#  Eigenphase shift decomposition of the RPA strength function based on the
Jost-RPA method

K. Mizuyama1,2, T. Dieu Thuy3, T. V. Nhan Hao3,4 1 Institute of Research and
Development, Duy Tan University, Da Nang 550000, Vietnam
2 Faculty of Natural Sciences, Duy Tan University, Da Nang 550000, Vietnam
3 Faculty of Physics, University of Education, Hue University, 34 Le Loi
Street, Hue City, Vietnam
4 Center for Theoretical and Computational Physics, University of Education,
Hue University, 34 Le Loi Street, Hue City, Vietnam  corresponding author:
<EMAIL_ADDRESS>
###### Abstract

The S-matrix which satisfies the unitarity, giving the poles as RPA excited
states, is derived using the extended Jost function within the framework of
the RPA theory. An analysis on the correspondence between the component
decomposition of the RPA strength function by the eigenphase shift obtained by
diagonalisation of the S-matrix and the S- and K-matrix poles was performed in
the calculation of the 16O quadrupole excitations. The results show the
possibility that the states defined by the eigenphase shift can be expressed
as RPA-excited eigenstates corresponding to the S-matrix poles in the
continuum region.

###### pacs:

21.60.Jz, 24.10.Eq, 24.10.Ht, 25.60.Bx, 25.60.Dz

##  I Introduction

In nuclear physics, which is a finite quantum many-body system, there are
mainly two phenomena called “resonance”. One is the “resonance” that occurs in
nuclear reactions and the other is the “resonance” that refers to a collective
excitation of the nuclear structure (e.g. the giant resonance). Originally,
“resonance” was referred to a vibration phenomenon in classical mechanics in
which the amplitude of a vibrating body increases when it is stimulated by an
external vibration equal to its eigenfrequency. The term “collective vibration
mode” in nuclear structures is sometimes called “resonance”, e.g. the giant
resonance, because it is considered to be analogous to “resonance” in
classical mechanics. Such quantum states are generally represented by a linear
combination of several different quantum states and are known as metastable
states. In scattering theory, a resonance as a metastable state appears as a
peak with width in the cross section, etc., and the peak energy and width are
considered to be given by the real and imaginary parts of the poles of the
S-matrix on the complex energy plane, respectively. This is due to the fact
that the resonance formula, such as the so-called Breit-Wigner type, has been
derived from models which reproduce the resonance state, and it has been
confirmed that the resonance energy and width are given by the real and
imaginary parts of the S-matrix poles. However, it is not strictly guaranteed
that the real part of the poles of the S-matrix coincides with the resonance
peak of the cross section, etc., due to the properties of the S-matrix as a
multivalued function of energy jost-class . Moreover, due to the special
quantum transition to continuum states and interference effects such as the
Fano effect jost-fano , the cross section can become concave or asymmetric
shape near the real part of the poles of the S-matrix. Another way of
determining the resonance state (resonance energy) is to use the K-matrix; it
is known that the poles of the K-matrix give the stationary wave solution of
the Schrödinger equation for one-dimensional scattering problems. The K-matrix
is defined as a Hermite matrix and its poles appear on the real axis of the
complex energy. The phase shift can be defined using the S-matrix. When
analyzing resonances using the phase shift $\delta$, it is considered that the
presence of a resonance significantly increases the phase shift $\delta$ near
the resonance energy and that the phase shift crosses $\delta=\frac{\pi}{2}$
at the resonance energy. The poles of the K-matrix are equal to the energy at
which $\delta=\frac{\pi}{2}$. The K-matrix can be represented using the
S-matrix, but the poles of each can be related to each other or independent
jost-class . Therefore, there are still many unclear aspects of resonance in
terms of its definition and classification.

The Jost function is a function of complex energy given as the coefficient
which connects the regular and irregular solutions in the fundamental
differential equation of a quantum system based on the Hamiltonian of the
system, such as the Schrödinger equation. It is known that the Jost function
can be used to define the S-matrix, and that the zeros of the Jost function on
the complex energy plane provide the complex eigenvalues of the system as the
poles of the S-matrix. The Jost function was originally formulated for the
single-channel system jost , but it can also be defined for multi-channel
systems Rakityansky ; jost-hfb ; JostRPA , since the Jost function can be
given as a coefficient function which connects the regular and irregular
solutions in systems where the fundamental equation is given by the
differential equation. In Ref.jost-hfb  we extended the Jost function within
the framework of the Hartree-Fock- Bogoliubov (HFB) method. By focusing on S-
and K-matrix poles in nucleon scattering targeting open-shell nuclei, we have
attempted to analyze and classify the resonances. As a result, it is found
that in nucleon scattering within the framework of HFB theory, there exist
“shape resonances” formed by centrifugal force potentials and two types of
“quasiparticle resonances” caused by pair-correlation effects. In such
“resonance states”, the S- and the K-matrix poles are found to exist
simultaneously. On the other hand, the S- and K-matrix poles can exist
independently, in which case the behavior of the scattering wave function is
not characteristic of a resonant state, especially inside the nucleus. In Ref.
JostRPA , the Jost function was further extended within the framework of
Random-Phase-Approximation (RPA) theory (hereafter this extended method will
be referred to as the Jost-RPA method), and applied to the calculation of the
$E1$ excitation modes of 16O. As a result, the poles corresponding to
collective excitation modes such as the giant resonance were successfully
found on the complex energy plane.

The S-matrix can be defined by using the Jost function due to the fact that
the wave function obtained by multiplying the regular solution of the system
by the inverse of the Jost function satisfies the scattering boundary
conditions. In Ref.JostRPA , it was shown that the poles of the inverse matrix
of the Jost function (the zero point of the determinant of the Jost function)
in the representation of the Green function which gives the RPA response
function correspond to the peak of the strength function of the $E1$
excitation mode, but the representation of the S-matrix using the Jost
function has not been explicitly shown. If we focus only on the asymptotic of
the wave function and the definition of the Jost function, it is most natural
to consider that the S-matrix is defined as the Jost function
($\mbox{\boldmath$\mathcal{J}$}^{(-)}$) as the coefficient of the outgoing
wave multiplied by the inverse of the Jost function as the coefficient of the
incoming wave ($\mbox{\boldmath$\mathcal{J}$}^{(+)}$) as
$\mbox{\boldmath$\mathcal{J}$}^{(+)-1}\mbox{\boldmath$\mathcal{J}$}^{(-)}$.
However, the S-matrix defined in this way does not satisfy unitarity. It is
essential to define the S-matrix which satisfies unitarity in order to analyze
and discuss the classification of resonances using S- and K-matrices. The
S-matrix which satisfies unitarity can be diagonalised to define eigenphase
shifts. It may be possible to examine the detailed properties of a physical
quantity (such as RPA strength functions, transition densities, wave
functions, etc.) by decomposing it into “states” defined by the eigenphase
shift Rakityansky ; Rodberg ; Carew ; Lee .

In this paper, we first present how to define and derive the S-matrix which
satisfies unitarity in the Jost-RPA method (Sec.II.1). Then, we show how to
decompose the RPA strength function by the states defined for each eigenphase
shift obtained by diagonalisation the S-matrix in Sec.II.2. However, the
eigenphase shift does not necessarily correspond to the RPA eigenstates, since
the original S-matrix is made to give boundary conditions that connect to free
particle states at $r\to\infty$ (scattering boundary conditions). Therefore,
in Sec.II.3, we propose how to give the eigenphase shifts corresponding to the
RPA eigenstates. In Sec.II.4, the expression of the formulae for the explicit
treatment of isospin dependence is explained. In Sec.II.5, the S-matrix poles
and eigenphase shift transition densities for isoscalar and isovector modes
(which are obtained independently when the Coulomb interaction is neglected)
are described. In Sec.III, we present numerical results of the application of
the method developed in this paper to the quadrupole excitation 16O.

##  II Theory

###  II.1 Derivation of the S-matrix in the Jost-RPA

As shown in Ref.JostRPA , the Jost function
$\mbox{\boldmath$\mathcal{J}$}^{(\pm)}$ in the extended Jost function method
within RPA theory (Jost-RPA method) is defined as the coefficient function
which connects $\mbox{\boldmath$\Phi$}^{(r)}$ and
$\mbox{\boldmath$\Phi$}^{(\pm)}$ as

$\displaystyle\mbox{\boldmath$\Phi$}^{(r)\mathsf{T}}=\frac{1}{2}\left(\mbox{\boldmath$\mathcal{J}$}^{(+)}\mbox{\boldmath$\Phi$}^{(-)\mathsf{T}}+\mbox{\boldmath$\mathcal{J}$}^{(-)}\mbox{\boldmath$\Phi$}^{(+)\mathsf{T}}\right)$
(1)

where $\mbox{\boldmath$\Phi$}^{(r)}$ and $\mbox{\boldmath$\Phi$}^{(\pm)}$ are
the regular and irregular solutions of the coordinate space representation RPA
simultaneous differential equations, respectively. When the configuration
number of the particle-hole excitation in the RPA is given by $N=N_{n}+N_{p}$,
where $N_{n}$ and $N_{p}$ are the configuration numbers for neutrons and
protons, respectively, $\mbox{\boldmath$\Phi$}^{(r)}$,
$\mbox{\boldmath$\Phi$}^{(\pm)}$ and $\mbox{\boldmath$\mathcal{J}$}^{(\pm)}$
are given in a $2N\times 2N$ matrix form. Factor $2$ is due to the fact that
the positive and negative energy solutions exist correlated with each other.
For example, a particle-hole excitation configuration of 16O quadrupole
excitation is shown in Table 2. In this case, $N_{n}=8$ and $N_{p}=8$ for
neutrons and protons respectively, which means $N=N_{n}+N_{p}=16$.

From the asymptotic in the limit of $r\to\infty$ satisfied by the irregular
solutions, it seems that
$\mbox{\boldmath$\mathcal{J}$}^{(+)-1}\mbox{\boldmath$\mathcal{J}$}^{(-)}$ can
be defined as an S-matrix if the scattering wave function
$\mbox{\boldmath$\Psi$}^{(+)}$ is defined as.

$\displaystyle\mbox{\boldmath$\Psi$}^{(+)\mathsf{T}}\equiv\frac{1}{2}\left[\mbox{\boldmath$\Phi$}^{(-)\mathsf{T}}+\left(\mbox{\boldmath$\mathcal{J}$}^{(+)-1}\mbox{\boldmath$\mathcal{J}$}^{(-)}\right)\mbox{\boldmath$\Phi$}^{(+)\mathsf{T}}\right],$
(2)

however,
$\mbox{\boldmath$\mathcal{J}$}^{(+)-1}\mbox{\boldmath$\mathcal{J}$}^{(-)}$
does not satisfy the unitarity (which the S-matrix must satisfy in the
scattering problem).

To solve this problem, we first focus on the process of deriving the RPA
Green’s function in the Jost-RPA method.

The RPA Green’s function in the Jost-RPA framework is represented by

$\displaystyle\mbox{\boldmath$\mathcal{G}$}^{(\pm)}(r,r^{\prime})$
$\displaystyle=\mp
i\frac{2m}{\hbar^{2}}\left[\theta(r-r^{\prime})\mbox{\boldmath$\Phi$}^{(\pm)}(r)\mbox{\boldmath$\mathcal{K}$}\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)\mbox{\boldmath$\Phi$}^{(r)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\theta(r^{\prime}-r)\mbox{\boldmath$\Phi$}^{(r)}(r)\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)^{\mathsf{T}}\mbox{\boldmath$\mathcal{K}$}\mbox{\boldmath$\Phi$}^{(\pm)\mathsf{T}}(r^{\prime})\right]$
(3)

$\mathcal{K}$ is a diagonal matrix with the momentum component, where the
momentum component is expressed as
$k_{1,\alpha}=\sqrt{\frac{2m}{\hbar^{2}}(e_{\alpha}+E)}$ or
$k_{2,\alpha}=\sqrt{\frac{2m}{\hbar^{2}}(e_{\alpha}-E)}$ corresponding to the
positive or negative energy solution, respectively, where $e_{\alpha}$ is the
hole state energy and $E$ is the excitation energy of the system, $\alpha$ is
the subscript which represents the p-h excitation configuration shown in Table
2.

