# Pauli resonance states in light nuclei: how they appear and how they can be
eliminated

N. Kalzhigitov<EMAIL_ADDRESS>Al-Farabi Kazakh National University,
Almaty, Kazakhstan  V. S. Vasilevsky<EMAIL_ADDRESS>Bogolyubov
Institute for Theoretical Physics,
Kyiv, 03143, Ukraine

###### Abstract

Systematic analysis of parameters and properties of the Pauli resonance states
are performed for light nuclei 6Li, 7Li, 8Be, 9Be and 10B, which are treated
as two-cluster systems. The Pauli resonance states are redundant solutions of
the resonating group method appearing when one try use more advanced
description of the internal structure of interacting clusters. Our
calculations are performed in a standard and advanced versions of the
resonating group method. The standard version employs wave functions of many-
particle shell model to describe internal motion of nucleons within each
cluster. The advanced version is based on three-cluster resonating group
method. As in the standard version, the internal wave functions of three
clusters are approximated by wave functions of many-particle shell model
model. However, in advanced version one of pair of clusters forms a bound
state, and third cluster is considered to interact with such state. It is
found that the Pauli resonance states in nuclei under consideration have
energy between 11 and 46 MeV, and their widths vary from 8 keV to 6.7 MeV.
Analysis of wave functions of Pauli resonance states and matrix elements of
norm kernel allowed us to formulate an effective method for eliminating Pauli
resonance states. It is demonstrated that this method effectively eliminate
all determined the Pauli resonance states.

Resonating group method, light nuclei, Pauli principle, Pauli resonance
states, Pauli allowed states, norm kernel

##  I Introduction

We are going to study properties of so-called Pauli resonance states, which
have been numerously observed in Refs. Clement  _et al._ (1974), Spitz  _et
al._ (1981), Kanada  _et al._ (1975a), Kanada  _et al._ (1975b) Fliessbach and
Walliser (1982), Kanada  _et al._ (1982), Stubeda  _et al._ (1982), Fujiwara
and Tang (1983), Walliser and Fliessbach (1983), Walliser  _et al._ (1985),
Kanada  _et al._ (1988), Kruglanski and Baye (1992) and many others. These
resonance states appear within the resonating group method (RGM) when one
tries to use more realistic description of interacting nuclei (clusters). As
the Pauli resonance states appear not in all realizations (versions) of the
resonating group method, we start with a short classification of main versions
of the RGM, which are relevant to the subject of the present paper. The main
difference of these methods is (consists) in the form of the wave function,
which is used to approximate cluster structure of a compound nucleus. The
standard version of the RGM suggests the following form of wave function of a
two-cluster system of $A$ nucleons for partition $A=A_{1}+A_{2}$

$\Psi\left(A\right)=\widehat{\mathcal{A}}\left\\{\Phi_{1}\left(A_{1},b\right)\Phi_{2}\left(A_{2},b\right)\psi\left(\mathbf{x}\right)\right\\},$
(1)

where $\mathbf{x}$ is a distance between center of mass of clusters,
$\Phi_{1}\left(A_{1},b\right)$ and $\Phi_{2}\left(A_{2},b\right)$ are wave
functions of the many-particle shell model describing motion of nucleons
within the first and second cluster, respectively. They are antisymmetric and
translationally invariant. Oscillator length $b$ determines effective size of
clusters.

For the sake of brevity, we omit all quantum numbers. They will be explicitly
indicated in Sec. II.

In second version which we call improved one wave function is chosen in the
form

$\Psi\left(A\right)=\widehat{\mathcal{A}}\left\\{\Phi_{1}\left(A_{1},b_{1}\right)\Phi_{2}\left(A_{2},b_{2}\right)\psi\left(\mathbf{x}\right)\right\\},$
(2)

where different oscillator length $b_{1}$ and $b_{2}$ are used to improve
description of the internal structure of each cluster. This version is
suitable for clusters with large difference of masses, i.e., for example, when
$A_{1}\gg A_{2}$.

The third version is called advanced version of the RGM and related to
advanced description of the internal structure of one

$\Psi\left(A\right)=\widehat{\mathcal{A}}\left\\{\Phi_{1}\left(A_{1},b_{1}\right)\Psi_{2}\left(A_{2},b\right)\psi\left(\mathbf{x}\right)\right\\},$
(3)

or two clusters

$\Psi\left(A\right)=\widehat{\mathcal{A}}\left\\{\Psi_{1}\left(A_{1},b\right)\Psi_{2}\left(A_{2},b\right)\psi\left(\mathbf{x}\right)\right\\},$
(4)

Contrary to wave function $\Phi_{\alpha}\left(A_{\alpha},b\right)$
($\alpha$=1,2), wave function $\Psi_{\alpha}\left(A_{\alpha},b\right)$ is a
solution of two-cluster Schrödinger equation with clusterization
$A_{\alpha}=A_{\alpha 1}+A_{\alpha 2}$ and is presented in the form similar to
(1). This version of the RGM suggests more correct description of compound
system and is appropriate when one or both clusters $A_{1}$ and $A_{2}$ have
explicit (evident,definite) two-cluster structure or, in other words, they
have weakly bound state(s) and thus can be easily split on two fragments. Many
light nuclei, such as $d$, 6Li, 7Li, 7Be, have such properties as their
separation energies are less than 3 MeV.

