Classification of resonances and pairing effects on $NA$-scattering within the HFB framework

The purpose of this subsection is to show that the K-matrix poles are the Sturm-Liouville eigen value. For simplicity, we start the discussion within the Hartree-Fock (HF) framework ignoring the pairing. The pairing effect is discussed in the latter section.

Within the HF framework, the Lippmann-Schwinger equation is given by

	$\displaystyle\psi_{0,lj}^{(+)}(r;\epsilon)$	$\displaystyle=$	$\displaystyle F_{l}(kr)$
			$\displaystyle+\int_{0}^{\infty}dr^{\prime}G_{F,l}^{(+)}(r,r^{\prime};k)U_{lj}(r^{\prime})\psi_{0,lj}^{(+)}(r^{\prime};\epsilon)$

with the HF potential $U_{lj}$, where $\epsilon$ is the incident energy ($\epsilon=\frac{\hbar^{2}k^{2}}{2m}$) and $G_{F,lj}(r,r^{\prime};\epsilon)$ is the free particle Green’s function defined by

	$\displaystyle G_{F,l}^{(\pm)}(r,r^{\prime};k)$	$\displaystyle\equiv$	$\displaystyle\mp i\frac{2mk}{\hbar^{2}}\left[\theta(r-r^{\prime})F_{l}(kr^{\prime})O^{(\pm)}_{l}(kr)\right.$		(2)
			$\displaystyle+\left.\theta(r^{\prime}-r)F_{l}(kr)O^{(\pm)}_{l}(kr^{\prime})\right].$

With the scaled riccati-spherical Bessel functions defined by $F_{l}(kr)=rj_{l}(kr)$ and $O^{(\pm)}_{l}(kr)=rh_{l}^{(\pm)}(kr)$.

It should be noted that $\psi_{0,lj}^{(+)}(r;\epsilon)$ is defined by

$\displaystyle\psi_{0,lj}^{(+)}(r;\epsilon)=\frac{u_{lj}(r;\epsilon)}{J_{0,lj}^{(+)}(\epsilon)},$

(3)

where $u_{lj}(r;\epsilon)$ is the regular solution of the HF equation for a given energy $\epsilon$.

$J_{0,lj}^{(+)}(\epsilon)$ is the HF Jost function. There are three types of the expressions of the Jost function as following.

	$\displaystyle J_{0,lj}^{(\pm)}(\epsilon)$	$\displaystyle=$	$\displaystyle 1\mp\frac{k}{i}\frac{2m}{\hbar^{2}}\int drO_{l}^{(\pm)}U_{lj}(r)u_{lj}(r;\epsilon)$		(4)
		$\displaystyle=$	$\displaystyle 1\mp\frac{k}{i}\frac{2m}{\hbar^{2}}\int drF_{l}^{(\pm)}U_{lj}(r)v^{(\pm)}_{lj}(r;\epsilon)$		(5)
		$\displaystyle=$	$\displaystyle\pm\frac{k}{i}W_{lj}(u,v^{(\pm)}).$		(6)

where $v^{(\pm)}_{lj}(r;\epsilon)$ is the HF irregular solution which satisfies the asymptotic boundary condition $\lim_{r\to\infty}v^{(\pm)}_{lj}(r;\epsilon)\to O^{(\pm)}_{l}(kr)$, and $W_{lj}(u,v^{(\pm)})$ is the Wronskian.

The S-matrix is defined as

$\displaystyle S_{lj}^{(0)}(\epsilon)\equiv\frac{J_{0,lj}^{(-)}(\epsilon)}{J_{0,lj}^{(+)}(\epsilon)},$

(7)

and the S-matrix pole $\epsilon_{R}^{0}$ is given by

$\displaystyle J_{0,lj}^{(+)}(\epsilon_{R}^{0})=0.$

(8)

It is, therefore, clear that Eq.(LABEL:LSHF) can be rewritten as

$\displaystyle u_{lj}(r;\epsilon_{R}^{0})$

$\displaystyle=$

$\displaystyle\int_{0}^{\infty}dr^{\prime}G_{F,l}^{(+)}(r,r^{\prime};k_{R})U_{lj}(r^{\prime})u_{lj}(r^{\prime};\epsilon_{R}^{0})$

(9)

for $\epsilon=\epsilon_{R}^{0}$.