The conditions for the Green’s function to be given in the form Eq.(3) require
that equations (A.6) and (A.7) given in the Appendix of JostRPA  are
satisfied. Therefore, the Jost function satisfies the following equation,

$\displaystyle\pm\frac{1}{i}\left[\mbox{\boldmath$\Phi$}^{(\pm)}(r)\mbox{\boldmath$\mathcal{K}$}\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)\frac{\partial\mbox{\boldmath$\Phi$}^{(r)\mathsf{T}}(r)}{\partial
r}\right.$
$\displaystyle\left.-\mbox{\boldmath$\Phi$}^{(r)}(r)\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)^{\mathsf{T}}\mbox{\boldmath$\mathcal{K}$}\frac{\partial\mbox{\boldmath$\Phi$}^{(\pm)\mathsf{T}}(r)}{\partial
r}\right]=-\mbox{\boldmath$1$}.$ (4)

and

$\displaystyle\left[\mbox{\boldmath$\Phi$}^{(\pm)}(r)\mbox{\boldmath$\mathcal{K}$}\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)\mbox{\boldmath$\Phi$}^{(r)\mathsf{T}}(r)\right.$
$\displaystyle\left.-\mbox{\boldmath$\Phi$}^{(r)}(r)\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)^{\mathsf{T}}\mbox{\boldmath$\mathcal{K}$}\mbox{\boldmath$\Phi$}^{(\pm)\mathsf{T}}(r)\right]=\mbox{\boldmath$0$}.$
(5)

Substituting the relationship between regular and irregular solutions (which
is the definition of the Jost function) into these equations and considering
the limit of $r\to\infty$, one can obtain the following equation

$\displaystyle\mbox{\boldmath$\mathcal{K}$}^{\frac{1}{2}}\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)\left(\mbox{\boldmath$\mathcal{J}$}^{(\mp)}\right)\mbox{\boldmath$\mathcal{K}$}^{-\frac{1}{2}}$
$\displaystyle=\left[\mbox{\boldmath$\mathcal{K}$}^{\frac{1}{2}}\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)-1}\right)\left(\mbox{\boldmath$\mathcal{J}$}^{(\mp)}\right)\mbox{\boldmath$\mathcal{K}$}^{-\frac{1}{2}}\right]^{\mathsf{T}}.$
(6)

This formula shows that the left-hand side is a symmetric matrix, which we
define as the “S-matrix” (although this is not the S-matrix which satisfies
the unitarity used in the scattering problem) as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}(\mbox{\boldmath$\mathcal{K}$})$
$\displaystyle\equiv$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}^{\frac{1}{2}}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}\mbox{\boldmath$\mathcal{J}$}^{(-)}\mbox{\boldmath$\mathcal{K}$}^{-\frac{1}{2}}=\mbox{\boldmath$\mathcal{S}$}^{\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$}).$
(7)

Note that the S-matrix defined by multiplying
$\mbox{\boldmath$\mathcal{K}$}^{\frac{1}{2}}$ and
$\mbox{\boldmath$\mathcal{K}$}^{-\frac{1}{2}}$ from both sides in this way is
sometimes called the “flux adjusted S-matrix” Rakityansky , and an “adjust
factor” for mass must also be applied to define the S-matrix if the mass of
the particle changes depending on the channel.

Using this $\mathcal{S}$ with the symmetric property and the regular and
irregular solutions redefined as

$\displaystyle\widehat{\mbox{\boldmath$\Phi$}}^{(r)\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$})$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}^{\frac{1}{2}}\mbox{\boldmath$\Phi$}^{(\pm)\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$})$
(8) $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\mbox{\boldmath$\mathcal{J}$}^{(+)}\widehat{\mbox{\boldmath$\Phi$}}^{(-)\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$})+\mbox{\boldmath$\mathcal{J}$}^{(-)}\widehat{\mbox{\boldmath$\Phi$}}^{(+)\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$})\right]$
$\displaystyle\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$})$
$\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}^{\frac{1}{2}}\mbox{\boldmath$\Phi$}^{(\pm)\mathsf{T}}(\mbox{\boldmath$\mathcal{K}$}),$
(9)

the Green’s function Eq.(3) can be expressed as

$\displaystyle\mbox{\boldmath$\mathcal{G}$}^{(+)}(r,r^{\prime})$
$\displaystyle=-i\frac{m}{\hbar^{2}}\left[\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\Phi$}}^{(+)}(r)\widehat{\mbox{\boldmath$\Phi$}}^{(-)\mathsf{T}}(r^{\prime})\right.$
(10)
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\Phi$}}^{(-)}(r)\widehat{\mbox{\boldmath$\Phi$}}^{(+)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\Phi$}}^{(+)}(r)\mbox{\boldmath$\mathcal{S}$}\widehat{\mbox{\boldmath$\Phi$}}^{(+)\mathsf{T}}(r^{\prime})\right]$

Since “the unitarity which the S-matrix must satisfy” and the “analytic
continuation between Riemann sheets of the complex energy plane” are closely
related, we next show the properties of the complex conjugate of the Green’s
function. In RPA theory, it is considered to have analytic continuation with
various Riemann sheets where resonances exist at branch-cut lines above
various thresholds in the positive and negative energy regions respectively.
In order to express explicitly the contributions and correlations relating to
the positive and negative energy solutions in the expression of the Green’s
function, we make the following equation transformations.

When $\mbox{\boldmath$\mathcal{K}$}_{1}$ and
$\mbox{\boldmath$\mathcal{K}$}_{2}$ are $N\times N$ diagonal matrices with
$k_{1,\alpha}$ and $k_{2,\alpha}$ as matrix components, respectively, and
$\mathcal{K}$ is represented as

$\displaystyle\mbox{\boldmath$\mathcal{K}$}=\begin{pmatrix}\mbox{\boldmath$\mathcal{K}$}_{1}&\mbox{\boldmath$0$}\\\
\mbox{\boldmath$0$}&\mbox{\boldmath$\mathcal{K}$}_{2}\end{pmatrix},$ (11)

we represent $\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)}$ as

$\displaystyle\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(\pm
1)}&\widehat{\mbox{\boldmath$\phi$}}^{(\pm 2)}\end{pmatrix}$ (12)

where $\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1)}$ and
$\widehat{\mbox{\boldmath$\phi$}}^{(\pm 2)}$ are $2N\times N$ matrices. The
Jost function can also be expressed as

$\displaystyle\mbox{\boldmath$\mathcal{J}$}^{(\pm)}=\begin{pmatrix}\mbox{\boldmath$\mathcal{J}$}_{11}^{(\pm)}&\mbox{\boldmath$\mathcal{J}$}_{12}^{(\pm)}\\\
\mbox{\boldmath$\mathcal{J}$}_{21}^{(\pm)}&\mbox{\boldmath$\mathcal{J}$}_{22}^{(\pm)}\end{pmatrix},$
(13)

and the “S-matrix” is expressed as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}=\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{11}&\mbox{\boldmath$\mathcal{S}$}_{12}\\\
\mbox{\boldmath$\mathcal{S}$}_{21}&\mbox{\boldmath$\mathcal{S}$}_{22}\end{pmatrix}$
(14)

with

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{11}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}_{1}^{\frac{1}{2}}\left(\mbox{\boldmath$\mathcal{J}$}^{(+)}_{11}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\right)^{-1}$
(15)
$\displaystyle\left(\mbox{\boldmath$\mathcal{J}$}^{(-)}_{11}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(-)}_{21}\right)\mbox{\boldmath$\mathcal{K}$}_{1}^{-\frac{1}{2}}$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{12}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}_{1}^{\frac{1}{2}}\left(\mbox{\boldmath$\mathcal{J}$}^{(+)}_{11}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\right)^{-1}$
(16)
$\displaystyle\left(\mbox{\boldmath$\mathcal{J}$}^{(-)}_{12}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(-)}_{22}\right)\mbox{\boldmath$\mathcal{K}$}_{2}^{-\frac{1}{2}}$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{21}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}_{2}^{\frac{1}{2}}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}$
(17)
$\displaystyle\left(\mbox{\boldmath$\mathcal{J}$}^{(-)}_{21}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\mbox{\boldmath$\mathcal{K}$}_{1}^{-\frac{1}{2}}\mbox{\boldmath$\mathcal{S}$}_{11}\mbox{\boldmath$\mathcal{K}$}_{1}^{\frac{1}{2}}\right)\mbox{\boldmath$\mathcal{K}$}_{1}^{-\frac{1}{2}}$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{22}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{K}$}_{2}^{\frac{1}{2}}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}$
(18)
$\displaystyle\left(\mbox{\boldmath$\mathcal{J}$}^{(-)}_{22}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\mbox{\boldmath$\mathcal{K}$}_{1}^{-\frac{1}{2}}\mbox{\boldmath$\mathcal{S}$}_{12}\mbox{\boldmath$\mathcal{K}$}_{2}^{\frac{1}{2}}\right)\mbox{\boldmath$\mathcal{K}$}_{2}^{-\frac{1}{2}}$

The $\det\mbox{\boldmath$\mathcal{J}$}^{(+)}$ which gives the poles of the
S-matrix can be expressed as

$\displaystyle\det\mbox{\boldmath$\mathcal{J}$}^{(+)}$
$\displaystyle=\det\mbox{\boldmath$\mathcal{J}$}^{(+)}_{22}\det\left(\mbox{\boldmath$\mathcal{J}$}^{(+)}_{11}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\right)$
(19)
$\displaystyle=\det\mbox{\boldmath$\mathcal{J}$}^{(+)}_{11}\det\left(\mbox{\boldmath$\mathcal{J}$}^{(+)}_{22}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{11}\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\right).$
(20)

In particular, the complex energy $E$ satisfying

$\displaystyle\det\left(\mbox{\boldmath$\mathcal{J}$}^{(+)}_{11}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(+)}_{21}\right)=0$
(21)

gives the S-matrix pole in the positive energy region.

By using these representation, the Green’s function is represented as

$\displaystyle\mbox{\boldmath$\mathcal{G}$}^{(+)}(r,r^{\prime})$
$\displaystyle=-i\frac{m}{\hbar^{2}}\left[\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\phi$}}^{(+1)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(-1)\mathsf{T}}(r^{\prime})\right.$
(22)
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\phi$}}^{(-1)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(+1)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle+\left.\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\phi$}}^{(+2)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(-2)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\phi$}}^{(-2)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(+2)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+1)}(r)\mbox{\boldmath$\mathcal{S}$}_{11}\widehat{\mbox{\boldmath$\phi$}}^{(+1)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+1)}(r)\mbox{\boldmath$\mathcal{S}$}_{12}\widehat{\mbox{\boldmath$\phi$}}^{(+2)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+2)}(r)\mbox{\boldmath$\mathcal{S}$}_{21}\widehat{\mbox{\boldmath$\phi$}}^{(+1)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+2)}(r)\mbox{\boldmath$\mathcal{S}$}_{22}\widehat{\mbox{\boldmath$\phi$}}^{(+2)\mathsf{T}}(r^{\prime})\right]$

In the RPA theory, solutions are symmetrically given between the positive and
negative energy regions, so the analytic continuation of the Riemann sheets of
the complex energy plane is also symmetric between the positive and negative
energy regions. Therefore, the properties of complex conjugation related to
the analytic continuation in the positive energy region are presented below.

In the positive energy region (Re $E>0$), when Re $E\leqq-e_{\alpha_{c}}$
($\alpha_{c}$ is a certain configuration), the momentum have the following
properties,

$\displaystyle k_{1,\alpha}(E^{*})$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{cc}k_{1,\alpha}^{*}(E)&\mbox{for
$\alpha\leqq\alpha_{c}$}\\\ -k_{1,\alpha}^{*}(E)&\mbox{for
$\alpha>\alpha_{c}$}\end{array}\right.$ (25) $\displaystyle
k_{2,\alpha}(E^{*})$ $\displaystyle=$ $\displaystyle-k_{2,\alpha}^{*}(E)\hskip
10.0pt\mbox{for all $\alpha(\in 1,2,\cdots N)$}.$ (26)

Although trivial, these properties of momentum indicate that on the real axis
of complex energy $E$, $k_{1,\alpha}$ for $\alpha>\alpha_{c}$ and
$k_{2,\alpha}$ are pure imaginary numbers and $k_{1,\alpha}$ for
$\alpha\leqq\alpha_{c}$ is a real number.

To express the nature of the complex conjugation of
$\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)}$ as a function of complex energy
$E$, the column components of $\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1)}$ in
Eq.(12) are delimited by $\alpha_{c}$ as

$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(\pm
1c)}&\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1d)}\end{pmatrix}$ (27)

where $\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1c)}$ and
$\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1d)}$ are $2N\times\alpha_{c}$ and
$2N\times(N-\alpha_{c})$ matrices, respectively.

Due to the properties of momentum given by Eqs.(25) and (26), the complex
conjugate of $\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)}$ as a function of
complex energy $E$ is given by

$\displaystyle\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)*}(E^{*})$
$\displaystyle=$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(\pm
1)*}(E^{*})&\widehat{\mbox{\boldmath$\phi$}}^{(\pm 2)*}(E^{*})\end{pmatrix}$
(28)

with

$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1c)*}(E^{*})$
$\displaystyle=$ $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\mp 1c)}(E)$
(29) $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm 1d)*}(E^{*})$
$\displaystyle=$ $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm
1d)}(E)i\mbox{\boldmath$\eta$}_{d}$ (30)
$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm 2)*}(E^{*})$
$\displaystyle=$ $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm
2)}(E)i\mbox{\boldmath$\eta$}$ (31)

where $\eta$ is a $N\times N$ diagonal matrix with $(-)^{l^{(\alpha)}}$ as a
matrix element ($l^{(\alpha)}$ is the particle angular momentum number of a
particle-hole excitation configuration component $\alpha$), and
$\mbox{\boldmath$\eta$}_{d}$ is the same, but
$(N-\alpha_{c})\times(N-\alpha_{c})$ diagonal matrix for $\alpha>\alpha_{c}$.