The Pauli resonance states have not seen in the standard version. Only shape
resonance states were detected within this version in a single-channel
approximation. As is well-known, shape resonances are created by the
centrifugal and/or Coulomb barriers, and thus they lie relative close to the
threshold of corresponding channel. The Pauli resonance states have been
detected in the improved and advanced versions.

The most spectacular demonstration of the Pauli resonance states was presented
in Refs. Walliser and Fliessbach (1983), Walliser  _et al._ (1985), where
elastic scattering of alpha particles on 16O has been calculated within
standard and improved versions. Set of narrow and wide resonance states was
emerged when different oscillator lengths (frequencies) were used for wave
functions describing internal structure of 16O and 4He nuclei. They spread
over a wide energy range from small to relatively high energies above the
16O+4He threshold. In light nuclei within the advanced version of the RGM
Clement  _et al._ (1974), Spitz  _et al._ (1981), Kanada  _et al._ (1975a),
Kanada  _et al._ (1975b), Kanada  _et al._ (1982), Stubeda  _et al._ (1982),
Fujiwara and Tang (1983), Kanada  _et al._ (1988) the Pauli resonance states
have been observed at relatively high energy region $E>$15 MeV. It was noticed
in Ref. Spitz  _et al._ (1981) that the Pauli resonance states manifest
themselves in the states with small values of the total orbital momentum $L$.

Despite that the different authors have used different names for such type of
resonances, such as ”positive energy bound states” Clement  _et al._ (1974),
”redundant” Fliessbach and Walliser (1982) or ”spurious states” Stubeda  _et
al._ (1982), it was widely recognized that the correctly treated Pauli
principle is origin of those states.

Let us recall the main types of resonance states observed in many-particle and
particular in nuclear systems. The first type is shape resonance states, they
are created by centrifugal or/and Coulomb barriers. The second type is
represented by the Feshbach resonance states. These resonance states appear
due to a weak coupling between open and closed channels. There are two
necessary conditions for creating the Feshbach resonances. A compound system
should have at least two channels with different threshold energies, and there
should be at least one bound state in the channel with larger threshold
energy, provided that this channel is considered separately from the channel
with lowest (lower, smaller) threshold energy.

The phenomenon which is called the Pauli resonance state cannot be explained
by two main factors of creating of resonance states and thus cannot be
attributed to the first or second types of resonances. It cannot be the
Feshbach resonance as such resonance states observed in single-channel cases
(calculations). It is impossible to relate (explain) the Pauli resonances by
centrifugal or Coulomb barrier as they appear in states with zero or very
small angular momenta, or they require very huge barrier.

The Pauli resonance states have been considered as redundant solutions of the
RGM equations, and thus one needs to use an algorithm (method) to eliminate
these states. They distort real physical quantities, such as phase shifts,
cross sections of different processes and so on. To our knowledge, there is
only one algorithm for eliminating the Pauli resonance states formulated in
Ref. Kruglanski and Baye (1992) and applied to the $\alpha$+16O system. We
refer to this method as the REV method, which is stand for removing of
eigenvalues of the norm kernel. It was suggested in Ref. Kruglanski and Baye
(1992) to omit so-called almost forbidden Pauli states. A criterion was
formulated how to distinguish such states from allowed states. This algorithm
has miraculously eliminated all Pauli resonance states from the elastic
scattering of alpha particles on 16O.

In the present paper we are going to examine continuous spectrum states of a
set of light nuclei such as 6Li, 7Li, 7Be, 8Be, 9Be and 10B. All these nuclei
are considered as a three-cluster configuration and treated within a three-
cluster model formulated in Ref. Vasilevsky  _et al._ (2009). The three-
cluster configuration is than reduced to three (if all three clusters are
different) or two (if two of three clusters are identical) binary channels.
With such reduction, one pair of clusters forms a bound state, which within
our method is described in a two-cluster approximation.

To study Pauli resonance states we will first of all analyze matrix of overlap
and its eigenvalues for some set of light nuclei. Based on this analysis, we
suggest an alternative method for eliminating the Pauli resonance states. This
new method we call as the ROF method, it means Removing of Oscillator
Functions. We will demonstrate that both method give close results and
completely eliminate all Pauli resonance states.

The structure of the present paper is the following. In Section II we give a
brief introduction to the methods applied to study properties of the Pauli
resonances in light nuclei. In Section III the choice of input parameters and
details of calculations are discussed. The manifestations of the Pauli
resonance states in various two-cluster systems are demonstrated in Section
III.1. The analysis of parameters of the Pauli resonance states and their wave
functions is carried out in this section. Then in the Section III.5 we analyze
behavior of matrix elements of the norm kernel and eigenvalues of the matrix.
In Section IV we briefly explain main ideas of eliminating the Pauli resonance
states suggested in Ref. Kruglanski and Baye (1992). Here we also demonstrate
its efficiency. In Section V we formulate an alternative method for
eliminating the Pauli resonance states and demonstrate how it works in two-
cluster systems under consideration. Concluding remarks are presented in
Section VI.