By introducing the standing wave Green’s function $PG_{F,l}$ defined by

	$\displaystyle G_{F,l}^{(\pm)}(r,r^{\prime};k)$	$\displaystyle=$	$\displaystyle PG_{F,l}(r,r^{\prime};k)$		(10)
			$\displaystyle\mp i\frac{2mk}{\hbar^{2}}F_{l}(kr)F_{l}(kr^{\prime}),$

Eq.(LABEL:LSHF) can be rewritten as

	$\displaystyle\psi_{0,lj}^{(+)}(r;\epsilon)$	$\displaystyle=$	$\displaystyle\left(1-iT^{(0)}_{lj}(\epsilon)\right)F_{l}(kr)$
			$\displaystyle+\int_{0}^{\infty}dr^{\prime}PG_{F,l}(r,r^{\prime};k)U_{lj}(r^{\prime})\psi_{0,lj}^{(+)}(r^{\prime};\epsilon)$

where $T^{(0)}_{lj}$ is the T-matrix within the HF framework which can be calculated by

$\displaystyle T^{(0)}_{lj}(\epsilon)$

$\displaystyle=$

$\displaystyle\frac{2mk}{\hbar^{2}}\int_{0}^{\infty}drF_{l}(kr)U_{lj}(r)\psi^{(+)}_{0,lj}(r;\epsilon)$

(12)

The K-matrix $K_{lj}^{(0)}$ is expressed by the T-matrix $T^{(0)}_{lj}$ as

$\displaystyle K_{lj}^{(0)}(\epsilon)$

$\displaystyle=$

$\displaystyle\frac{T^{(0)}_{lj}(\epsilon)}{1-iT^{(0)}_{lj}(\epsilon)},$

(13)

and the standing wave function $\psi_{0,lj}^{(S)}(r;\epsilon)$ is defined by

	$\displaystyle\psi_{0,lj}^{(S)}(r;\epsilon)$	$\displaystyle\equiv$	$\displaystyle\left(1+iK_{lj}^{(0)}(\epsilon)\right)\psi_{0,lj}^{(+)}(r;\epsilon)$		(14)
		$\displaystyle\equiv$	$\displaystyle\frac{1}{1-iT_{lj}^{(0)}(\epsilon)}\psi_{0,lj}^{(+)}(r;\epsilon)$		(15)

Eq.(LABEL:LSHF2) can be rewritten as

	$\displaystyle\psi_{0,lj}^{(S)}(r;\epsilon)$	$\displaystyle=$	$\displaystyle F_{l}(kr)$
			$\displaystyle+\int_{0}^{\infty}dr^{\prime}PG_{F,l}(r,r^{\prime};k)U_{lj}(r^{\prime})\psi_{0,lj}^{(S)}(r^{\prime};\epsilon).$

If there is an energy $\epsilon_{n}^{0}$ which is given by

$\displaystyle 1-iT^{(0)}_{lj}(\epsilon_{n}^{0})=0,$

(17)

then Eq.(LABEL:LSHF3) can be rewritten as

$\displaystyle\psi_{0,lj}^{(+)}(r;\epsilon_{n}^{0})$

$\displaystyle=$

$\displaystyle\int_{0}^{\infty}dr^{\prime}PG_{F,l}(r,r^{\prime};k_{n}^{0})U_{lj}(r^{\prime})\psi_{0,lj}^{(+)}(r^{\prime};\epsilon_{n}^{0}).$

using Eq.(15), also $\epsilon_{n}^{0}$ is a pole of the K-matrix by Eq.(13). Eq.(17) can be represented by using the Jost function as

$\displaystyle J_{0,lj}^{(+)}(\epsilon_{n}^{0})+J_{0,lj}^{(-)}(\epsilon_{n}^{0})=0.$

(19)

because the K-matrix can be represented by using the Jost function as

$\displaystyle K_{lj}^{(0)}(\epsilon)=i\left(\frac{J_{0,lj}^{(-)}(\epsilon)-J_{0,lj}^{(+)}(\epsilon)}{J_{0,lj}^{(-)}(\epsilon)+J_{0,lj}^{(+)}(\epsilon)}\right)$

(20)

It can be proved that the Jost function has a symmetric property $J_{0,lj}^{(-)}(\epsilon)=J_{0,lj}^{(+)*}(\epsilon^{*})$. This requires $\epsilon_{n}^{0}$ to be a real number.

