of emitted neutrons from knockout reactions are calculated, in good agreements to recent experiments without adjustment to final results. The continuum effects are significant; the difference from bound-state calculation is more than 1 MeV for the cases studied. We point out that the 1 + -2 + spacing of 24 O is smaller in the CCSM than in the bound-state calculation, and that the CCSM result indeed agrees well with the recent experiment [7]. This suggests that the appropriate treatment of the nuclear force is still important in the continuum. We also calculated the resonance states by a phase shift approach and the Gamow Shell Model (GSM) using the same Hamiltonian (Eq. (3)). The peak energies agree with those of CCSM doorway-state emission calculation rather will, although the doorway-state emission and the neutron-scattering resonance are very different processes in principle. The width of the resonance states were obtained to be smaller than the case of CCSM doorway-state-emission calculation. This difference originates partly in the differences between the doorway and the resonance states. On the other hand, the differences among the CCSM (doorway), CCSM (phase shift) and GSM appear rather minor, particularly as far as the peak energies are concerned.
The present work suggests that the high single-particle energies for the bound 0d 3/2 , which are used for shellmodel calculations for bound states, are indeed consistent with observed continuum properties. If they were not high enough, the continuum spectra would become too low compared to experiments. There should be a strong shell evolution mechanism so that this orbit comes down in F isotopes [2,3,35]. As the CCSM idea naturally extends the successful SM interaction to continuum states (or open quantum systems), its future developments into full configurations and more applications to exotic nuclei are intriguing issues. For instance, the continuum "ground state" of 26,28 O attracts much interest as a possible "mother of pearl" of neutron nugget/matter. The continuum effect was not taken into account for bound states in this paper for simplicity. There may be some cases where the continuum couplings can be crucial even for bound states, e.g., 11 Li.
We thank Prof. K. Kato for valuable comments. This work has been supported in part by JSPS core-to-core program, EFES and by grant-in-aid for Scientific Research (A) 20244022. KT thanks JSPS for a fellowship.