Also the complex conjugate of the regular solution
$\widehat{\mbox{\boldmath$\Phi$}}^{(r)}(E)$ is given by

$\displaystyle\widehat{\mbox{\boldmath$\Phi$}}^{(r)*}(E^{*})$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(r1)*}(E^{*})&\widehat{\mbox{\boldmath$\phi$}}^{(r2)*}(E^{*})\end{pmatrix}$
(32)

with

$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r1)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(r1c)}&\widehat{\mbox{\boldmath$\phi$}}^{(r1d)}\end{pmatrix}$
(33)

and

$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r1c)*}(E^{*})$
$\displaystyle=$ $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r1c)}(E)$
(34) $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r1d)*}(E^{*})$
$\displaystyle=$
$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r1d)}(E)i\mbox{\boldmath$\eta$}_{d}$
(35) $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r2)*}(E^{*})$
$\displaystyle=$
$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(r2)}(E)i\mbox{\boldmath$\eta$},$
(36)

By applying these properties of solutions to the definition of the Jost
function, we can obtain the complex conjugation properties of the “S-matrix”.
To show the complex conjugate nature of the “S-matrix”,
$\mbox{\boldmath$\mathcal{S}$}_{11}$, $\mbox{\boldmath$\mathcal{S}$}_{12}$ and
$\mbox{\boldmath$\mathcal{S}$}_{21}$ are divided into block matrices bounded
by $\alpha=\alpha_{c}$ as follows.

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{11}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{11}^{cc}&\mbox{\boldmath$\mathcal{S}$}_{11}^{cd}\\\
\mbox{\boldmath$\mathcal{S}$}_{11}^{dc}&\mbox{\boldmath$\mathcal{S}$}_{11}^{dd}\end{pmatrix},\hskip
10.0pt\mbox{\boldmath$\mathcal{S}$}_{12}=\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{12}^{cd}\\\
\mbox{\boldmath$\mathcal{S}$}_{12}^{dd}\end{pmatrix}$ (37)
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{21}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{21}^{dc}&\mbox{\boldmath$\mathcal{S}$}_{21}^{dd}\end{pmatrix}$
(38)

where $\mbox{\boldmath$\mathcal{S}$}_{11}^{cc}$ is the
$\alpha_{c}\times\alpha_{c}$ matrix, $\mbox{\boldmath$\mathcal{S}$}_{11}^{cd}$
is the $\alpha_{c}\times(N-\alpha_{c})$ matrix,
$\mbox{\boldmath$\mathcal{S}$}_{11}^{dc}$ is the
$(N-\alpha_{c})\times\alpha_{c}$ matrix,
$\mbox{\boldmath$\mathcal{S}$}_{11}^{dd}$ is the
$(N-\alpha_{c})\times(N-\alpha_{c})$ matrix,
$\mbox{\boldmath$\mathcal{S}$}_{12}^{cd}$ is the $\alpha_{c}\times N$ matrix,
$\mbox{\boldmath$\mathcal{S}$}_{12}^{dd}$ is the $(N-\alpha_{c})\times N$
matrix, $\mbox{\boldmath$\mathcal{S}$}_{21}^{dc}$ is the $N\times\alpha_{c}$
matrix and $\mbox{\boldmath$\mathcal{S}$}_{21}^{dd}$ is the
$N\times(N-\alpha_{c})$ matrix.

To distinguish between above-threshold terms, below-threshold terms and their
correlated terms in the “S-matrix”, The block matrices are then further
rearranged as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}=\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{cc}&\mbox{\boldmath$\mathcal{S}$}_{cd}\\\
\mbox{\boldmath$\mathcal{S}$}_{dc}&\mbox{\boldmath$\mathcal{S}$}_{dd}\end{pmatrix}$
(39)
$\displaystyle\left\\{\begin{array}[]{ll}\mbox{\boldmath$\mathcal{S}$}_{cc}\equiv\mbox{\boldmath$\mathcal{S}$}_{11}^{cc}&\mbox{($\alpha_{c}\times\alpha_{c}$
matrix)}\\\
\mbox{\boldmath$\mathcal{S}$}_{cd}\equiv\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{11}^{cd}&\mbox{\boldmath$\mathcal{S}$}_{12}^{cd}\end{pmatrix}&\mbox{($\alpha_{c}\times(2N-\alpha_{c})$
matrix)}\\\
\mbox{\boldmath$\mathcal{S}$}_{dc}\equiv\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{11}^{dc}\\\
\mbox{\boldmath$\mathcal{S}$}_{21}^{dc}\end{pmatrix}&\mbox{($(2N-\alpha_{c})\times\alpha_{c}$
matrix)}\\\
\mbox{\boldmath$\mathcal{S}$}_{dd}\equiv\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{11}^{dd}&\mbox{\boldmath$\mathcal{S}$}_{12}^{dd}\\\
\mbox{\boldmath$\mathcal{S}$}_{21}^{dd}&\mbox{\boldmath$\mathcal{S}$}_{22}^{dd}\end{pmatrix}&\mbox{($(2N-\alpha_{c})\times(2N-\alpha_{c})$
matrix)}\end{array}\right.$ (44)

The complex conjugates of these rearranged block matrices are given as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}^{*}_{cc}(E^{*})$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}^{-1}_{cc}(E)$ (45)
$\displaystyle\mbox{\boldmath$\mathcal{S}$}^{*}_{cd}(E^{*})$ $\displaystyle=$
$\displaystyle-\mbox{\boldmath$\mathcal{S}$}^{-1}_{cc}(E)\mbox{\boldmath$\mathcal{S}$}_{cd}(E)i\widetilde{\mbox{\boldmath$\eta$}}$
(46) $\displaystyle\mbox{\boldmath$\mathcal{S}$}^{*}_{dc}(E^{*})$
$\displaystyle=$
$\displaystyle-i\widetilde{\mbox{\boldmath$\eta$}}\mbox{\boldmath$\mathcal{S}$}_{dc}(E)\mbox{\boldmath$\mathcal{S}$}^{-1}_{cc}(E)$
(47) $\displaystyle\mbox{\boldmath$\mathcal{S}$}^{*}_{dd}(E^{*})$
$\displaystyle=$
$\displaystyle\widetilde{\mbox{\boldmath$\eta$}}\mbox{\boldmath$\mathcal{S}$}_{dd}(E)\widetilde{\mbox{\boldmath$\eta$}}$
(48)
$\displaystyle-\widetilde{\mbox{\boldmath$\eta$}}\mbox{\boldmath$\mathcal{S}$}_{dc}(E)\mbox{\boldmath$\mathcal{S}$}^{-1}_{cc}(E)\mbox{\boldmath$\mathcal{S}$}_{cd}(E)\widetilde{\mbox{\boldmath$\eta$}}$

respectively, where $\widetilde{\mbox{\boldmath$\eta$}}$ is the
$(2N-\alpha_{c})\times(2N-\alpha_{c})$ diagonal matrix which is defined by

$\displaystyle\widetilde{\mbox{\boldmath$\eta$}}=\begin{pmatrix}\mbox{\boldmath$\eta$}_{d}&\mbox{\boldmath$0$}\\\
\mbox{\boldmath$0$}&\mbox{\boldmath$\eta$}\end{pmatrix}.$ (49)

If the irregular solution is also rearranged as

$\displaystyle\widehat{\mbox{\boldmath$\Phi$}}^{(\pm)}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(\pm
c)}&\widehat{\mbox{\boldmath$\phi$}}^{(\pm d)}\end{pmatrix}$ (50)

with

$\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm c)}$
$\displaystyle\equiv$ $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm
1c)}$ (51) $\displaystyle\widehat{\mbox{\boldmath$\phi$}}^{(\pm d)}$
$\displaystyle\equiv$
$\displaystyle\begin{pmatrix}\widehat{\mbox{\boldmath$\phi$}}^{(\pm
1d)}&\widehat{\mbox{\boldmath$\phi$}}^{(\pm 2)}\end{pmatrix}$ (52)

(note that $\widehat{\mbox{\boldmath$\phi$}}^{(\pm c)}$ and
$\widehat{\mbox{\boldmath$\phi$}}^{(\pm d)}$ are $2N\times\alpha_{c}$ and
$2N\times(2N-\alpha_{c})$ matrices, respectively), the Green’s function
Eq.(22) can be expressed as

$\displaystyle\mbox{\boldmath$\mathcal{G}$}^{(+)}(r,r^{\prime})$
$\displaystyle=-i\frac{m}{\hbar^{2}}\left[\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\phi$}}^{(+c)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(-c)\mathsf{T}}(r^{\prime})\right.$
(53)
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\phi$}}^{(-c)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(+c)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(-d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\phi$}}^{(-d)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+c)}(r)\mbox{\boldmath$\mathcal{S}$}_{cc}\widehat{\mbox{\boldmath$\phi$}}^{(+c)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+c)}(r)\mbox{\boldmath$\mathcal{S}$}_{cd}\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\mbox{\boldmath$\mathcal{S}$}_{dc}\widehat{\mbox{\boldmath$\phi$}}^{(+c)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\mbox{\boldmath$\mathcal{S}$}_{dd}\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right],$

and its complex conjugate is given by

$\displaystyle\mbox{\boldmath$\mathcal{G}$}^{(+)*}(r,r^{\prime};E^{*})$
$\displaystyle=i\frac{m}{\hbar^{2}}\left[\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\phi$}}^{(-c)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(+c)\mathsf{T}}(r^{\prime})\right.$
(54)
$\displaystyle\left.+\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\phi$}}^{(+c)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(-c)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle-\left.\theta(r-r^{\prime})\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(-d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.-\theta(r^{\prime}-r)\widehat{\mbox{\boldmath$\phi$}}^{(-d)}(r)\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(-c)}(r)\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}\widehat{\mbox{\boldmath$\phi$}}^{(-c)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(-c)}(r)\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}\mbox{\boldmath$\mathcal{S}$}_{cd}\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\mbox{\boldmath$\mathcal{S}$}_{dc}\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}\widehat{\mbox{\boldmath$\phi$}}^{(-c)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.-\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\mbox{\boldmath$\mathcal{S}$}_{dd}\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right.$
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r)\mbox{\boldmath$\mathcal{S}$}_{dc}\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}\mbox{\boldmath$\mathcal{S}$}_{cd}\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r^{\prime})\right].$

Using the properties of the Green’s function, the S-matrix and the complex
conjugate of the irregular solution shown above, one can obtain the
“imaginary” part of the Green’s function as

$\displaystyle\frac{1}{2i}\left[\mbox{\boldmath$\mathcal{G}$}^{(+)}(r,r^{\prime};E)-\mbox{\boldmath$\mathcal{G}$}^{(+)*}(r,r^{\prime};E^{*})\right]$
$\displaystyle=-\frac{2m}{\hbar^{2}}\widehat{\mbox{\boldmath$\psi$}}^{(+c)}(r;E)\widehat{\mbox{\boldmath$\psi$}}^{(+c)\dagger}(r^{\prime};E^{*})$
(55)

with $\widehat{\mbox{\boldmath$\psi$}}^{(+c)}(r)$ which is defined as

$\displaystyle\widehat{\mbox{\boldmath$\psi$}}^{(+c)}(r;E)$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left[\widehat{\mbox{\boldmath$\phi$}}^{(-c)}(r;E)+\widehat{\mbox{\boldmath$\phi$}}^{(+c)}(r;E)\mbox{\boldmath$\mathcal{S}$}_{cc}(E)\right.$
(56)
$\displaystyle\left.+\widehat{\mbox{\boldmath$\phi$}}^{(+d)}(r;E)\mbox{\boldmath$\mathcal{S}$}_{dc}(E)\right]$

where this wave function is given as the $2N\times\alpha_{c}$ matrix. The
complex conjugate of $\widehat{\mbox{\boldmath$\psi$}}^{(+c)}(r)$ can be
expressed as

$\displaystyle\widehat{\mbox{\boldmath$\psi$}}^{(+c)\dagger}(r;E^{*})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\widehat{\mbox{\boldmath$\phi$}}^{(+c)\mathsf{T}}(r;E)+\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}(E)\widehat{\mbox{\boldmath$\phi$}}^{(-c)\mathsf{T}}(r;E)\right.$
(57)
$\displaystyle\left.+\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}(E)\mbox{\boldmath$\mathcal{S}$}_{cd}(E)\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r;E)\right].$

due to the complex conjugation properties of $\mathcal{S}$ and irregular
solutions.

Eqs.(7) and (45) show that $\mbox{\boldmath$\mathcal{S}$}_{cc}$ is a unitary
matrix and satisfies
$\mbox{\boldmath$\mathcal{S}$}_{cc}^{\dagger}\mbox{\boldmath$\mathcal{S}$}_{cc}=\mbox{\boldmath$\mathcal{S}$}_{cc}\mbox{\boldmath$\mathcal{S}$}_{cc}^{\dagger}=\mbox{\boldmath$1$}$
where $1$ is an $\alpha_{c}\times\alpha_{c}$ unit matrix.