The Sturm-Liouville theory is the theory of the second-order differential equations of the form:

$\displaystyle\frac{\partial}{\partial r}\left[p(r)\frac{\partial\hat{\phi}(r)}{\partial r}\right]+q(r)\hat{\phi}(r)=-\nu w(r)\hat{\phi}(r),$

(21)

where $p(r)$, $q(r)$ and $w(r)$ are positive definite coefficient functions. For the HF equation, the coefficient functions are given by

	$\displaystyle p(r)$	$\displaystyle=$	$\displaystyle\frac{\hbar^{2}}{2m}$		(22)
	$\displaystyle q(r)$	$\displaystyle=$	$\displaystyle\epsilon-\frac{\hbar^{2}l(l+1)}{2mr^{2}}$		(23)
	$\displaystyle w(r)$	$\displaystyle=$	$\displaystyle-U_{lj}(r),$		(24)

when $\epsilon>0$. $\nu$ is called the eigen value of the Sturm-Liouville equation. According to the Sturm-Liouville theory, there are infinite number of real eigenvalues $\nu_{n}$ which can be numbered so that $\nu_{1}<\nu_{2}<\cdots\nu_{\infty}$ (Mercer’s theorem Mercer ). An eigen function $\hat{\phi}_{n,lj}(r)$ has $n-1$ nodes inside the potential. Note that the orthogonality of $\hat{\phi}_{n,lj}$ is given by

$\displaystyle\int_{0}^{\infty}dr\hat{\phi}_{m,lj}(r)w(r)\hat{\phi}_{n,lj}(r)=\delta_{mn}.$

(25)

Completeness is given by

	$\displaystyle\sum_{n=1}^{\infty}w(r)\hat{\phi}_{n,lj}(r)\hat{\phi}_{n,lj}(r^{\prime})$
	$\displaystyle=\sum_{n=1}^{\infty}\hat{\phi}_{n,lj}(r)\hat{\phi}_{n,lj}(r^{\prime})w(r^{\prime})$
	$\displaystyle=\delta(r-r^{\prime}).$		(26)

The eigensolution $\hat{\phi}_{n,lj}$ satisfies

	$\displaystyle\hat{\phi}_{n,lj}(r)$
	$\displaystyle=\int_{0}^{\infty}dr^{\prime}PG_{F,l}(r,r^{\prime};k)\nu_{n}U_{lj}(r^{\prime})\hat{\phi}_{n,lj}(r^{\prime}).$		(27)

If we suppose that there is an eigen value which satisfies $\nu_{n}(\epsilon=\epsilon^{0}_{n})=1$, it is very easy to notice that Eq.(LABEL:LSHF3-en) and Eq.(27) are equivalent.

Therefore, the K-matrix pole has the physical meaning as the standing wave eigen solution of the Sturm-Liouville eigen value problem, and it is slightly different from the S-matrix pole. Because the condition for S-matrix pole(Eq.(8)) is not required to have the condition for the K-matrix pole (Eq.(19)). Eqs.(8) and (19) are independent each other. This implies that the S-matrix pole and K-matrix pole may have the different properties.

In order to discuss the paring dependence of poles of the S- and K-matrix, it is necessary to introduce the definition of the S- and K-matrix which are described within the HFB framework. In this section, we shall derive the definition of the K-matrix with the help of the HFB Jost function.

By using the HFB Jost function, the S-matrix is expressed as

	$\displaystyle S_{lj}(E)=\sum_{s=1,2}\left(\mathcal{J}_{lj}^{(+)}(E)\right)^{-1}_{1s}\left(\mathcal{J}_{lj}^{(-)}(E)\right)_{s1}$
	$\displaystyle=\frac{(\mathcal{J}_{lj}^{(+)}(E))_{22}(\mathcal{J}_{lj}^{(-)}(E))_{11}-(\mathcal{J}_{lj}^{(+)}(E))_{12}(\mathcal{J}_{lj}^{(-)}(E))_{21}}{(\mathcal{J}_{lj}^{(+)}(E))_{22}(\mathcal{J}_{lj}^{(+)}(E))_{11}-(\mathcal{J}_{lj}^{(+)}(E))_{12}(\mathcal{J}_{lj}^{(+)}(E))_{21}}$
	$\displaystyle=\frac{\det\mathcal{J}_{lj}^{(+)*}(E^{*})}{\det\mathcal{J}_{lj}^{(+)}(E)}$		(28)

where $E$ is the complex quasiparticle energy. The incident energy $\epsilon$ and the quasiparticle energy $E$ are related as

$\displaystyle\epsilon=\lambda+E$

(29)

where $\lambda(<0)$ is the chemical potential (Fermi energy) given in the HFB framework.