##  II Method

In this paper we will use two types of two-cluster functions and thus two
realizations of the RGM. The first type of functions realizes the standard
form of the resonating group method and the second type realizes the advanced
form of the RGM. Wave function of the first type for partition $A=A_{1}+A_{2}$
are

$\Psi_{E,J}\left(A\right)=\widehat{\mathcal{A}}\left\\{\left\\{\left[\Phi_{1}\left(A_{1},L_{1},S_{1},b\right)\Phi_{2}\left(A_{2},L_{2},S_{2},b\right)\right]_{S}\psi_{E,l,L,J}\left(x\right)Y_{l}\left(\widehat{\mathbf{x}}\right)\right\\}_{L}\right\\}_{J}$
(5)

and wave functions of the second type for partition
$A=A_{1}+A_{2}=A_{1}+\left(A_{21}+A_{22}\right)$ are

$\Psi_{E,J}\left(A\right)=\widehat{\mathcal{A}}\left\\{\left\\{\left[\Phi_{1}\left(A_{1},L_{1},S_{1},b\right)\Psi_{2}\left(A_{2},L_{2},S_{2},b\right)\right]_{S}\psi_{E,l,J}\left(x\right)Y_{l}\left(\widehat{\mathbf{x}}\right)\right\\}_{L}\right\\}_{J}$
(6)

where $\Psi_{2}\left(A_{2},S_{2},L_{2},b\right)$ is wave function of a bound
state of two-cluster subsystem with partition $\left(A_{21}+A_{22}\right)$

$\Psi_{2}\left(A_{2},L_{2},S_{2},b\right)=\widehat{\mathcal{A}}\left\\{\left[\Phi_{1}\left(A_{21},S_{21},b\right)\Phi_{2}\left(A_{22},S_{22},b\right)\right]_{S_{2}}g_{\mathcal{E},\lambda,J}\left(y\right)Y_{\lambda}\left(\widehat{\mathbf{y}}\right)\right\\}_{J}$
(7)

Recall, that in this paper we use capital letter $\Phi$ to denote wave
functions that are not solutions of corresponding Schrödinger equation, they
are wave functions of the many-particle shell model. These functions can be
constructed as Slater determinates from a single-particle oscillator orbitals.
The capital and small letters $\Psi$ and $\psi$ represent solutions of the
many-particle Schrödinger equation or corresponding integro-differential
Wheeler equation Wheeler (1937a), Wheeler (1937b).

One can see that we use $LS$ coupling scheme when the total spin $S$ is a
vector sum of spins of clusters, and total orbital momentum $L$ is a vector
sum of the orbital momenta of both clusters $L_{1}$ and $L_{2}$ and orbital
moment of relative motion of clusters $l$.

In the present paper we consider special case of the advanced version of the
RGM. The usage of the special case is justified by employing of three-cluster
model for investigating of a compound nucleus and a set of reactions
proceeding through the nucleus. In this special case only one of two functions
is the solution of two-cluster Schrödinger equation, and other function is the
many-body shell model wave functions. A four-cluster model will allow one to
consider a general case with two wave functions $\Psi_{1}$ and $\Psi_{2}$ to
be solutions of two-cluster Schrödinger equations.

To realize the advanced model, we employ of a three-model which was formulated
in Refs. Vasilevsky  _et al._ (2009), Nesterov  _et al._ (2009). Within this
model a three-cluster configuration is transformed into a set of binary
channels, i.e. in several pairs of interacting nuclei, and one of the
interacting nuclei is considered as a two-cluster system. In Refs. Vasilevsky
_et al._ (2009), Nesterov  _et al._ (2009) the model has been applied to study
nuclei 7Be and 7Li with three-cluster configurations $\alpha+d+n$ and
$\alpha+d+p$ respectively. The structure of the 10B nucleus has been
investigated in Ref. Nesterov  _et al._ (2014) by employing three-cluster
configurations $\alpha+\alpha+d$. Recently, the model which involves two
three-cluster configurations $\alpha+p+n$ and $t+d+p$ has been used in Ref.
Kalzhigitov  _et al._ (2021) to study resonance states of 6Li in a wide energy
range.

The model involves the Gaussian basis to determine bound-state wave functions
of two-cluster subsystems and oscillator basis to describe scattering of the
third cluster on a bound state of two-cluster subsystem. The abbreviation
AMGOB is used to distinguish this model. In the AMGOB, two-cluster (7) and
three-cluster (6) wave functions are represented as

$\displaystyle\Psi_{2}\left(A_{2},S_{2},L_{2},b\right)=\sum_{\nu=1}^{N_{G}}D_{\nu}^{E,L,J}\widehat{\mathcal{A}}\left\\{\left[\Phi_{1}\left(A_{21},S_{21},b\right)\Phi_{2}\left(A_{22},S_{22},b\right)\right]_{S}G_{L}\left(x,b_{\nu}\right)Y_{L}\left(\widehat{\mathbf{x}}\right)\right\\}_{J},$
(8)
$\displaystyle\Psi_{E,J}\left(A\right)=\sum_{n=0}^{N_{O}}C_{nL}^{E,J}\widehat{\mathcal{A}}\left\\{\left[\Phi_{1}\left(A_{1},S_{1},b\right)\Psi_{2}\left(A_{2},S_{2},L_{2},b\right)\right]_{S,L_{2}}\Phi_{n,L}\left(y,b\right)Y_{L}\left(\widehat{\mathbf{y}}\right)\right\\}_{J},$
(9)