Since $\widehat{\mbox{\boldmath$\psi$}}^{(-c)}$ is defined by
$\widehat{\mbox{\boldmath$\psi$}}^{(-c)}\equiv\widehat{\mbox{\boldmath$\psi$}}^{(+c)*}(E^{*})$,
$\widehat{\mbox{\boldmath$\psi$}}^{(-c)}$ can be expressed as

$\displaystyle\widehat{\mbox{\boldmath$\psi$}}^{(-c)\mathsf{T}}(r;E)$
$\displaystyle=$
$\displaystyle\widehat{\mbox{\boldmath$\psi$}}^{(+c)\dagger}(r;E^{*})$ (59)
$\displaystyle=$
$\displaystyle\frac{1}{2}\left[\widehat{\mbox{\boldmath$\phi$}}^{(+c)\mathsf{T}}(r;E)+\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}(E)\widehat{\mbox{\boldmath$\phi$}}^{(-c)\mathsf{T}}(r;E)\right.$
$\displaystyle\left.+\mbox{\boldmath$\mathcal{S}$}_{cc}^{-1}(E)\mbox{\boldmath$\mathcal{S}$}_{cd}(E)\widehat{\mbox{\boldmath$\phi$}}^{(+d)\mathsf{T}}(r;E)\right],$

therefore one can find that $\widehat{\mbox{\boldmath$\psi$}}^{(+c)}$ and
$\widehat{\mbox{\boldmath$\psi$}}^{(-c)}$ are related as
$\widehat{\mbox{\boldmath$\psi$}}^{(+c)\mathsf{T}}(E)=\mbox{\boldmath$\mathcal{S}$}_{cc}(E)\widehat{\mbox{\boldmath$\psi$}}^{(-c)\mathsf{T}}(E)$.
This is the same relationship as that between the scattering wavefunction and
the S-matrix in the scattering problem.

As shown in Ref.JostRPA , the RPA response function $R(r,r^{\prime};E)$ is
expressed using a $2N$-dimensional vector $\vec{\mbox{\boldmath$\varphi$}}(r)$
defined by the hole state wave function and a Green’s function
$\mbox{\boldmath$\mathcal{G}$}^{(+)}(r,r^{\prime};E)$ as

$\displaystyle R(r,r^{\prime};E)$ $\displaystyle=$
$\displaystyle\vec{\mbox{\boldmath$\varphi$}}^{\mathsf{T}}(r)\mbox{\boldmath$\mathcal{G}$}^{(+)}(r,r^{\prime};E)\vec{\mbox{\boldmath$\varphi$}}(r^{\prime})$
(60)

Applying Eq. (55) to the definition of the strength function $S_{F}(E)$ for
the external field $f(r)$ expressed using the RPA response function
$R(r,r^{\prime};E)$, the strength function can be expressed as

$\displaystyle S_{F}(E)$ $\displaystyle=-\frac{1}{\pi}\mbox{ Im }\int\int
drdr^{\prime}f(r)R(r,r^{\prime};E)f(r^{\prime})$
$\displaystyle=\left[\sqrt{\frac{1}{\pi}\frac{2m}{\hbar^{2}}}\int
dr\vec{\mbox{\boldmath$\varphi$}}^{\mathsf{T}}(r)f(r)\widehat{\mbox{\boldmath$\psi$}}^{(+c)}(r;E)\right]$
$\displaystyle\times\left[\sqrt{\frac{1}{\pi}\frac{2m}{\hbar^{2}}}\int
dr^{\prime}\widehat{\mbox{\boldmath$\psi$}}^{(+c)\dagger}(r^{\prime};E^{*})f(r^{\prime})\vec{\mbox{\boldmath$\varphi$}}(r^{\prime})\right]$
(61)

From the above, we can say that in the framework of RPA theory,
$\widehat{\mbox{\boldmath$\psi$}}^{(\pm c)}$ is the scattering wave function
representing the transition to continuum including resonant states, and
$\mbox{\boldmath$\mathcal{S}$}_{cc}$ is the S-matrix which satisfies the
unitarity.

Table 1: The ground state properties of 16O: The bound single-particle levels (unit:MeV) and the r.m.s radius (unit:fm) for neutron and proton. |  | Neutron | Proton |
---|---|---|---|---
hole states
| $s_{1/2}$ | -36.17 | -31.16 |
| $p_{3/2}$ | -21.31 | -16.84 |
| $p_{1/2}$ | -16.38 | -11.95 |
particle states
| $d_{5/2}$ | -6.81 | -2.95 |
| $s_{1/2}$ | -4.90 | -1.43 |
r.m.s radius
| $\sqrt{\langle r^{2}\rangle}$ | 2.41 | 2.46 |
Table 2: RPA configuration for the $J^{\pi}=2^{+}$ excitations of 16O |  $q=n$ or $p$ | $\alpha$ | $(lj)$ | $(lj)^{-1}$ | $\epsilon_{\alpha}$ |
---|---|---|---|---|---|---
|  | 1 | $p_{3/2}$ | $(p_{1/2})^{-1}$ | -16.38 |
|  | 2 | $f_{5/2}$ | $(p_{1/2})^{-1}$ | -16.38 |
|  | 3 | $f_{5/2}$ | $(p_{3/2})^{-1}$ | -21.31 |
| $n$ | 4 | $f_{7/2}$ | $(p_{3/2})^{-1}$ | -21.31 |
|  | 5 | $p_{1/2}$ | $(p_{3/2})^{-1}$ | -21.31 |
|  | 6 | $p_{3/2}$ | $(p_{3/2})^{-1}$ | -21.31 |
|  | 7 | $d_{3/2}$ | $(s_{1/2})^{-1}$ | -36.17 |
|  | 8 | $d_{5/2}$ | $(s_{1/2})^{-1}$ | -36.17 |
|  | 9 | $p_{3/2}$ | $(p_{1/2})^{-1}$ | -11.95 |
|  | 10 | $f_{5/2}$ | $(p_{1/2})^{-1}$ | -11.95 |
|  | 11 | $f_{5/2}$ | $(p_{3/2})^{-1}$ | -16.84 |
| $p$ | 12 | $f_{7/2}$ | $(p_{3/2})^{-1}$ | -16.84 |
|  | 13 | $p_{1/2}$ | $(p_{3/2})^{-1}$ | -16.84 |
|  | 14 | $p_{3/2}$ | $(p_{3/2})^{-1}$ | -16.84 |
|  | 15 | $d_{3/2}$ | $(s_{1/2})^{-1}$ | -31.16 |
|  | 16 | $d_{5/2}$ | $(s_{1/2})^{-1}$ | -31.16 |

###  II.2 Eigenphase shift decomposition of the RPA strength function and
transition density

If we define the transition density
$\vec{\mbox{\boldmath$\rho$}}^{(\pm)\mathsf{T}}(r;E)$ as an
$\alpha_{c}$-dimensional vector as,

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm)}(r;E)$ $\displaystyle\equiv$
$\displaystyle\sqrt{\frac{1}{\pi}\frac{2m}{\hbar^{2}}}\widehat{\mbox{\boldmath$\psi$}}^{(\pm
c)\mathsf{T}}(r;E)\vec{\mbox{\boldmath$\varphi$}}(r),$ (62)

it can be found that the strength function Eq.(61) can be expressed as

$\displaystyle S_{F}(E)$ $\displaystyle=\int
drf(r)\vec{\mbox{\boldmath$\rho$}}^{(+)\dagger}(r;E^{*})\int
dr^{\prime}f(r^{\prime})\vec{\mbox{\boldmath$\rho$}}^{(+)}(r^{\prime};E)$ (63)
$\displaystyle=\sum_{\alpha=1}^{\alpha_{c}}\int
drf(r)\rho^{(+\alpha)*}(r;E^{*})\int
dr^{\prime}f(r^{\prime})\rho^{(+\alpha)}(r^{\prime};E)$ (64)

by using the transition density, where $\rho^{(+\alpha)}$ is a component for
$\alpha$ of the transition density vector
$\vec{\mbox{\boldmath$\rho$}}^{(+)}$. Therefore, the strength function on the
real axis of the complex excitation energy $E$ can be expressed as

$\displaystyle S_{F}(E)$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{\alpha_{c}}\left|\int
drf(r)\rho^{(+\alpha)}(r;E)\right|^{2}$ (65)

The relationship between $\vec{\mbox{\boldmath$\rho$}}^{(+)}$ and
$\vec{\mbox{\boldmath$\rho$}}^{(-)}$ is given by

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(+)}(r;E)$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}(E)\vec{\mbox{\boldmath$\rho$}}^{(-)}(r;E)$
(66) $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}(E)\vec{\mbox{\boldmath$\rho$}}^{(+)*}(r;E^{*})$
(67)

using the S-matrix.
The S-matrix $\mbox{\boldmath$\mathcal{S}$}_{cc}$, which is a unitary matrix,
can be diagonalised using the unitary matrix defined by $\mathcal{U}$ its
eigenvectors as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{U}$}\mbox{\boldmath$\mathcal{S}$}_{\delta}\mbox{\boldmath$\mathcal{U}$}^{\dagger}$
(68)

where $\mbox{\boldmath$\mathcal{S}$}_{\delta}$ is a diagonal matrix with the
complex eigenvalues of $\mbox{\boldmath$\mathcal{S}$}_{cc}$ as matrix
elements, and since $|\det\mbox{\boldmath$\mathcal{S}$}_{\delta}|=1$,
$\mbox{\boldmath$\mathcal{S}$}_{\delta}$ can be expressed as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{\delta}=\begin{pmatrix}e^{2i\delta_{1}}&&&&\\\
&e^{2i\delta_{2}}&&\text{\huge{0}}&\\\ &&\ddots&&\\\
&\text{\huge{0}}&&\ddots&\\\ &&&&e^{2i\delta_{\alpha_{c}}}\end{pmatrix}.$ (69)

with $\delta_{\alpha}$ which is so-called as the eigenphase shift .

If the “eigenphase transition density”
$\vec{\mbox{\boldmath$\rho$}}_{\delta}^{(\pm)}(r;E)$ is defined by the unitary
transformation as

$\displaystyle\vec{\mbox{\boldmath$\rho$}}_{\delta}^{(\pm)}(r;E)\equiv\mbox{\boldmath$\mathcal{U}$}^{\dagger}\vec{\mbox{\boldmath$\rho$}}^{(\pm)}(r;E),$
(70)

Note that the vector component of
$\vec{\mbox{\boldmath$\rho$}}_{\delta}^{(\pm)}$, $\rho_{\delta}^{(+\alpha)}$,
is a linear combination of the vector component of
$\vec{\mbox{\boldmath$\rho$}}^{(\pm)}$, $\rho^{(\pm\alpha)}$, and the
coefficients are given by the matrix components of the unitary matrix
$\mbox{\boldmath$\mathcal{U}$}^{\dagger}$, which diagonalises the S-matrix
$\mbox{\boldmath$\mathcal{S}$}_{cc}$.

The strength function with $\rho_{\delta}^{(+\alpha)}$ is just an expression
in which $\rho^{(+\alpha)}$ is replaced by $\rho_{\delta}^{(+\alpha)}$ in
Eq.(65) as

$\displaystyle S_{F}(E)$ $\displaystyle=$
$\displaystyle\sum_{\alpha=1}^{\alpha_{c}}\left|\int
drf(r)\rho_{\delta}^{(+\alpha)}(r;E)\right|^{2}$ (71)

because the unitary transformation Eq.(70) does not change the strength
function.

The T-matrix $\mbox{\boldmath$\mathcal{T}$}_{cc}$ and K-matrix
$\mbox{\boldmath$\mathcal{K}$}_{cc}$ are defined by using the S-matrix
$\mbox{\boldmath$\mathcal{S}$}_{cc}$ as

$\displaystyle\mbox{\boldmath$\mathcal{T}$}_{cc}(E)$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left(\mbox{\boldmath$\mathcal{S}$}_{cc}(E)-\mbox{\boldmath$1$}\right)$
(72) $\displaystyle\mbox{\boldmath$\mathcal{K}$}_{cc}(E)$ $\displaystyle=$
$\displaystyle
i\left(\mbox{\boldmath$\mathcal{S}$}_{cc}(E)+\mbox{\boldmath$1$}\right)^{-1}\left(\mbox{\boldmath$\mathcal{S}$}_{cc}(E)-\mbox{\boldmath$1$}\right),$
(73)

the component for $\alpha$ of the diagonalized T-matrix
$\mbox{\boldmath$\mathcal{T}$}_{\delta}$ and K-matrix
$\mbox{\boldmath$\mathcal{K}$}_{\delta}$ are therefore expressed by using the
eigenphase shift as

$\displaystyle\mathcal{T}_{\delta_{\alpha}}(E)$ $\displaystyle=$
$\displaystyle-e^{i\delta_{\alpha}}\sin\delta_{\alpha}$ (74)
$\displaystyle\mathcal{K}_{\delta_{\alpha}}(E)$ $\displaystyle=$
$\displaystyle-\tan\delta_{\alpha}.$ (75)

###  II.3 Eigenphase shift corresponding to RPA eigenstate

In the RPA theory, collective excitation modes of nuclei, such as giant
resonances, are thought to be caused by the effects of residual interaction.
As given in Ref.JostRPA , the Hamiltonian of the RPA equation as a
simultaneous differential equation in coordinate space representation is given
by

$\mathcal{H}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
r^{2}}\mbox{\boldmath$1$}+\mbox{\boldmath$\mathcal{V}$}_{MF}+\mbox{\boldmath$\mathcal{V}$}_{res}$
(76)

where the Hamiltonian is given by the $2N\times 2N$ matrix form,
$\mbox{\boldmath$\mathcal{V}$}_{MF}$ is the mean field part which is given as
the diagonal matrix, and $\mbox{\boldmath$\mathcal{V}$}_{res}$ is the residual
interaction. When the Hamiltonian is decomposed as

$\mathcal{H}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{H}$}_{MF}+\mbox{\boldmath$\mathcal{V}$}_{res}$
(77)

with

$\displaystyle\mbox{\boldmath$\mathcal{H}$}_{MF}$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}{\partial
r^{2}}\mbox{\boldmath$1$}+\mbox{\boldmath$\mathcal{V}$}_{MF}$ (78)

and $\mbox{\boldmath$\mathcal{S}$}_{MF}$ is defined as the S-matrix calculated
using $\mbox{\boldmath$\mathcal{H}$}_{MF}$.