The denominator of Eq.(28) is the determinant of the HFB Jost function, i.e.

	$\displaystyle\det\mathcal{J}_{lj}^{(+)}(E)$
	$\displaystyle=(\mathcal{J}_{lj}^{(+)}(E))_{22}(\mathcal{J}_{lj}^{(+)}(E))_{11}-(\mathcal{J}_{lj}^{(+)}(E))_{12}(\mathcal{J}_{lj}^{(+)}(E))_{21}.$

and the numerator is also expressed as

	$\displaystyle\det\mathcal{J}_{lj}^{(+)*}(E^{*})$
	$\displaystyle=(\mathcal{J}_{lj}^{(+)}(E))_{22}(\mathcal{J}_{lj}^{(-)}(E))_{11}-(\mathcal{J}_{lj}^{(+)}(E))_{12}(\mathcal{J}_{lj}^{(-)}(E))_{21}$

due to the symmetric properties of the HFB Jost function given by

	$\displaystyle(\mathcal{J}_{lj}^{(\pm)*}(E^{*}))_{11}$	$\displaystyle=$	$\displaystyle(\mathcal{J}_{lj}^{(\mp)}(E))_{11}$		(32)
	$\displaystyle(\mathcal{J}_{lj}^{(\pm)*}(E^{*}))_{12}$	$\displaystyle=$	$\displaystyle(-)^{l}(\mathcal{J}_{lj}^{(\pm)}(E))_{12}$		(33)
	$\displaystyle(\mathcal{J}_{lj}^{(\pm)*}(E^{*}))_{21}$	$\displaystyle=$	$\displaystyle(-)^{l}(\mathcal{J}_{lj}^{(\mp)}(E))_{21}$		(34)
	$\displaystyle(\mathcal{J}_{lj}^{(\pm)*}(E^{*}))_{22}$	$\displaystyle=$	$\displaystyle(\mathcal{J}_{lj}^{(\pm)}(E))_{22}$		(35)

The S-matrix pole $E_{R}$ is determined by

$\displaystyle\det\mathcal{J}_{lj}^{(+)}(E_{R})=0.$

(36)

The T-matrix is given by

	$\displaystyle T_{lj}(E)$	$\displaystyle=$	$\displaystyle\frac{i}{2}\left(S_{lj}(E)-1\right)$		(37)
		$\displaystyle=$	$\displaystyle\frac{i}{2}\left(\frac{\det\mathcal{J}_{lj}^{(+)*}(E^{*})-\det\mathcal{J}_{lj}^{(+)}(E)}{\det\mathcal{J}_{lj}^{(+)}(E)}\right),$		(38)

and the K-matrix is

	$\displaystyle K_{lj}(E)$	$\displaystyle=$	$\displaystyle\frac{T_{lj}(E)}{1-iT_{lj}(E)}$		(39)
		$\displaystyle=$	$\displaystyle i\left(\frac{\det\mathcal{J}_{lj}^{(+)*}(E^{*})-\det\mathcal{J}_{lj}^{(+)}(E)}{\det\mathcal{J}_{lj}^{(+)*}(E^{*})+\det\mathcal{J}_{lj}^{(+)}(E)}\right)$		(40)

The K-matrix pole $E_{n}$ is determined by

$\displaystyle\det\mathcal{J}_{lj}^{(+)*}(E_{n})+\det\mathcal{J}_{lj}^{(+)}(E_{n})=0.$

(41)

In order to analyze the pairing effect, the two potential formula may be useful. Here, we shall derive the two potential formula for the K-matrix. As is introduced in jost-fano , the S-matrix and T-matrix can be divided into the HF and pairing parts as

	$\displaystyle S_{lj}(E)$	$\displaystyle=$	$\displaystyle S^{(0)}_{lj}(E)S^{(1)}_{lj}(E)$		(42)
	$\displaystyle T_{lj}(E)$	$\displaystyle=$	$\displaystyle T^{(0)}_{lj}(E)+T^{(1)}_{lj}(E)S^{(0)}_{lj}(E)$		(43)