where

$G_{L}\left(x,b_{\nu}\right)=\frac{1}{b_{\nu}^{3/2}}\sqrt{\frac{2}{\Gamma\left(L+3/2\right)}}\rho^{L}\exp\left\\{-\frac{1}{2}\rho^{2}\right\\},\qquad\left(\rho=\frac{x}{b_{\nu}}\right),$
(10)

is the Gaussian function and

$\displaystyle\Phi_{n,L}\left(y,b\right)$
$\displaystyle=\left(-1\right)^{n}\mathcal{N}_{nL}~{}b^{-3/2}\rho^{L}e^{-\frac{1}{2}\rho^{2}}L_{n}^{L+1/2}\left(\rho^{2}\right),\quad$
(11) $\displaystyle\rho$
$\displaystyle=\frac{y}{b},\quad\mathcal{N}_{nL}=\sqrt{\frac{2\Gamma\left(n+1\right)}{\Gamma\left(n+L+3/2\right)}},$

is the oscillator function. In Eqs. (10) and (11), $b_{\nu}$ and $b$ denote
oscillator lengths. Motivation for usage of these functions can be found in
Ref. Vasilevsky  _et al._ (2009). The expansion coefficients $D_{\nu}^{E,L,J}$
and $C_{nL}^{E,J}$ are solutions of a set of linear equations originated from
corresponding Schrödinger equations. This is a system of equations for the
expansion coefficients $D_{\nu}^{E,L,J}$

$\sum_{\widetilde{\nu}=0}\left[\left\langle\nu,L\left|\widehat{H}^{\left(2\right)}\right|\widetilde{\nu},L\right\rangle-E\left\langle\nu,L|\widetilde{\nu},L\right\rangle\right]D_{\widetilde{\nu}}^{E,L,J}=0$
(12)

and here is a system of equations for the expansion coefficients
$C_{nL}^{E,J}$

$\sum_{\widetilde{n}=0}\left[\left\langle
n,L\left|\widehat{H}\right|\widetilde{n},L\right\rangle-E\left\langle
n,L|\widetilde{n},L\right\rangle\right]C_{\widetilde{n}L}^{E,J}=0$ (13)

System of equations (12) involves matrix elements of the two-cluster
Hamiltonian
$\left\langle\nu,L\left|\widehat{H}^{\left(2\right)}\right|\widetilde{\nu},L\right\rangle$
and unit operator (norm kernel)
$\left\langle\nu,L|\widetilde{\nu},L\right\rangle$ between cluster Gaussian
functions

$\left|\nu,L\right\rangle=\widehat{\mathcal{A}}\left\\{\left[\Phi_{1}\left(A_{1},S_{1},b\right)\Psi_{2}\left(A_{2},S_{2},L_{2},b\right)\right]_{S,L_{2}}G_{L}\left(x,b_{\nu}\right)Y_{L}\left(\widehat{\mathbf{y}}\right)\right\\}_{J}.$
(14)

while system of equations (13) involves matrix elements of the three-cluster
Hamiltonian $\left\langle
n,L\left|\widehat{H}\right|\widetilde{n},L\right\rangle$ and unit operator
$\left\langle n,L|\widetilde{n},L\right\rangle$ between cluster oscillator
functions

$\left|n,L\right\rangle=\widehat{\mathcal{A}}\left\\{\left[\Phi_{1}\left(A_{1},S_{1},b\right)\Psi_{2}\left(A_{2},S_{2},L_{2},b\right)\right]_{S,L_{2}}\Phi_{n,L}\left(y,b\right)Y_{L}\left(\widehat{\mathbf{y}}\right)\right\\}_{J}.$
(15)

We will also use another basis of cluster oscillator functions

$\left|n,L\right\rangle_{0}=\widehat{\mathcal{A}}\left\\{\left[\Phi_{1}\left(A_{1},S_{1},b\right)\Phi_{2}\left(A_{2},S_{2},L_{2},b\right)\right]_{S,L_{2}}\Phi_{n,L}\left(y,b\right)Y_{L}\left(\widehat{\mathbf{y}}\right)\right\\}_{J}$
(16)

to expand wave functions of two-cluster systems in the standard version of the
RGM (5). It is obvious, that the wave functions $\left|n,L\right\rangle_{0}$
are partial case of wave functions $\left|n,L\right\rangle$ when the second
cluster has the most compact shape.

Appearance of the matrix $\left\|\left\langle
n,L|\widetilde{n},L\right\rangle\right\|$ in Eq. (13) indicates that cluster
oscillator basis (15) is not orthonormal, despite that all functions to the
right of the antisymmetrization operator in Eq. (15) are normalized to unity
on the corresponding part of coordinate space. This matrix plays an important
role in cluster models. It reflects effects of the Pauli principle. If one
neglects the total antisymmetrization by putting $\widehat{\mathcal{A}}=1$,
one obtains unit matrix $\left\|\left\langle
n,L|\widetilde{n},L\right\rangle\right\|$. When effects of the Pauli principle
are small, than the diagonal matrix elements are close to unity and off-
diagonal matrix elements tend to zero. Such behavior of matrix elements
$\left\langle n,L|\widetilde{n},L\right\rangle$ is observed for large values
of $n$ and $\widetilde{n}$. This region of quantum numbers $n$ and
$\widetilde{n}$ corresponds to large distances between clusters and thus is
called as asymptotic region.