$\mbox{\boldmath$\mathcal{S}$}^{(1)}$ is defined by

$\displaystyle\mbox{\boldmath$\mathcal{S}$}^{(1)}\equiv\mbox{\boldmath$\mathcal{S}$}_{cc}\mbox{\boldmath$\mathcal{S}$}_{MF}^{\dagger}$
(79)

The excited states of the system obtained by RPA theory are the eigenstates of
the Hamiltonian $\mathcal{H}$, which are created by the superposition of the
basis defined by the $\mbox{\boldmath$\mathcal{H}$}_{MF}$ (particle-hole
excited state configurations) with the effect of the residual interaction
$\mbox{\boldmath$\mathcal{V}$}_{res}$. Based on this and the discussion in
Ref. jost-class , we can assume that the RPA eigenstates are the
$\mbox{\boldmath$\mathcal{S}$}^{(1)}$ poles and that the strength function, by
its definition, reflects the $\mbox{\boldmath$\mathcal{S}$}^{(1)}$ poles in
its peak structure. The poles of the $\mbox{\boldmath$\mathcal{K}$}^{(1)}$
defined using $\mbox{\boldmath$\mathcal{S}$}^{(1)}$ can be interpreted as the
RPA eigenvalues as a Sturm-Liouville problem and obtained on the real axis of
energy as the energy when the eigenphase shift $\delta^{(1)}$ becomes $\pi/2$,
obtained by diagonalising $\mbox{\boldmath$\mathcal{S}$}^{(1)}$. As stated in
Ref.jost-class , although the S- and K- matrices are related to each other,
their poles are not guaranteed to exist at the same time and to have a one-to-
one correspondence. However, the existence of poles of the K-matrix
corresponding to the poles of the S-matrix is a condition for the resonance
property to be satisfied.

If $\vec{\mbox{\boldmath$\rho$}}^{(+)}_{\delta^{(1)}}$ is defined as

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(+)}_{\delta^{(1)}}\equiv\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}\vec{\mbox{\boldmath$\rho$}}^{(+)}$
(80)

using $\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}$, which diagonalises
$\mbox{\boldmath$\mathcal{S}$}^{(1)}$ as,

$\displaystyle\mbox{\boldmath$\mathcal{S}$}^{(1)}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{U}$}^{(1)}\mbox{\boldmath$\mathcal{S}$}_{\delta}^{(1)}\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}$
(81)

since $\mbox{\boldmath$\mathcal{U}$}^{(1)}$ is a unitary matrix, the RPA
strength function which is represented by Eq.(63) can be expressed as

$\displaystyle S_{F}(E)$ $\displaystyle=\int
drf(r)\vec{\mbox{\boldmath$\rho$}}^{(+)\dagger}_{\delta^{(1)}}(r;E^{*})\int
dr^{\prime}f(r^{\prime})\vec{\mbox{\boldmath$\rho$}}^{(+)}_{\delta^{(1)}}(r^{\prime};E)$
(82)

###  II.4 Isoscalar and Isovector strength

So far, for simplicity, the isospin dependence has not been specified, but the
strength function, which clearly shows the isospin dependence, is expressed as

$\displaystyle S_{F,\tau\tau^{\prime}}(E)$ $\displaystyle=\int
drf(r)\vec{\mbox{\boldmath$\rho$}}^{(+)\dagger}_{\tau}(r;E^{*})\int
dr^{\prime}f(r^{\prime})\vec{\mbox{\boldmath$\rho$}}^{(+)}_{\tau^{\prime}}(r^{\prime};E)$
(83)

with

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm)}_{\tau}(r;E)$
$\displaystyle\equiv$
$\displaystyle\sqrt{\frac{1}{\pi}\frac{2m}{\hbar^{2}}}\widehat{\mbox{\boldmath$\psi$}}^{(\pm
c)\mathsf{T}}(r;E)\vec{\mbox{\boldmath$\varphi$}}_{\tau}(r).$ (84)

where $\vec{\mbox{\boldmath$\varphi$}}_{\tau}$ for $\tau=n,p$ are a
$2N$-dimensional vector defined by the hole-state wavefunction
$\varphi_{\alpha}^{(q=n,p)}$ as,

$\displaystyle\vec{\mbox{\boldmath$\varphi$}}_{n}=\begin{pmatrix}\begin{pmatrix}\varphi_{1}^{(n)}\\\
\vdots\\\ \varphi_{N_{n}}^{(n)}\end{pmatrix}\\\
\begin{pmatrix}\varphi_{1}^{(n)}\\\ \vdots\\\
\varphi_{N_{n}}^{(n)}\end{pmatrix}\\\ \begin{pmatrix}0\\\ \vdots\\\
0\end{pmatrix}\\\ \begin{pmatrix}0\\\ \vdots\\\
0\end{pmatrix}\end{pmatrix},\hskip
10.0pt\vec{\mbox{\boldmath$\varphi$}}_{p}=\begin{pmatrix}\begin{pmatrix}0\\\
\vdots\\\ 0\end{pmatrix}\\\ \begin{pmatrix}0\\\ \vdots\\\ 0\end{pmatrix}\\\
\begin{pmatrix}\varphi_{1}^{(p)}\\\ \vdots\\\
\varphi_{N_{n}}^{(p)}\end{pmatrix}\\\ \begin{pmatrix}\varphi_{1}^{(p)}\\\
\vdots\\\ \varphi_{N_{n}}^{(p)}\end{pmatrix}\end{pmatrix},$ (85)

and the ones for $\tau=0$(isoscalar) and $\tau=1$(isovector) are defined by

$\displaystyle\vec{\mbox{\boldmath$\varphi$}}_{0}$ $\displaystyle\equiv$
$\displaystyle\vec{\mbox{\boldmath$\varphi$}}_{n}+\vec{\mbox{\boldmath$\varphi$}}_{p}=\begin{pmatrix}\begin{pmatrix}\varphi_{1}^{(n)}\\\
\vdots\\\ \varphi_{N_{n}}^{(n)}\end{pmatrix}\\\
\begin{pmatrix}\varphi_{1}^{(n)}\\\ \vdots\\\
\varphi_{N_{n}}^{(n)}\end{pmatrix}\\\ \begin{pmatrix}\varphi_{1}^{(p)}\\\
\vdots\\\ \varphi_{N_{p}}^{(p)}\end{pmatrix}\\\
\begin{pmatrix}\varphi_{1}^{(p)}\\\ \vdots\\\
\varphi_{N_{p}}^{(p)}\end{pmatrix}\end{pmatrix}$ (86)

and

$\displaystyle\vec{\mbox{\boldmath$\varphi$}}_{1}$ $\displaystyle\equiv$
$\displaystyle\vec{\mbox{\boldmath$\varphi$}}_{n}-\vec{\mbox{\boldmath$\varphi$}}_{p}=\begin{pmatrix}\begin{pmatrix}\varphi_{1}^{(n)}\\\
\vdots\\\ \varphi_{N_{n}}^{(n)}\end{pmatrix}\\\
\begin{pmatrix}\varphi_{1}^{(n)}\\\ \vdots\\\
\varphi_{N_{n}}^{(n)}\end{pmatrix}\\\ -\begin{pmatrix}\varphi_{1}^{(p)}\\\
\vdots\\\ \varphi_{N_{p}}^{(p)}\end{pmatrix}\\\
-\begin{pmatrix}\varphi_{1}^{(p)}\\\ \vdots\\\
\varphi_{N_{p}}^{(p)}\end{pmatrix}\end{pmatrix}$ (87)

###  II.5 Decomposition of Isoscalar (IS) and Isovector (IV) modes in the
absence of Coulomb interaction

Decompose $\vec{\mbox{\boldmath$\rho$}}^{(\pm)}_{\tau}(r;E)$, given as an
$\alpha_{c}$-dimensional vector in Eq.(84), as

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm)}_{\tau}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\vec{\mbox{\boldmath$\rho$}}^{(\pm 1)}_{\tau}\\\
\vec{\mbox{\boldmath$\rho$}}^{(\pm 2)}_{\tau}\\\ \vdots\\\
\vec{\mbox{\boldmath$\rho$}}^{(\pm\alpha_{c})}_{\tau}\end{pmatrix}=\begin{pmatrix}\vec{\mbox{\boldmath$\rho$}}^{(\pm
n)}_{\tau}\\\ \vec{\mbox{\boldmath$\rho$}}^{(\pm p)}_{\tau}\end{pmatrix}$ (88)

where $\vec{\mbox{\boldmath$\rho$}}^{(\pm q)}_{\tau}$ (for $q=n$ or $p$) is
the $\alpha_{c}^{(q)}$-dimensional vector which is defined by

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm q)}_{\tau}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\vec{\mbox{\boldmath$\rho$}}^{(\pm 1)}_{\tau}\\\
\vec{\mbox{\boldmath$\rho$}}^{(\pm 2)}_{\tau}\\\ \vdots\\\
\vec{\mbox{\boldmath$\rho$}}^{(\pm\alpha_{c}^{(q)})}_{\tau}\end{pmatrix}$ (89)

with $\alpha_{c}=\alpha_{c}^{(n)}+\alpha_{c}^{(p)}$.
$\mbox{\boldmath$\mathcal{S}$}_{cc}$, given as an $\alpha_{c}\times\alpha_{c}$
matrix, is similarly decomposed as

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}=\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}&\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}\\\
\mbox{\boldmath$\mathcal{S}$}_{cc}^{pn}&\mbox{\boldmath$\mathcal{S}$}_{cc}^{pp}\end{pmatrix}$
(90)

where $\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}$,
$\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}$,
$\mbox{\boldmath$\mathcal{S}$}_{cc}^{pn}$ and
$\mbox{\boldmath$\mathcal{S}$}_{cc}^{pp}$ are
$\alpha_{c}^{(n)}\times\alpha_{c}^{(n)}$,
$\alpha_{c}^{(n)}\times\alpha_{c}^{(p)}$,
$\alpha_{c}^{(p)}\times\alpha_{c}^{(n)}$ and
$\alpha_{c}^{(p)}\times\alpha_{c}^{(p)}$ matrices, respectively.