By inserting Eq.(43) into Eq.(39), we obtain

$\displaystyle K_{lj}(E)$

$\displaystyle=$

$\displaystyle\frac{K^{(0)}_{lj}(E)+K^{(1)}_{lj}(E)}{1-K^{(0)}_{lj}(E)K^{(1)}_{lj}(E)}$

(44)

with

	$\displaystyle K^{(0)}_{lj}(E)$	$\displaystyle=$	$\displaystyle\frac{T^{(0)}_{lj}(E)}{1-iT^{(0)}_{lj}(E)}$		(45)
	$\displaystyle K^{(1)}_{lj}(E)$	$\displaystyle=$	$\displaystyle\frac{T^{(1)}_{lj}(E)}{1-iT^{(1)}_{lj}(E)}.$		(46)

As shown in jost-hfb , we can find that the partial cross section for $s_{1/2}$, $p_{1/2}$, $p_{3/2}$, $d_{5/2}$, $f_{5/2}$ and $g_{9/2}$ have the contribution for the total cross section of the neutron elastic scattering at the zero pairing limit ($\langle\Delta\rangle=0$ MeV).

In Figs.1-3, we show the square of the T-matrix $|T^{(0)}_{lj}|^{2}$ in the panel (a), this is a quantity which is directly related with the partial component of the cross section. The inverse of the K-matrix is shown in the panel (b) in order to check the existence of the K-matrix pole on the real axis of the incident energy. In panel (c), the phase shift which is determined by the S-matrix as $S_{lj}^{(0)}=e^{2i\delta^{0}_{lj}}$ is plotted as a function of the incident neutron energy $\epsilon$. The square of the Jost function $|J_{0,lj}(\epsilon)|^{2}$ are shown on the complex-$\epsilon$ plane in order to show the S-matrix pole.

In Fig.1, a peak of $|T^{(0)}_{lj}|^{2}$ is found at $\epsilon^{0}_{n}=7.81$ MeV for $f_{5/2}$, $\epsilon^{0}_{n}=9.48$ MeV for $g_{9/2}$, respectively. The energy of peak is located at the energy which satisfies $1/K^{(0)}_{lj}(\epsilon^{0}_{n})=0$. At $\epsilon=\epsilon^{0}_{n}$, the phase shift is obviously $\delta_{0}=\frac{\pi}{2}$ by definition. In the panel (d), we can find the S-matrix pole at $\epsilon_{R}^{0}=6.75-i1.63$ MeV for $f_{5/2}$ and $\epsilon_{R}^{0}=9.06-i1.14$ MeV for $g_{9/2}$, respectively. However, $\epsilon_{n}$ is slightly different from different from the real part of the S-matrix pole $\epsilon^{0,r}_{R}$.

When there is a S-matrix pole, $J_{0,lj}^{(+)}(\epsilon)$ is expressed by the first order Taylor expansion around $\epsilon=\epsilon_{R}^{0}$ as

$\displaystyle J_{0,lj}^{(+)}(\epsilon)\sim(\epsilon-\epsilon_{R}^{0})\left.\frac{dJ_{0,lj}^{(+)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0}}$

(47)

where $\epsilon_{R}^{0}=\epsilon_{R}^{0,r}-i\epsilon_{R}^{0,i}$. $\epsilon_{R}^{0,i}$ is necessary to be small for this approximation.

The S-matrix is also approximately expressed as

$\displaystyle S^{(0)}_{lj}(\epsilon)$

$\displaystyle=$

$\displaystyle\frac{J_{0,lj}^{(-)}(\epsilon)}{J_{0,lj}^{(+)}(\epsilon)}\sim\frac{(\epsilon-\epsilon_{R}^{0*})\left.\frac{dJ_{0,lj}^{(-)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0*}}}{(\epsilon-\epsilon_{R}^{0})\left.\frac{dJ_{0,lj}^{(+)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0}}}$

(48)

Therefore we can obtain

$\displaystyle S^{(0)}_{lj}(\epsilon=\epsilon_{R}^{0,r})\sim-\frac{\left.\frac{dJ_{0,lj}^{(-)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0,*}}}{\left.\frac{dJ_{0,lj}^{(+)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0}}}$

(49)

Since the S-matrix $S^{(0)}_{lj}(\epsilon=\epsilon_{R}^{0,r})$ is expressed as $S^{(0)}_{lj}(\epsilon=\epsilon_{R}^{0,r})=e^{2i\delta_{R}^{0}}$ by using the phase-shift $\delta_{R}^{0}$ at $\epsilon=\epsilon_{R}^{0,r}$, we can obtain