In the standard version of the RGM, the matrix $\left\|\left\langle
n,L|\widetilde{n},L\right\rangle\right\|$ is diagonal for two $s$-clusters.
Within advanced version of the RGM, as will be demonstrated later, the matrix
$\left\|\left\langle n,L|\widetilde{n},L\right\rangle\right\|$ is not
diagonal, however the largest matrix elements are situated on the main
diagonal of the matrix.

It worthwhile noticing that the wave functions $\left\\{C_{nL}^{E,J}\right\\}$
obtained by solving system of equations (13) are normalized by the conditions

$\sum_{n,n=0}C_{nL}^{E_{\alpha},J}\left\langle
n,L|\widetilde{n},L\right\rangle
C_{\widetilde{n}L}^{E_{\alpha},J}=\delta_{\alpha\beta}$ (17)

for states of discrete spectrum and

$\sum_{n,n=0}C_{nL}^{E,J}\left\langle n,L|\widetilde{n},L\right\rangle
C_{\widetilde{n}L}^{\widetilde{E},J}=\delta\left(E-\widetilde{E}\right)$ (18)

for continuous spectrum states. An important consequence of Eqs. (17) and (18)
is that the value $\left|C_{nL}^{E,J}\right|^{2}$ does not determine the
contribution (weight) of the oscillator functions $\left|n,L\right\rangle$ to
the norm of a bound state or continuous spectrum state.

To solve equation (13) for a finite number of basis functions ($n$=0, 1, 2, …,
$N_{O}-1$), one needs to analyze the $N_{O}\times N_{O}$ matrix
$\left\|\left\langle n,L|\widetilde{n},L\right\rangle\right\|$, whether this
matrix contains redundant states, which are called the Pauli forbidden states.
For this aim the diagonalization procedure is usually employed which yields
the eigenvalues $\Lambda_{\alpha}$ ($\alpha$=1, 2, …, $N_{O}$) and
corresponding eigenfunctions $\left\|U_{n}^{\alpha}\right\|$ of the matrix
$\left\|\left\langle n,L|\widetilde{n},L\right\rangle\right\|$. Eigenstates
with $\Lambda_{\alpha}=0$ are called the Pauli forbidden states and have to be
removed from the space. Eigenstates with small values of $\Lambda_{\alpha}$
are called the partially or almost forbidden states. There are usually a large
number of eigenstates with $\Lambda_{\alpha}=1$. These states are not affected
by the antisymmetrization. Besides, the matrix $\left\|\left\langle
n,L|\widetilde{n},L\right\rangle\right\|$ can have eigenvalues with
$\Lambda_{\alpha}>1$, they are called the super allowed states. It will be
shown later that eigenfunctions with small values of $\Lambda_{\alpha}$
($\Lambda_{\alpha}<1)$ and with large values of $\Lambda_{\alpha}$
($\Lambda_{\alpha}>1)$ describes the most compact two-cluster configurations,
i.e. states with minimal distance between interacting clusters.

Note that the construction of the Pauli allowed states is a key problem for
many-cluster systems. Many algorithms have been formulated (see, for example,
Horiuchi (1977), Fujiwara and Horiuchi (1980), Kato  _et al._ (1988),) to
construct and to select Pauli allowed states.

Actually (In fact), in our hands we have two different discrete
representations of the Schrödinger equation. The first representation is the
oscillator basis representation and will be referred as $n-$representation.
The second representation is formed by eigenvalues of the norm kernel matrix
and will be referred as $\alpha-$representation. It is necessary to recall
that both representations are related by the orthogonal matrix
$\left\|U_{n}^{\alpha}\right\|$.

In the $\alpha-$representation the set of equations (13) is transformed to the
form

$\sum_{\widetilde{\alpha}=1}^{N_{O}}\left[\left\langle\alpha,L\left|\widehat{H}\right|\widetilde{\alpha},L\right\rangle-E\Lambda_{\alpha}\delta_{\alpha,\widetilde{\alpha}}\right]C_{\widetilde{\alpha}L}^{E,J}=0,$
(19)

where

$\left\langle\alpha,L\left|\widehat{H}\right|\widetilde{\alpha},L\right\rangle=\sum_{n,\widetilde{n}=0}^{N_{O}}U_{n}^{\alpha}\left\langle
n,L\left|\widehat{H}\right|\widetilde{n},L\right\rangle
U_{\widetilde{n}}^{\widetilde{\alpha}}.$ (20)

If the cluster system under consideration contains no the Pauli forbidden
states, then one may use the set of equations (13) or (19), both sets give the
same spectrum but different wave functions. One has to use the set of
equations (19) when there are one or more the Pauli forbidden states.

To study effects of the Pauli principle we will analyze the overlap matrix
$\left\|\left\langle n,L|\widetilde{n},L\right\rangle\right\|$, we will also
analyze the eigenvalues and eigenfunctions of the matrix.