Since $\mbox{\boldmath$\mathcal{S}$}_{cc}$ is defined as in Eq.(44) using
$\mbox{\boldmath$\mathcal{S}$}_{11}$ and $\mbox{\boldmath$\mathcal{S}$}_{11}$
is expressed as in Eq.(15), we obtain

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}$ $\displaystyle=$
$\displaystyle\left[\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{pp}\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\right)^{-1}\right.$
(91)
$\displaystyle\left.\left(\mbox{\boldmath$\mathcal{M}$}^{(-)}_{nn}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{pp}\mbox{\boldmath$\mathcal{M}$}^{(-)}_{pn}\right)\right]_{cc}$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}$ $\displaystyle=$
$\displaystyle\left[\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{pp}\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\right)^{-1}\right.$
(92)
$\displaystyle\left.\left(\mbox{\boldmath$\mathcal{M}$}^{(-)}_{np}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{pp}\mbox{\boldmath$\mathcal{M}$}^{(-)}_{pp}\right)\right]_{cc}$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}^{pn}$ $\displaystyle=$
$\displaystyle\left[\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pp}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{nn}\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\right)^{-1}\right.$
(93)
$\displaystyle\left.\left(\mbox{\boldmath$\mathcal{M}$}^{(-)}_{pn}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{nn}\mbox{\boldmath$\mathcal{M}$}^{(-)}_{nn}\right)\right]_{cc}$
$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}^{pp}$ $\displaystyle=$
$\displaystyle\left[\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pp}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{nn}\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\right)^{-1}\right.$
(94)
$\displaystyle\left.\left(\mbox{\boldmath$\mathcal{M}$}^{(-)}_{pp}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{nn}\mbox{\boldmath$\mathcal{M}$}^{(-)}_{np}\right)\right]_{cc}$

where $\mbox{\boldmath$\mathcal{M}$}^{(\pm)}_{qq^{\prime}}$ is the block
matrices of $\mbox{\boldmath$\mathcal{M}$}^{(\pm)}$ which is defined by

$\displaystyle\mbox{\boldmath$\mathcal{M}$}^{(\pm)}\equiv\mbox{\boldmath$\mathcal{K}$}_{1}^{\frac{1}{2}}\left(\mbox{\boldmath$\mathcal{J}$}^{(\pm)}_{11}-\mbox{\boldmath$\mathcal{J}$}^{(+)}_{12}\mbox{\boldmath$\mathcal{J}$}^{(+)-1}_{22}\mbox{\boldmath$\mathcal{J}$}^{(\pm)}_{21}\right)\mbox{\boldmath$\mathcal{K}$}_{1}^{-\frac{1}{2}}.$
(95)

The complex energy satisfying $\det\mbox{\boldmath$\mathcal{M}$}^{(+)}=0$
gives the poles of $\mbox{\boldmath$\mathcal{S}$}_{cc}$, and
$\det\mbox{\boldmath$\mathcal{M}$}^{(+)}$ can also be expressed as

$\displaystyle\det\mbox{\boldmath$\mathcal{M}$}^{(+)}$
$\displaystyle=\det\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pp}\det\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{pp}\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\right)$
(96)
$\displaystyle=\det\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}\det\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pp}-\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}\mbox{\boldmath$\mathcal{M}$}^{(+)-1}_{nn}\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\right)$
(97)

When there is no Coulomb interaction, the isospin symmetry leads

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm n)}_{n}$ $\displaystyle=$
$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm p)}_{p}$ (98)
$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm n)}_{p}$ $\displaystyle=$
$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm p)}_{n}$ (99)
$\displaystyle\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pp}$ (100)
$\displaystyle\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}$ $\displaystyle=$
$\displaystyle\mbox{\boldmath$\mathcal{M}$}^{(+)}_{pn}$ (101)

therefore, we have the relation for the isoscalar and isovector transition
density as

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm n)}_{0}$ $\displaystyle=$
$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm p)}_{0}$ (102)
$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(\pm n)}_{1}$ $\displaystyle=$
$\displaystyle-\vec{\mbox{\boldmath$\rho$}}^{(\pm p)}_{1}.$ (103)

Using these relations, we can obtain

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(+q)}_{0}(r;E)$ $\displaystyle=$
$\displaystyle\left(\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}(E)+\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}(E)\right)\vec{\mbox{\boldmath$\rho$}}^{(-q)}_{0}(r;E)$
(104) $\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(+q)}_{1}(r;E)$
$\displaystyle=$
$\displaystyle\left(\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}(E)-\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}(E)\right)\vec{\mbox{\boldmath$\rho$}}^{(-q)}_{1}(r;E)$
(105)

and also

$\displaystyle\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}\pm\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}$
$\displaystyle=\left[\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}\pm\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\right)^{-1}\left(\mbox{\boldmath$\mathcal{M}$}^{(-)}_{nn}\pm\mbox{\boldmath$\mathcal{M}$}^{(-)}_{np}\right)\right]_{cc}.$
(106)

The poles of
$\mbox{\boldmath$\mathcal{S}$}_{cc}^{nn}\pm\mbox{\boldmath$\mathcal{S}$}_{cc}^{np}$
can be represented by the zeros of

$\displaystyle\det\left(\mbox{\boldmath$\mathcal{M}$}^{(+)}_{nn}\pm\mbox{\boldmath$\mathcal{M}$}^{(+)}_{np}\right)=0$
(107)

on the complex energy plane.

Expressing the isospin dependence explicitly, Eqs.(80) and (81) are expressed
in block matrix form as

$\displaystyle\begin{pmatrix}\vec{\mbox{\boldmath$\rho$}}^{(+n)}_{\delta^{(1\alpha)},\tau}\\\
\vec{\mbox{\boldmath$\rho$}}^{(+p)}_{\delta^{(1\beta)},\tau}\end{pmatrix}=\begin{pmatrix}\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}&\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}\\\
\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\beta}&\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\beta}\end{pmatrix}\begin{pmatrix}\vec{\mbox{\boldmath$\rho$}}^{(+n)}_{\tau}\\\
\vec{\mbox{\boldmath$\rho$}}^{(+p)}_{\tau}\end{pmatrix}$ (108)

and

$\displaystyle\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}^{(1)}_{nn}&\mbox{\boldmath$\mathcal{S}$}^{(1)}_{np}\\\
\mbox{\boldmath$\mathcal{S}$}^{(1)}_{pn}&\mbox{\boldmath$\mathcal{S}$}^{(1)}_{pp}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\mbox{\boldmath$\mathcal{U}$}^{(1)}_{n\alpha}&\mbox{\boldmath$\mathcal{U}$}^{(1)}_{n\beta}\\\
\mbox{\boldmath$\mathcal{U}$}^{(1)}_{p\alpha}&\mbox{\boldmath$\mathcal{U}$}^{(1)}_{p\beta}\end{pmatrix}\begin{pmatrix}\mbox{\boldmath$\mathcal{S}$}_{\delta}^{(1\alpha)}&0\\\
0&\mbox{\boldmath$\mathcal{S}$}_{\delta}^{(1\beta)}\end{pmatrix}\begin{pmatrix}\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}&\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}\\\
\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\beta}&\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\beta}\end{pmatrix}.$
(109)

When there is no Coulomb interaction, we can obtain the isoscalar and
isovector eigenphase transition density for
$\mbox{\boldmath$\mathcal{S}$}^{(1)}$ as

$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(+n)}_{\delta^{(1\alpha)},0}$
$\displaystyle=$
$\displaystyle\left(\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}+\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}\right)\vec{\mbox{\boldmath$\rho$}}^{(+n)}_{0}$
(110)
$\displaystyle\vec{\mbox{\boldmath$\rho$}}^{(+n)}_{\delta^{(1\alpha)},1}$
$\displaystyle=$
$\displaystyle\left(\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}-\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}\right)\vec{\mbox{\boldmath$\rho$}}^{(+n)}_{1}$
(111)

from Eq.(108), and also we can obtain

$\displaystyle\mbox{\boldmath$\mathcal{S}$}^{(1)}_{nn}\pm\mbox{\boldmath$\mathcal{S}$}^{(1)}_{np}=\mbox{\boldmath$\mathcal{U}$}^{(1)}_{n\alpha}\mbox{\boldmath$\mathcal{S}$}_{\delta}^{(1\alpha)}\left(\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}\pm\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}\right)$
(112)

from Eq.(109). These results show that when diagonalising
$\mbox{\boldmath$\mathcal{S}$}^{(1)}$, the eigenphase shift
$\delta^{(1\alpha)}$ with
$\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}=\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}$
gives the Isoscalar mode and the eigenphase shift $\delta^{(1\alpha)}$ with
$\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{n\alpha}=\mbox{\boldmath$\mathcal{U}$}^{(1)\dagger}_{p\alpha}$
gives the Isovector mode. Each mode is obtained independently when there is no
Coulomb interaction. When a Coulomb interaction is taken into account, the
isospin symmetry is broken, so the isoscalar and isovector modes are not
independent solutions to each other. However, from the discussion in the
absence of the Coulomb interaction, the eigenphase shift obtained by
diagonalising $\mbox{\boldmath$\mathcal{S}$}^{(1)}$ can be interpreted as
corresponding to the poles of $\mbox{\boldmath$\mathcal{S}$}^{(1)}$ which are
given by zeros of the Jost function on the complex energy plane.

Figure 1: (Color online) The isoscalar and isovector RPA strength functions
for the quadrupole excitation of 16O were shown by the solid blue and red
curves in the top and middle panels, respectively. The dashed line shows the
RPA strength function calculated with $\kappa_{np}=0$, multiplied by factors
$0.05$ and $0.4$ in the top and middle panels, respectively. The trajectories
of poles (a)-(f) obtained by solving Eq.(21) varying the constant parameters
$f_{1}$ and $f_{2}$ multiplied to the residual interaction are shown in the
bottom panel. The open circle ($\circ$), cross ($\times$) and filled circle
($\bullet$) symbols show the position of poles which are calculated by the RPA
(i.e., $f_{1}=1$, $f_{2}=1$), RPA with $\kappa_{np}=0$ ($f_{1}=1$, $f_{2}=0$)
and unperturbed($f_{1}=0$, $f_{2}=0$) solutions, respectively.(See Table.3.)

##  III Results and discussions

In the numerical calculations in this paper the same model and parameters as
in Ref.JostRPA  are used for the Woods-Saxon potential and the residual
interaction. The target nucleus was also chosen as a relatively light
spherical nucleus, 16O as before, in order to understand the details of the
method and to proceed with the analysis. In this paper, the RPA quadrupole
excitation is calculated by adopting $f(r)=r^{2}$ as the external field; the
distinction between isoscalar(IS) and isovector(IV) modes is given by the sign
of the hole state vector, as given in Eqs. (86) and (87). The ground state
properties, the single particle levels and r.m.s radius for neutron and proton
of 16O are shown in Table 1. The configuration of the quadrupole excitations
of 16O and subscription $\alpha$ which is used to describe the matrix elements
of the Jost function are shown in Table 2. A minor change regarding the
numerical solution method from Ref.JostRPA , but the Newton-Raphson method
Rakityansky ; NRM  is adopted in this paper to solve Eq.(21) and find the
S-matrix poles on the complex energy plane.

Table 3: Values of the S-matrix poles shown in the bottom panel of Fig.1. “Sheet” is the Riemann sheet on which the poles are found. The energy regions given in parentheses are the branch-cut lines of analytical continuation on the real axis with the first Riemann sheet. The “origin” gives the configuration of the single-particle excitation of the poles, calculated with $f_{1}=f_{2}=0.0$. |  | S-matrix pole [MeV] |
---|---|---|---
Sheet (branch-cut [MeV]) | No. | $f_{1}=f_{2}=1.0$ | $f_{1}=1.0,f_{2}=0.0$ | $f_{1}=f_{2}=0.0$ | origin
|  | ($\circ$ in Fig.1) | ($\times$ in Fig.1) | ($\bullet$ in Fig.1) |
Sheet 1 ($11.95\leq E\leq 16.38$) | – | – | – | – | –
Sheet 2 ($16.38\leq E\leq 16.84$) | (a) | $16.76-i0.11$ | $25.62-i1.10$ | $25.65-i1.05$ | $\pi[f_{7/2}\otimes(p_{3/2})^{-1}]$
Sheet 3 ($16.84\leq E\leq 21.31$) | – | – | – | – | –
Sheet 4 ($21.31\leq E\leq 31.16$) | (b) | $27.02-i1.50$ | $26.91-i0.77$ | $26.89-i0.70$ | $\nu[f_{7/2}\otimes(p_{3/2})^{-1}]$
(c) | $28.48-i0.57$ | $28.78-i0.17$ | $28.21$ | $\pi[d_{5/2}\otimes(s_{1/2})^{-1}]$
(d) | $30.47-i0.66$ | $30.01-i0.19$ | $29.36$ | $\nu[d_{5/2}\otimes(s_{1/2})^{-1}]$
Sheet 5 ($31.16\leq E\leq 36.17$) | (e) | $34.75-i0.59$ | $34.91-i0.37$ | $34.69-i0.26$ | $\pi[d_{3/2}\otimes(s_{1/2})^{-1}]$
Sheet 6 ($36.17\leq E$) | (f) | $36.61-i0.42$ | $36.67-i0.10$ | $36.36-i0.001$ | $\nu[d_{3/2}\otimes(s_{1/2})^{-1}]$

In the top and middle panels of Fig.1, the RPA isoscalar and isovector
strength functions are shown by the blue and red solid curves, respectively.
The dashed curve shows the RPA strength function calculated by ignoring the
residual interaction between neutrons and protons. To compare the peak
structures of the solid and dashed strength functions in the energy region
above 20 MeV, the dashed strength functions were multiplied by 0.05 (top
panel) and 0.4 (middle panel). The bottom panel of Fig.1 shows the S-matrix
poles and their trajectories obtained by solving Eq.(21). The trajectories
were obtained by varying the constants multiplied by the residual interaction
in the same way as done in Ref.JostRPA . Depending on the poles, the Riemann
sheet to which the poles belong is different; the values of the S-matrix poles
and the Riemann sheets to which they belong are shown in Table 3.

Figure 2: (Color online) The same figure with Fig.1 but calculated without
Coulomb interaction. The trajectories of poles are calculated by solving
Eq.(107), the positive and negative signs for the isoscalar and isovector
modes, respectively. (See. Table.4.)

Basically, isoscalar (IS) and isovector (IV) modes are excitation modes caused
by neutron-proton coupling. Therefore, there is no distinction between IS and
IV modes in the strength function indicated by the dashed curve in the upper
and middle panels, and it can be easily confirmed that the S-matrix poles
marked with cross($\times$) symbols in the bottom panel correspond to the
peaks of the strength function indicated by the dashed curve.