$\displaystyle\left.\frac{dJ_{0,lj}^{(-)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0,*}}\sim-e^{2i\delta_{R}^{0}}\left.\frac{dJ_{0,lj}^{(+)}(\epsilon)}{d\epsilon}\right|_{\epsilon=\epsilon_{R}^{0}}$

(50)

Applying Eqs.(47) and (50) to the condition Eq.(19), we can obtain

$\displaystyle\epsilon_{n}^{0}$

$\displaystyle\sim$

$\displaystyle\epsilon_{R}^{0,r}+\epsilon_{R}^{0,i}\cot\delta_{R}^{0}$

(51)

Using Eqs.(48) and (51), the approximated expression of the T-matrix can be obtained as

	$\displaystyle T^{(0)}_{lj}(\epsilon)$
	$\displaystyle=\frac{i}{2}\left(S^{(0)}_{lj}(\epsilon)-1\right)$
	$\displaystyle\sim\frac{-i}{\epsilon-\epsilon^{0}_{n}+\epsilon_{R}^{0,i}\frac{e^{i\delta_{R}^{0}}}{\sin\delta_{R}^{0}}}\left[(\epsilon-\epsilon_{n}^{0})e^{i\delta_{R}^{0}}\cos\delta_{R}^{0}+\epsilon_{R}^{0,i}\frac{e^{i\delta_{R}^{0}}}{\sin\delta_{R}^{0}}\right]$
			(52)

If $\delta_{R}^{0}$ is supposed to be $\delta_{R}^{0}\sim\frac{\pi}{2}$, then Eq.(52) becomes

$\displaystyle T^{(0)}_{lj}(\epsilon)\sim\frac{\epsilon_{R}^{0,i}}{\epsilon-\epsilon_{n}^{0}+i\epsilon_{R}^{0,i}}$

(53)

because

	$\displaystyle\frac{e^{i\delta_{R}^{0}}}{\sin\delta_{R}^{0}}$	$\displaystyle\sim$	$\displaystyle i$		(54)
	$\displaystyle\cos\delta_{R}^{0}$	$\displaystyle\sim$	$\displaystyle 0.$		(55)

Eq.(53) is the Breit-Wigner resonance formula. In the panel (a) of Fig.1, the square of T-matrices calculated with Eqs.(52) and (53) are plotted by the dashed blue and dotted green curves, respectively. One can see that Eq.(52) can express the asymmetric shape of the square of the T-matrix very well. The difference between the exact result and approximated one is due to the higher order contribution which are dropped in Eq.(50). The higher order contribution seems to be more important for the description of the shape at higher energy. The difference between $\epsilon_{n}^{0}$ and $\epsilon_{R}^{0,r}$ is due to the phase shift $\delta_{R}^{0}$ which is determined by the S-matrix at $\epsilon=\epsilon_{R}^{0,r}$. The imaginary part of the S-matrix pole $\epsilon_{R}^{0,i}$ represents the half of the width of the Breit-Wigner type peak of the square of the T-matrix.

In Fig.2, we can find the S-matrix poles at $\epsilon=0.54-i0.56$ and $2.59-i4.63$ MeV for $p_{1/2}$ and $d_{5/2}$, respectively. However, no K-matrix is found in the panel (b). In the panel (a), we can see the broad peak shape of square of the T-matrix, but obviously the peak energy is quite different from $\epsilon_{R}^{0,r}$, this is due to the values of the phase shift at $\epsilon=\epsilon_{R}^{0,r}$ which differ from $\frac{\pi}{2}$.

In Fig.3, we can find the peak of the square of the T-matrix at $\epsilon=3.70$ and $4.12$ MeV for $s_{1/2}$ and $p_{3/2}$, respectively. And we can confirm that those peaks are poles of the K-matrix from the panel (b). However, no S-matrix pole is found in the panel (d). Therefore, it may be possible to interpret that those peaks are the standing wave eigen solutions which don’t have the lifetime.