##  III Results and discussions

As was pointed above, we consider a set of light nuclei. In Table 1 we list
these nuclei and present details of the model and calculations. Here 3C stands
for a three-cluster configuration which is taken into consideration, BC
indicates binary channels which are studied. The Minnesota potential Thompson
_et al._ (1977) is used as a nucleon-nucleon potential. Oscillator length $b$
is chosen to minimize energy of three-cluster threshold. The exchange
parameter $u$ of the MP is usually selected to reproduce energy of the ground
state of a compound system, accounted from the lowest two- or three-body
thresholds.

We employ four Gaussian function to obtain the energy and wave functions of
two-cluster subsystems, and 100 oscillator functions to describe scattering of
the third cluster on two-cluster subsystem. It was checked numerously, that
such number of oscillator functions is sufficient to obtain the bound state
energies of a compound nucleus and scattering parameters with acceptable
precision.

Table 1: List of nuclei to be considered, their three-cluster configurations (3C) and binary channels (BC), and input parameters of calculations: oscillator length $b$, echange parameter $m$ or $u$ of the nucleon-nucleon potential Nucleus | 3C | BC | $b$ | $u$ | Source
---|---|---|---|---|---
6Li | $\alpha+p+n$ | $\alpha+d$ | 1.285 | 0.863 | Kalzhigitov  _et al._ (2021)
| $t+d+p$ |  $t+^{3}$He |  |  |
7Li | $\alpha+d+n$ |  $\alpha+^{3}$H, 6Li$+n$ | 1.311 | 0.956 | Nesterov  _et al._ (2009)
7Be | $\alpha+d+p$ |  $\alpha+^{3}$He, 6Li$+p$ | 1.311 | 0.956 | Vasilevsky  _et al._ (2009)
8Be | $\alpha+d+d$ |  6Li$+d$ | 1.311 | 0.956 |
9Be | $\alpha+t+d$ |  $t+^{6}$Li | 1.285 | 0.950 | Lashko  _et al._ (2023)
10B | $\alpha+\alpha+d$ |  $d+^{8}$Be, $\alpha+^{6}$Li | 1.298 | 0.900 | Nesterov  _et al._ (2014)

To consider properties of the Pauli resonances in more detail, we restricted
ourselves to a single-channel approximation. Moreover, we do not consider
mixture of states with different values of the total orbital momentum $L$ and
total spin $S$, thus in our present model $L$ and $S$ are additional quantum
numbers to the angular momentum $J$ and parity $\pi$ of a compound system. In
this paper we will not consider many-channel cases. This is a subject for our
next investigation.

###  III.1 Manifestation of the Pauli resonance states

In this section we show how the Pauli resonance states manifest themselves in
nuclei under consideration. For this aim we consider phase shifts. The most
typical picture is shown in Fig. 1 where phase shifts are displayed for four
different $J^{\pi}$ states of the elastic $\alpha+t$ scattering. These phase
shifts exhibit resonance states of two different types. The first type is the
shape resonance states, they are created in the states 7/2- and 5/2-. These
resonance states are formed by combination of a huge centrifugal barrier at
the $L$=3 state and Coulomb barrier. The shape resonance states lie close to
the $\alpha+t$ threshold. The second type is the Pauli resonances which
exhibit themselves in the states 3/2- and 1/2-. Energies of the Pauli
resonances are $E$=25.8 MeV for 3/2- state and $E$=29.0 MeV for 1/2- state.
The total orbital momentum $L$=1 is responsible for these resonance states. It
means that in these states the centrifugal barrier is approximately 6 smaller
than in the 7/2- and 5/2-states. Thus the centrifugal barrier cannot be
responsible for the Pauli resonance states. The phase shifts displayed in Fig.
1 are similar to the phase shifts which were shown in Fig. 1 of Ref. Clement
_et al._ (1974). The improved version of the RGM has been used in Ref. Clement
_et al._ (1974), and two different oscillator lengths were chosen for an alpha
particle and a triton.

Figure 1: Phase shifts of the $\alpha+t$ elastic scattering in the states
$J^{\pi}$ = 1/2-, 3/2-, 5/2- and 7/2-

Let us consider the Pauli resonance states in the 1+ ($L$=0, $S$=1) and 2-
($L$=1, $S$=1)states of 6Li, which exhibit themselves in the channels 4He+$d$
and 3He+$t$. It is necessary to recall, that wave functions of deuteron and
triton are obtained in two-cluster (two-body) approximations as $p+n$ and
$d+n$, respectively. Such advanced description of deuteron and triton
stipulate appearance of the Pauli resonance states, shown in Fig. 2. Only one
Pauli resonance state is found in each channel. Energies and widths of these
resonances depend on the total orbital momentum $J$.

Figure 2: Phase shifts of the elastic 4He+$d$ and 3He+$t$ calculated for the
1+ and 2- states in the advanced version of RGM

Another example of the Pauli resonance manifestation is shown in Fig. 3 for
the 6Li+$\alpha$ scattering. This case demonstrates that two-cluster system
may have two Pauli resonance states, they are located at energy range 10$\leq
E\leq$45 MeV. A sharp growing of the 1+ phase shifts around 13.4 MeV indicates
that there is very narrow resonance state with the widths $\Gamma$=56 keV.
Other Pauli resonances are significantly wider.