However, in RPA calculations considering neutron-proton coupling, it is known
that IS and IV modes are mixed when the neutron and proton density
distributions are different. (Mixing of IS and IV modes occurs even in Z=N
nuclei, such as 16O, due to isospin symmetry breaking by the presence of the
Coulomb interaction.)sagawa . It is therefore difficult to find a clear
correspondence between the poles and the peaks of the RPA strength function,
except for a pole (a), which are clearly isolated from the other poles.

Table 4: Almost the same table as Table 3 for Fig.2. The “mode” shown in the last column indicates the excitation mode (IS: Isoscalar or IV: Isovector) for each pole. |  | S-matrix pole [MeV] |  |
---|---|---|---|---
Sheet (branch-cut [MeV]) | No. | $f_{1}=f_{2}=1.0$ | $f_{1}=1.0,f_{2}=0.0$ | $f_{1}=f_{2}=0.0$ | origin | mode
|  | ($\circ$ in Fig.2) | ($\times$ in Fig.2) | ($\bullet$ in Fig.2) |  |
Sheet 1 ($16.38\leq E\leq 21.31$) | (a’) | $17.33-i0.19$ | $26.90-i0.77$ | $26.89-i0.70$ | $[f_{7/2}\otimes(p_{3/2})^{-1}]$ | IS
Sheet 2 ($21.31\leq E\leq 36.17$) | (b’) | $27.86-i1.11$ | IV
(c’) | $29.14-i0.54$ | $30.02-i0.20$ | $29.36$ | $[d_{5/2}\otimes(s_{1/2})^{-1}]$ | IS
(d’) | $30.50-i0.65$ | IV
Sheet 3 ($36.17\leq E$) | (e’) | $36.20-i0.35$ | $36.68-i0.11$ | $36.36-i0.001$ | $[d_{3/2}\otimes(s_{1/2})^{-1}]$ | IS
(f’) | $36.92-i0.42$ | IV

For $Z=N$ nuclei, the IS and IV modes can be obtained as independent solutions
without mixing if the Coulomb interaction is neglected. Fig.2 is the same as
Fig.1 but ignoring the Coulomb interaction. Using Eq.(107), the S-matrix poles
can also be obtained independently for the IS and IV modes. The results of
Fig.2 and Table 4 show that (a’), (c’) and (e’) are the poles of the IS mode
and (b’), (d’) and (f’) are the poles of the IV mode. (a’) and (b’), (c’) and
(d’), (e’) and (f’) are IS and IV modes, respectively, originating from the
same state due to the effect of residual interaction. In other words, in the
case of a pair of poles (a’) of an IS mode and a pole (b’) of an IV mode, for
example, the origin of both poles is a pole at $26.89-i0.70$ MeV. Firstly,
this becomes a pole existing at $26.90-i0.77$ MeV due to the effect of
residual interaction between homologous nucleons. This is then further split
into an independent IS-mode pole($17.33-i0.19$ MeV) and an IV-mode
pole($27.86-i1.11$ MeV) by the effect of residual interaction between neutrons
and protons.

Comparing the positions of the S-matrix poles indicated by open circle
symbols($\circ$) in Fig.2 and the peaks of the strength function represented
by solid curves, it seems that the real part of the S-matrix poles generally
corresponds to the peaks of the strength function, except the IS mode (c’). A
comparison of the S-matrix poles in Figs.1 and 2 shows that there is a
correspondence. In the presence of the Coulomb interaction, the origin pole
separates into two poles due to isospin symmetry breaking, as shown in Fig.1
and Table 3. In the example of the previous pair (a’) and (b’), the origin
pole at $26.89-i0.70$ MeV separates into $25.65-i1.05$ MeV and $26.89-i0.70$
MeV poles, which become pole (a) and pole (b) respectively due to residual
interaction.

Figure 3: (Color online) Eigenphase shift $\delta^{(1)}_{\alpha}$ obtained by
diagonalising the $\mbox{\boldmath$\mathcal{S}$}^{(1)}$ matrix using Eq.(81)
(bottom panel) and decomposition of the RPA strength function by the
eigenphase shift components (second and third panels). (a)-(f) in the upper
figure are the S-matrix poles shown in Fig.1 and Table 3. The imaginary part
of the poles is represented by error bars. (i)-(v) show the positions of the
K-matrix poles appearing on the real axis of energy. The energy of the
K-matrix pole is equivalent to the energy when the eigenphase shift
$\delta^{(1)}_{\alpha}$ crosses $\pi/2$. (See Table.5.) Figure 4: (Color
online) The same figure with Fig.3 but calculated without Coulomb interaction.

However, as discussed in Ref.jost-fano ; jost-class , it does not always
guarantee that the position of the real part of the S-matrix poles in general
reflects or corresponds to the position of the peak of a physical quantity
such as the cross section or strength function. This is because the Jost
function is essentially a multivalued function of complex energy, so there can
be energy shifts caused by the nature of the multivalued function, and if
several poles exist in close proximity to each other, there is a superposition
of the contributions of the individual poles. There are also background
contributions arising from non-resonant continuum state contributions, because
interference effects, including Fano effects, also affect the peak structure.
Furthermore, there is a mixture of IS and IV modes, so it is not easy to
derive conclusions about the correspondence between the peak structure of the
strength function and the S-matrix poles and their contributions.

The Gamow shell modelgamow1 ; gamow2  and the complex scaling methodCSM1 ;
CSM2  exist as methods which can decompose the contribution of the S-matrix
poles in the strength function and cross section. Further extension of the
method is underway to apply these methods to our Jost function method. In this
paper, we attempt the alternative way for the detailed analysis.

In the previous section of this paper, we derived the S-matrix which satisfies
the unitarity associated with the description of the RPA strength function
within the framework of RPA theory using the Jost function. The S-matrix which
satisfies unitarity can be diagonalised using the unitary matrices as shown in
Eq.(81), which provides the eigenphase shifts. Since the unitary
transformation is an isometric transformation, the unitary matrix, which
diagonalises the S-matrix, can decompose the RPA intensity function into
components for each eigenphase shift without changing its magnitude. The poles
of the K-matrix, defined using the S-matrix, appear on the real axis of energy
and are equal to the energy at which the eigenphase shift crosses $\pi/2$,
referred to as the eigenvalue of the Sturm-Liouville problem.

Figure 5: (Color online) Magnified view of the energy region from $16$ MeV to
$19$ MeV in Fig.3. Figure 6: (Color online) Magnified view of the energy
region from $16$ MeV to $19$ MeV in Fig.4.

Eigenphase shifts $\delta^{(1)}_{\alpha}$ obtained by diagonalising
$\mbox{\boldmath$\mathcal{S}$}^{(1)}$ are shown in the bottom panel of Fig. 3.
The first and third rows of the figure show the strength function decomposed
into the components giving the eigenphase shifts using Eq.(82). In the top
panel of Fig.3, the S-matrix poles (a)-(f) are shown with their imaginary
parts represented by error-bars, since it is known that the real part of the
S-matrix pole is the resonance energy and the imaginary part is the half-width
if it is an isolated pole. (i)-(v) show the location of the K-matrix poles on
the real axis of energy. As is clear from the derivation of the S-matrix in
the previous section, the size of the S-matrix depends on the energy region of
the Riemann sheet shown in Table 3. For example, in the energy region of Sheet
2 ($16.38\leq E\leq 16.84$ MeV), the S-matrix is a $4\times 4$ matrix because
$\alpha=1,2,9$ and $10$ are open in the configuration channels given in Table
3. In the case of Sheet 4 ($21.31\leq E\leq 31.16$ MeV), the S-matrix is a
$4\times 4$ matrix because $\alpha=1-6,9-14$ channels are open. Energy regions
are indicated by dotted vertical lines in Fig. 3. Despite the fact that the
eigenphase shift is obtained by diagonalising S-matrices of different sizes in
each energy region, the eigenphase shift is obtained as a function of energy,
which is continuously connected at the boundaries of the energy region. The
components of the strength function per eigenphase shift are also obtained as
a function of continuous energy at the boundaries of the energy region, and
the sum of all those components reproduces the total strength function (shown
as grey-filled area). Note that the “total strength function” is the same as
the strength function which can be obtained by using the cRPA method shlomo .
The same figure with Fig.3 but calculated without Coulomb interaction is shown
in Fig.4. In order to show the first quadrupole exited state clearly, the
magnified view of the energy region from $16$ MeV to $19$ MeV of Figs.3 and 4
are shown in Figs.5 and 6, respectively.

If the Coulomb interaction is neglected, the eigenphase shifts shown in Fig.4
can be obtained independently for the IS and IV modes using Eq.(112). As a
result, it can be seen that $\delta^{(1)}_{1}$, $\delta^{(1)}_{4}$ and
$\delta^{(1)}_{5}$ are the eigenphase shifts of the IS mode, while
$\delta^{(1)}_{2}$, $\delta^{(1)}_{3}$ and $\delta^{(1)}_{6}$ are those of the
IV mode. These results are also consistent with the results of the component
decomposition of the strength function which are shown above the eigenphase
shifts. The energy at which the eigenphase shift crosses $\pi/2$ is the
K-matrix pole; it is known that if the S-matrix pole is given by a single
pole, the energy derivative $\frac{d\delta^{(1)}_{\alpha}}{dE}$ of the phase
shift corresponds to the imaginary part of the pole (resonance width) at the
energy at which the phase shift crosses $\pi/2$. It is therefore reasonable to
consider (ii’) shown in Fig.4 as an independent K-matrix pole which is not
associated with the S-matrix pole. For the K-matrix poles, except for (ii’),
the correspondence with the S-matrix poles can be found from the difference in
energy positions and modes. The results of the correspondence are summarized
in the lower half of Table 5.

Table 5: Values of the K-matrix poles and corresponding eigenphase shifts shown in Figs. 3, 5, 4 and 6. | No. | Eigenphase | K-mat. pole | Corr. | mode |
---|---|---|---|---|---|---
|  |  | [MeV] | S-mat. |  |
with Coulomb |  |
| (i) | $\delta_{1}^{(1)}$ | $16.78$ | (a) | IS |
| (ii) | $\delta_{1}^{(1)}$ | $27.27$ | – | ind. |
| (iii) | $\delta_{1}^{(1)}$ | $28.05$ | (c) |  |
| (iv) | $\delta_{3}^{(1)}$ | $30.31$ | (d) |  |
| (v) | $\delta_{4}^{(1)}$ | $36.80$ | (e),(f) |  |
without Coulomb |  |
| (i’) | $\delta_{1}^{(1)}$ | $17.40$ | (a’) | IS |
| (ii’) | $\delta_{1}^{(1)}$ | $27.23$ | – | ind. |
| (iii’) | $\delta_{1}^{(1)}$ | $28.72$ | (c’) | IS |
| (iv’) | $\delta_{3}^{(1)}$ | $30.86$ | (d’) | IV |
| (v’) | $\delta_{4}^{(1)}$ | $35.95$ | (e’) | IS |
| (vi’) | $\delta_{6}^{(1)}$ | $37.27$ | (f’) | IV |

As seen in Fig. 6, the eigenphase shift $\delta^{(1)}_{1}$ gives a very large
strength function of the IS first quadrupole excited state around $E=17.40$
MeV of the K-matrix pole (i’); the value of the imaginary part of the S-matrix
pole corresponding to $\delta^{(1)}_{1}$ (given in Table 4 shows that the
width of this peak is $0.38$ MeV ($=0.19\times 2$). In contrast, despite the
presence of the corresponding S-matrix pole (c’) for the K-matrix pole (iii’),
$\delta^{(1)}_{5}$ makes only a minor contribution to the strength function of
the IS mode around the energy of (iii’) $E=28.72$ MeV. Near this energy, the
main contribution is given by the eigenphase shift $\delta^{(1)}_{5}$ (non-
resonant continuum state), which has no K-matrix pole and is close to zero.
However, the basic shape of the strength function near the K-matrix pole
(iii’) is given by the strength function component contributed by
$\delta^{(1)}_{1}$. The shape of this strength function $\delta^{(1)}_{1}$
component is asymmetric, which means that quantum interference effects such as
the Fano effect may exist. The Fano effect (or Fano resonance) is known as a
special quantum interference effect that is universal to quantum many-body
systems and is known in atomic physics as the effect that causes
autoionisation phenomena fano ; Chu ; moiseyev ; Desrier . As far as the RPA
strength function is concerned, its contribution is very small and negligible
compared to the first excited state, but as the K-matrix pole (iii’) is a
state with a corresponding S-matrix pole (c’), it is expected to give a clear
contribution to quantities related to the scattering cross section such as the
imaginary part of the T-matrix. However, as our numerical calculations and
analysis in this paper focus on the RPA strength function, we can only point
out the possible existence of the Fano effect.

The $\delta^{(1)}_{2}$ component in Fig.4 is considered to correspond to the
S-matrix pole (b’) because it gives the IV mode contribution from the strength
function, but there is no corresponding K-matrix pole. Therefore, the S-matrix
pole (d’) is considered to be an independent S-matrix pole which forms the
first peak of the IV mode around $E=27$ MeV, but without a corresponding
K-matrix pole.

The eigenphase shifts $\delta^{(1)}_{3}$ and $\delta^{(1)}_{6}$ give the
K-matrix poles (iv’) and (vi’) with corresponding IV-mode S-matrix poles (d’)
and(f’) respectively. These are then the dominant components of the peaks of
the IV strength function near the respective K-matrix poles.