The square of the wave functions $|\psi_{0,lj}^{(+)}|^{2}$ are shown in Fig.4 together with the potential $V_{lj}(r)=U_{lj}(r)+\hbar^{2}l(l+1)/2mr^{2}$ (the solid black curve). For $f_{5/2}$ and $g_{9/2}$, the amplitude at the internal region of the potential is larger than the one at the outside of the potential. The wave function which exhibits the internal structure in the nucleus is connected with the free particle states outside the nucleus asymptotically. This is a behavior which can be interpreted as a metastable state. Both the S- and K-matrix poles for $f_{5/2}$ and $g_{9/2}$ are found near the top of the centrifugal barrier of the potential. These properties exhibit the typical properties of the so-called “shape resonance” written in many textbooks. It should be noted that the K-matrix pole exists in the half of the width given by $\epsilon_{R}^{0,i}$ centered on $\epsilon_{R}^{0,r}$. The K-matrix poles of $s_{1/2}$ and $p_{3/2}$ are found at the continuum energy region. The behavior of the wave function exhibits a typical behavior of the continuum states, and the wave length of the wave function inside of the potential is reflected by the potential depth. The outer amplitude is larger than the inner one. The S-matrix poles for $p_{1/2}$ and $d_{5/2}$ are found near the top of the centrifugal barrier although the height of the barrier is very low. The amplitude of square of the wave function outside the potential is much larger than the one inside the potential, even though the imaginary part of the S-matrix pole for $p_{1/2}$ is small. As one can see in Fig.2, the phase shift is very small at $\epsilon=\epsilon_{R}^{0,r}$ for $p_{1/2}$ and $d_{5/2}$. This means that the interference between the scattering wave and outgoing wave without scattering is very small. Namely, the incident wave can not enter the nucleus at $\epsilon=\epsilon_{R}^{0,r}$ for $p_{1/2}$ and $d_{5/2}$ even though the centrifugal barrier is very small.

From the analysis shown above, basically the S- and K-matrix poles are independent physical solutions, and the existence of a K-matrix pole in the width of the S-matrix pole is necessary in order to form a resonance which has the property as a metastable state.

Thereafter, we shall call a S-matrix pole which has the corresponding the K-matrix pole nearby as a “resonance” About other solutions, we shall call a K-matrix pole to which has no corresponding S-matrix pole as a “standing wave solution”, and a S-matrix pole to which has no corresponding the K-matrix pole as just a (independent) “S-matrix pole”.

In the next section, we discuss the pairing dependence of these three types of scattering solutions including the quasiparticle resonance.

In Figs.5, 6 and 7, standing wave solutions $E_{n}$ and S-matrix poles $E_{R}(=E_{R}^{r}-iE_{R}^{i})$ are plotted as a function of the mean pairing gap $\langle\Delta\rangle$ within the range $0\leq\langle\Delta\rangle\leq 10$ MeV. The $E_{n}$ and $E_{R}$ are obtained as the numerical solution of Eqs.(36) and (41), respectively. The standing wave eigen solutions(K-matrix poles) $E_{n}$ are shown by the filled blue circles which are connected by solid curve, the S-matrix poles $E_{R}(=E_{R}^{r}-iE_{R}^{i})$ are shown by the unfilled red circles (for $E_{R}^{r}$) with the error-bars(for $E_{R}^{i}$). The results for the stable target of nucleus($\lambda=-8.0$ MeV) are shown in Fig.5. Figs.6 and 7 are results for the unstable target of nucleus ($\lambda=-1.0$ MeV).

The pairing dependence of the shape resonances for $f_{5/2}$ and $g_{9/2}$ with $\lambda=-8.0$ MeV (stable nucleus) and $\lambda=-1.0$ MeV (unstable nucleus) are shown at the top panels in Figs.5 and 6, respectively. It is very easy to notice that the standing wave solution is more sensitive for the pairing than the S-matrix pole in the sense of the energy shift. The standing wave solution $E_{n}$ is shifted to the higher energy as the pairing gap increases. The pairing gap has a tendency to makes $E_{R}^{i}$ (the imaginary part of the S-matrix pole $E_{R}$) larger, but the effect is small, $E_{R}^{r}$ is stable for the variation of the pairing gap. The shape resonance is a resonance with an $S^{(0)}_{lj}$ pole corresponding to the $K^{(0)}_{lj}$ pole. The S- and K-matrix are expressed by Eqs.(42) and (44) in the HFB framework, respectively. If Eq.(48) is good approximation for $S^{(0)}_{lj}$ and $S^{(0)}_{lj}$ has no pole, then it is very clear that the pole of $S_{lj}$ is given by the pole of $S^{(0)}_{lj}$ from Eq.(42), i.e. $S^{(1)}_{lj}$ has no effect on the pole of $S_{lj}$. If the higher order terms of the Taylor expansion of the Jost function, $S^{(1)}_{lj}$ can be seen as the effect of the pairing on the pole of $S_{lj}$. If Eq.(48) is good approximation for $S^{(0)}_{lj}$, we can obtain an approximated expression for $T^{(0)}_{lj}$ as given by Eq.(52). Then the approximated $K^{(0)}_{lj}$ is given by