Figure 3: Phase shifts of the elastic 6Li+$\alpha$ scattering as a function of
energy $E$.

In Fig. 4 we show phase shifts of the elastic 6Li+$d$ scattering with the
total orbital momentum $L$=1 and total spin $S$=2. Due to the spin-orbital
potential we obtain phase shifts for three states of the total angular
momentum $J^{\pi}$ =3-, 2- and 1-. Thus difference in behavior of phase shifts
is totally originated from the spin-orbit potential. Fig. 4 demonstrates that
energy and width of the Pauli resonance states depends on the spin-orbit
components of nucleon-nucleon interaction. Indeed, we obtained that $E$=22.636
MeV, $\Gamma$=0.952 MeV for $J$=1-, $E$=20.981 MeV, $\Gamma$=0.402 MeV for
$J$=2-, $E$=18.523 MeV, $\Gamma$=0.008 MeV for $J$=3-. We see that spin-
orbital potential substantially changes energy and width of the Pauli
resonance states.

Figure 4: Phase shifts of the elastic 6Li+$d$ scattering as a function of
energy. They obtained for $L$=1, $S$=2, and three values of the total angular
momentum $J$.

Let us consider how central part of the MP affects energy and width of the
Pauli resonance states. That can be done by varying the exchange parameter $u$
of that potential. This parameter affects interaction of nucleons in odd
states, they also affects cluster-cluster interaction. The smaller is $u$, the
smaller is interaction between clusters. When the parameter $u$ approaches to
unity, interaction between clusters is increased. Influence of parameter $u$
variation is carried out for the 3/2- state of 7Li considered as two-cluster
system $\alpha+t$. Results of variation $u$ is demonstrated for the ground
state energy $E_{GS}$, and for energy and width of the Pauli resonance. On can
see in Fig. 5 that parameter $u$ changes energy of the ground state. Moreover,
when $u<$0.86, the nucleus 7Li has no bound state. By varying parameter $u$
from 0.86 to 1, we change the ground state energy from -0.038 to -1.78 MeV.
However, variation of $u$ from 0.8 to 1, reduces significantly energy of the
Pauli resonance from 30.67 to 24.44 MeV. Such a variation of $u$ slightly
changes width of the Pauli resonance state from 13 to 33 keV.

Figure 5: Dependence of the ground state energy $E_{GS}$ of 7Li, energy and
width of the Pauli resonance state in the 3/2- state of the $\alpha+t$ channel
on the exchange parameter $u$ of the MP

###  III.2 Special case for $d+\alpha$ system

Taking into account peculiarities of our model, we decided to carry out an
additional investigation of the $d+\alpha$ system. For this specific case our
model allows to realize not only standard and advanced, but also improved
version of the RGM. If we take only one Gaussian function in expansion of
deuteron wave function, and select parameters $b_{0}$ (see Eq. (8)) to
minimize the bound state energy of a deuteron, we realize therefore the
improved version of the RGM. One Gaussian function with the optimal value of
$b_{0}=$1.512 fm creates bound state of deuteron with energy $E$ =-0.132 MeV,
while four Gaussian functions with optimal values of $b_{0}$ and $q$ generates
the deuteron bound state with energy $E$ =-2.020 MeV.

To locate the Pauli resonance state in the approximation in the energy range
below 50 MeV, we have to change the exchange parameter $u$ and take $u$=1.0.
In Fig. 6 we show phase shift of the $d+\alpha$ scattering in $L$=0, $S$=1
$J^{\pi}=1^{+}$ state obtained in three approximations. Standard version of
the RGM does not generate the Pauli resonance state in this case. The Pauli
resonance states appear in the improved (I) and advanced (A) versions.
Parameters of the Pauli resonance states substantially depends on a wave
function describing internal structure of deuteron. More realistic wave
function significantly increases width of the Pauli resonance (from
$\Gamma=$0.001 MeV to $\Gamma=$1.718 MeV), and dramatically changes the energy
(from $E=$47.55 MeV to $E=$22.49 MeV) of the resonance state in the state
$J^{\pi}=1^{+}$.

Figure 6: Phase shifts of the elastic $\alpha+d$ scattering in the state
$L=0$, $S=1$, $J^{\pi}=$1+, obtained in three differen approximations of the
RGM

###  III.3 Main properties of the Pauli resonance states

In Table 2 we collect information on parameters of the Pauli resonance states
detected in nuclei under consideration. It is detected 28 Pauli resonance
states. The energy of resonance states is reckoned from the threshold of the
channel indicated in column ”Channel” of Table 2 and is varying from 11 to 46
MeV. There are 10 narrow resonance states with $\Gamma<$1 MeV, six of them are
very narrow resonance states with the width $\Gamma<$0.1 MeV. The rest 18
resonance states are wider ones, their widths exceed 1 MeV. One can see that
in most cases two-cluster system with fixed quantum numbers $L$, $S$ and
$J^{\pi}$ has only one Pauli resonance state. However, there are some cases
when two Pauli resonance states are observed. The larger is energy of the
resonance state, the larger is the total width. Energy of the second resonance
state in 7Li and 8Be is approximately 15 MeV larger than the energy of the
first resonance state. In 10B the energy difference is more than 25 MeV.