The K-matrix pole (v’) given by $\delta^{(1)}_{4}$ has a corresponding
S-matrix pole (e’) of the IS mode, forming a sharp IS mode strength function
peak which seems to be embedded in the background of the non-resonant
continuum state produced by $\delta^{(1)}_{5}$. The $\delta^{(1)}_{4}$
component of the IS strength function also exhibits a very asymmetric shape.
It is possible that the Fano effect exists in this case as well, and if it
exists, it needs to be clarified in future studies, including what specific
phenomena it may be related to.

Figure 7: (Color online) The isovector transition density
$\rho_{\delta,\tau=1}^{(1)}$ and isoscalar transition density
$\rho_{\delta,\tau=0}^{(1)}$ at the first K-matrix pole energy of the
eigenphase shift giving the K-matrix poles ((i) and (i’)). Figure 8: (Color
online) The isovector transition density $\rho_{\delta,\tau=1}^{(1)}$ and
isoscalar transition density $\rho_{\delta,\tau=0}^{(1)}$ at the K-matrix pole
energy of the eigenphase shift giving the K-matrix poles ((ii)-(v) and
(ii’)-(vi’)).

The transition density of the eigenphase shift component can be calculated
using Eq. (80). Figs.7 and 8 show the transition densities of the eigenphase
shift components which give their K-matrix poles at the K-matrix pole energy.
The transition densities when neglecting the Coulomb interaction associated
with Figs.4 and 6 are shown in the right-hand panels of Figs.7 and 8. Note
that the absolute square of the eigenphase shift component of the transition
density integrated with respect to $r$ gives the eigenphase shift component of
the strength function in Figs.3 and 4, and the sum of all the eigenphase shift
components gives the total strength function, by definition (Eq.(82)).

The transition densities of the $\delta^{(1)}_{1}$ component of the K-matrix
pole (i’) in Fig. 7, the $\delta^{(1)}_{3}$ component of (iv’) and the
$\delta^{(1)}_{6}$ component of (vi’) in Fig. 8 does not have nodes and has a
peak near or slightly outside the surface of the nucleus. This shows that the
density distribution of neutrons and protons in the nucleus is collectively
vibrating, which is one of the typical properties of so-called collective
vibration modes.

Since $\delta^{(1)}_{1}$ component of (i’) has only the IS transition density
amplitude and $\delta^{(1)}_{3}$ component of (iv’) and $\delta^{(1)}_{6}$
component of (vi’) only the IV transition density amplitude, respectively,
this means that (i’) is an IS collective vibration mode and (iv’) and (vi’)
are IV collective vibration modes.

The $\delta^{(1)}_{1}$ component of the transition density at the K-matrix
pole (ii’) is the transition density with an independent K-matrix pole, the
$\delta^{(1)}_{1}$ component of (iii’) and the $\delta^{(1)}_{4}$ component of
(v’) with corresponding S-matrix poles (c’) and (e’) respectively. All these
transition densities give contributions to the IS mode strength function only.
Despite the $\delta^{(1)}_{1}$ component of (iii’) and the $\delta^{(1)}_{4}$
component of (v’) having corresponding S-matrix poles (c’) and (e’)
respectively, the transition densities have nodes and do not show clear
characteristics of collective vibration modes. In particular, even though the
$\delta^{(1)}_{4}$ component of (v’) gives a sharp peak in the shape of the
strength function of the IS mode, indicating the possible existence of a clear
resonance, the behavior of the transition density does not show any
characteristic of collective vibration modes. This could be due to the
possibility that resonance and collective mode do not necessarily coincide, or
it could be caused by special quantum interference effects such as the Fano
effect. This may need to be clarified in future research.

Figure 9: (Color online) The $\delta^{(1)}_{2}$ and $\delta^{(1)}_{5}$
components of the transition densities (IS and IV modes) at K-matrix poles
(ii), (iii), (ii’) and (iii’) respectively.

Although $\delta^{(1)}_{2}$ and $\delta^{(1)}_{5}$ are eigenphase shifts which
do not give K-matrix poles, they make the main contribution to the IV and IS
strength functions near the energies of the K-matrix poles (ii’) and (iii’),
respectively. The transition densities of $\delta^{(1)}_{2}$ and
$\delta^{(1)}_{5}$ at energies of the K-matrix poles (ii’) and (iii’) are
shown in the right-hand panels of Fig. 9. The eigenphase shift
$\delta^{(1)}_{2}$ corresponds to the corresponding S-matrix pole (b’),
although it does not have a K-matrix pole, and gives the main contribution to
the first peak of the IV strength function around 27 MeV, whose transition
density exhibits the characteristics of the IV type collective vibration. The
eigenphase shift $\delta^{(1)}_{5}$ has neither K-matrix poles nor S-matrix
poles but makes the main contribution to the IS strength function above $20$
MeV, and its transition density shows typical non-resonant continuum state
properties with nodes and widely spread amplitudes outside the nucleus.

Based on the knowledge gained from the analysis of calculations ignoring the
Coulomb interaction that has been carried out so far, the analysis of the RPA
strength function including the Coulomb interaction will be carried out from
here onwards. The difference caused by the presence or absence of the Coulomb
interaction is the change in the single-particle level structure of neutrons
and protons due to isospin symmetry breaking and the associated change in
density distribution, and the mixing of IS and IV modes caused by it.
Comparing the strength functions in Figs.3 and 4, it can be seen that the IS
and IV strength functions each contain a mixture of eigenphase shift
components which were not present when the Coulomb interaction was ignored.
There are also significant changes in the peak structure of the strength
function associated with changes in the eigenphase shift.

When the Coulomb interaction is ignored, the sharp peak of the IS strength
function, which was formed around $E=36$ MeV with the contribution of the
eigenphase shift $\delta^{(1)}_{4}$, is split into two peaks. Focusing on the
change in the eigenphase shift $\delta^{(1)}_{4}$ shown in the bottom panels
of Figs.3 and 4, $\delta^{(1)}_{4}$ couples with $\delta^{(1)}_{3}$ and
$\delta^{(1)}_{6}$ and the K-matrix pole (v’) that existed below the $E=36.17$
MeV threshold disappears. A new K-matrix pole (v) is then formed near the
K-matrix pole (vi’) which was formed by $\delta^{(1)}_{6}$ when the Coulomb
interaction was ignored. The eigenphase shift $\delta^{(1)}_{4}$ then gives
contributions to both IS and IV strength. In fact, the transition density of
$\delta^{(1)}_{4}$ at the K-matrix pole (v), shown on the left-hand side of
Fig. 8, now has contributions to both IS and IV modes. The peak around $E=35$
MeV of the IS strength function is then formed by the independent S-matrix
pole (e), which no longer has a corresponding K-matrix pole. A new peak around
$E=35$ MeV also appears in the IV strength function, which is formed by the
independent S-matrix pole (e) and is caused by the coupling of the eigenphase
shift $\delta^{(1)}_{3}$ with $\delta^{(1)}_{4}$. The fact that the eigenphase
shift $\delta^{(1)}_{3}$ is coupled to $\delta^{(1)}_{4}$ can be seen from the
transition densities. The transition density of $\delta^{(1)}_{3}$ in the
K-matrix pole (iv) shown in the left-hand panel of Fig. 8 now includes
characteristics of the transition density of $\delta^{(1)}_{4}$ in (v’) in the
right-hand panel, which gives contributions to both IS and IV strength.

The new peak around $E=28$ MeV in the IV strength function is caused by the
contribution of the eigenphase shift $\delta^{(1)}_{1}$. This is because
$\delta^{(1)}_{1}$ couples with $\delta^{(1)}_{2}$, so that $\delta^{(1)}_{1}$
now gives a contribution to the IV strength. Conversely, by coupling
$\delta^{(1)}_{2}$ with $\delta^{(1)}_{1}$, $\delta^{(1)}_{2}$ also makes the
contribution to the IS strength when there is the Coulomb interaction, whereas
originally it only made the contribution to the IV strength when there was no
Coulomb interaction. The same can be seen from the $\delta^{(1)}_{1}$ and
$\delta^{(1)}_{2}$ transition densities shown in Figs. 8 and 9. The effect of
the Coulomb interaction on the eigenphase shift $\delta^{(1)}_{5}$ component,
which is the main component of the IS strength in the energy region above $20$
MeV, is very small. It can be seen from the $\delta^{(1)}_{5}$ transition
density shown in Fig. 9 that it hardly changes at all.

The effect of the Coulomb interaction on the IS strength created by the
eigenphase shift $\delta^{(1)}_{1}$, which appears as a large and sharp peak
at the lowest energy (K-matrix poles (i) and (i’)) shown in Figs.6 and 5, is
mainly in peak energy and width. As far as Fig.5 is concerned, there is a
slight mixing of the IV mode associated with the coupling with
$\delta^{(1)}_{1}$, but the mixing effect with the IV mode is small because it
is isolated away from the other poles. The transition density (Fig.7) shows no
effect of the mixing of IV modes on this pole, only the characteristics of the
IS collective vibration mode.

##  IV Summary and conclusion

In this paper, we first derived the S-matrix which satisfies the unitarity
using the Jost function extended within RPA theory. Considering the properties
required to derive the RPA Green’s function in Ref.JostRPA  and the symmetric
properties of the RPA Green’s function and wave functions in complex energy
space in the definition of the RPA strength function, we were able to obtain
the “definition of the S-matrix with Jost function” (Eqs. (15) and (44)) which
satisfies the unitarity and its associated “scattering wave function” (Eq.
(56)). The S-matrix which satisfies the unitarity can be diagonalised by using
the unitary matrices, and the diagonal components of the diagonalised S-matrix
are expressed in terms of eigenphase shifts. However, the original S-matrix
gives a scattering boundary condition where the scattering wavefunction leads
through the S-matrix to a free particle state at $r\to\infty$ (the boundary
condition imposed to obtain irregular solutions). Since RPA theory describes
the excited states of the system in terms of the superposition of particle-
hole configurations defined by the mean-field caused by the effects of the
residual interaction, Eq.(79) was adopted as the definition of the S-matrix
from the idea of the two-potential problem to give the eigenphase shift
meaning to describe the RPA excited states.

Applying this method to the quadrupole excitation of the 16O, the strength
function was then decomposed into components for each eigenphase shift, and
the correspondence between the eigenphase shift components of the strength
function and the S-matrix poles and eigenphase shifts was analyzed. The
results show that the components of the RPA strength function corresponding to
the eigenphase shift obtained by diagonalising the S-matrix given by Eq.(79)
correspond to the S-matrix poles found on the complex energy plane as the
zeros of the determinant of the Jost function. Even though the peaks of the
strength function correspond to S-matrix poles, there are not necessarily
corresponding K-matrix poles, and there are also independent S-matrix poles
that do not have K-matrix poles. Also, in the case of poles with simultaneous
S- and K-matrix poles, if the eigenphase shift component of the strength
function corresponding to the pole exhibits an asymmetric shape, such that the
presence of special interference effects such as Fano resonances is suspected,
the transition density of that component does not have the characteristics of
a collective mode.

The Fano effect is a special quantum interference effect which occurs when a
bound or resonant state is coupled to a continuum, and is known in atomic
physics as the effect responsible for the phenomenon of autoionisation. In
Ref.jost-class , it is shown that in nucleon-nucleus scattering, even in a
resonant state where the S- and K-matrix poles exist simultaneously, the
scattering wavefunction loses its resonant feature (i.e. the feature that the
scattering amplitude is enhanced inside the nucleus) due to the presence of
the Fano effect. If a mechanism such as the Fano effect exists in RPA excited
states, it is possible that the transition density loses its collective mode
character due to this effect. It remains possible, however, that the
(resonance) states defined by the S- and K-matrix poles are an independent
concept and not directly related to the RPA collective mode. In order to
clarify this point, the further study is needed.

It is known that the solutions of the IS and IV modes can be obtained
independently in RPA theory for $Z=N$ nuclei such as 16O if the Coulomb
interaction are neglected, and it is confirmed that the Jost function, its
determinant and the S-matrix derived in this paper can also be divided into IS
and IV modes respectively. Comparing calculations with and without the Coulomb
interaction, it can be seen that they cause various effects, such as the
coupling of the eigenphase shifts of the IS and IV modes, the disappearance of
the K-matrix poles, and the creation of new peaks in the strength function and
the splitting of the originally existing peak due to these effects. The
eigenphase shift component of the transition density corresponding to a given
S-matrix pole is found to give contributions to both IS and IV modes, as the
S-matrix pole can no longer be divided into IS and IV modes when the Coulomb
interaction is present.

Judging comprehensively from these analyses, the eigenphase shifts obtained by
diagonalising the S-matrix derived in this paper can be considered to
represent the “RPA excited states” corresponding to the S-matrix poles.

## Acknowledgments

This work was supported by Hue University under the Core Research Program,
Grant No. NCM.DHH.2018.09. T.D.T. was funded by the Master, PhD Scholarship
Programme of Vingroup Innovation Foundation (VINIF), code no.
VINIF.2023.TS127.

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