$\displaystyle K^{(0)}_{lj}(E)$

$\displaystyle\sim$

$\displaystyle\frac{1}{E+\lambda-\epsilon_{n}^{0}}\left(\frac{\epsilon_{R}^{0,i}}{\sin^{2}\delta^{0}_{R}}\right)+\cot\delta^{0}_{R}.$

(56)

The standing wave eigen value $E_{n}$ is determined by $1/K_{lj}(E_{n})=0$, i.e.

$\displaystyle K^{(1)}_{lj}(E_{n})=1/K^{(0)}_{lj}(E_{n}).$

(57)

This is derived from Eq.(44).

By inserting Eq.(56) into Eq.(57), we can obtain

$\displaystyle E_{n}+\lambda\sim\epsilon_{n}^{0}+\frac{K^{(1)}_{lj}(E_{n})\epsilon_{R}^{0,i}}{\sin^{2}\delta^{0}_{R}(1-K^{(1)}_{lj}(E_{n})\cot\delta^{0}_{R})}.$

(58)

Since $\delta^{0}_{R}\sim\pi/2$ for the shape resonances of $f_{5/2}$ and $g_{9/2}$ as shown in Fig.1, Eq.(58) can be expressed as

$\displaystyle E_{n}+\lambda\sim\epsilon_{n}^{0}+K^{(1)}_{lj}(E_{n})\epsilon_{R}^{0,i}.$

(59)

This is the reason why $E_{n}$ of $f_{5/2}$ is more sensitive for the pairing than the one of $g_{9/2}$. In Fig.8, we show $K^{(1)}_{lj}(E_{n})$ as a function of $\langle\Delta\rangle$. The dotted curves are the fitted function curve by using $a\langle\Delta\rangle^{3}+b\langle\Delta\rangle^{2}+c\langle\Delta\rangle$. The adjusted parameters for $f_{5/2}$ and $g_{9/2}$ are shown in Table.1 and Table.2, respectively. From the results of adjusted parameters, we can see that $\langle\Delta\rangle^{2}$ term is dominant to express the pairing gap dependence since $b$ is largest value. We can see the clear dependence of $b$ on $\lambda$, but no difference between $f_{5/2}$ and $g_{9/2}$.

As is shown in Fig.2, the independent S-matrix poles are found for $p_{1/2}$ and $d_{5/2}$ at the zero pairing limit. The pairing effect on those S-matrix poles at the middle panels of Figs.5 and 6. There is almost no effect of the pairing for the S-matrix poles of $p_{1/2}$ with $\lambda=-8.0$ MeV and $d_{5/2}$ with $\lambda=-1.0$ MeV. The hole-type quasiparticle resonances are found at higher energy than the S-matrix poles, except $d_{5/2}$ with $\lambda=-8.0$ MeV. When the hole-type quasiparticle resonance exists, $T^{(1)}_{lj}(E)$ is given by

$\displaystyle T_{lj}^{(1)}(E)$

$\displaystyle=$

$\displaystyle\frac{\Gamma_{lj}(E)/2}{E-\lambda+e_{h}-F_{lj}(E)+i\Gamma_{lj}(E)/2},$

(60)

where

	$\displaystyle F_{lj}(E)$	$\displaystyle=$	$\displaystyle\frac{2m}{\hbar^{2}}\frac{2}{\pi}P\int_{0}^{\infty}dk^{\prime}k^{\prime 2}\frac{|\langle\psi_{0,lj}^{(+)}(k^{\prime})|\Delta|\phi_{h,lj}\rangle|^{2}}{k^{2}_{1}(E)-k^{\prime 2}},$
	$\displaystyle\Gamma_{lj}(E)/2$	$\displaystyle=$	$\displaystyle\frac{2mk_{1}(E)}{\hbar^{2}}|\langle\psi_{0,lj}^{(+)}(k_{1}(E))|\Delta|\phi_{h,lj}\rangle|^{2},$		(62)

as is derived in jost-fano .