Table 2: Parameters of the Pauli resonance states in 6Li, 7Li and 10B Nucleus | Channel | $L$ | $S$ | $J^{\pi}$ |  $E$, MeV |  $\Gamma$, MeV
---|---|---|---|---|---|---
6Li | $\alpha+d$ | 0 | 1 | 1+ | 24.218 | 1.165
|  | 1 | 1 | 2- | 32.370 | 6.755
|  3He$+t$ | 0 | 1 | 1+ | 31.844 | 0.209
|  | 1 | 1 | 2- | 22.403 | 0.618
7Li | $\alpha+t$ | 1 | 1/2 | 1/2- | 29.002 | 2.144
|  | 1 | 1/2 | 3/2- | 25.810 | 0.027
|  | 0 | 1/2 | 1/2+ | 20.148 | 2.589
|  | 0 | 1/2 | 1/2+ | 34.444 | 4.702
|  6Li$+n$ | 0 | 1/2 | 1/2+ | 12.863 | 3.332
|  | 0 | 3/2 | 3/2+ | 18.895 | 0.196
10B |  6Li$+\alpha$ | 1 | 1 | 0- | 11.090 | 3.198
|  | 1 | 1 | 0- | 35.834 | 4.600
|  | 1 | 1 | 1- | 11.098 | 3.424
|  | 1 | 1 | 1- | 36.167 | 5.105
|  | 0 | 1 | 1+ | 13.427 | 0.056
|  | 0 | 1 | 1+ | 41.144 | 2.751

In Table 3 we present parameters of the Pauli resonance states obtained in
different state of 6Li+$d$ and 7Li+$d$ scattering.

Table 3: Energies and widths of the Pauli resonance states in 8Be and 9Be obsereved in the channels 6Li+$d$ and 7Li+$d$, respectively Nucleus | Channel | $L$ | $S$ | $J^{\pi}$ |  $E$, MeV |  $\Gamma$, MeV
---|---|---|---|---|---|---
8Be |  6Li+$d$ | 0 | 0 | $0^{+}$ | 17.233 | 3.553
|  | 0 | 1 | $1^{+}$ | 14.989 | 1.011
|  | 0 | 1 | $1^{+}$ | 25.724 | 4.628
|  | 0 | 2 | $2^{+}$ | 20.656 | 0.008
|  | 1 | 0 | $1^{-}$ | 18.253 | 0.058
|  | 1 | 1 | $2^{-}$ | 45.555 | 6.097
|  | 1 | 1 | $2^{-}$ | 18.523 | 0.008
|  | 1 | 2 | $3^{-}$ | 18.531 | 0.013
|  | 1 | 2 | $2^{-}$ | 20.981 | 0.402
9Be |  7Li+$d$ | 1 | 1/2 | 1/2- | 13.733 | 1.003
|  | 0 | 1/2 | 1/2+ | 15.717 | 5.796
|  | 0 | 1/2 | 1/2+ | 27.958 | 1.836

By analyzing results presented in Tables 2 and 3, we came to the conclusion
that the Pauli resonance states in light nuclei have energy more than 11 MeV,
their widths are mainly large ($\Gamma>$0.9 MeV), however a few very narrow
resonance state were found. The most populated area of resonance states lies
in the interval 16$<E<$21 MeV, as it is demonstrated in Fig. 7, left panel.
Two dense area of widths of resonance states are located in intervals 0.008
$<\Gamma<$0.22 MeV and 0.9$<\Gamma<$1.2 MeV (Fig. 7, right panel). In many
cases only one Pauli resonance state appeared in a binary channel. We also
determined several cases with two resonance states.

Figure 7: Density of resonance state energies (left panel) and widths (right
panel) of all determined Pauli resonance states

In Fig. 8 we display spectrum of the Pauli resonances of positive parity
states with the total orbital momentum $L=0$. These resonance states emerge in
nuclei 7Li, 8Be, 9Be and 10B with clusterization 6Li+$A_{2}$, where $A_{2}$
stands for neutron, deuteron, triton and alpha particle. This Figure shows
that the energies of the first Pauli resonance states are quite close for all
nuclei. It also shows that there are two Pauli resonance states in the channel
6Li$+d$ with the total spin $S$=1 and in the channel 6Li$+t$ with the total
spins $S$=1/2 and $S$=3/2. One can see that the larger is the second cluster,
the larger is the energy of the highest Pauli resonance state. Indeed, it
growths from 19 MeV in 6Li$+n$ channel to 41 MeV in the channel 6Li$+\alpha$.

Figure 8: Spectrum of Pauli resonances for $L$=0 state of nuclei 7Li, 8Be,
9Be, 10B detected in the channels 6Li+$n$, 6Li+$d$, 6Li+$t$ and 6Li+$\alpha$,
correspondingly

The Pauli resonance states of negative parity created in the channel
6Li+$A_{2}$ ($A_{2}$ =$d$, $t$, $\alpha$) with the total orbital momentum
$L$=1 are show in Fig. 9. We did not find any resonance state in the channel
6Li+$n$. In this channel, the Pauli resonance states do not appear neither in
states with total spin $S$=1/2 nor in the states $S$=3/2. Five Pauli resonance
states are found in 8Be and 9Be, and four resonances are detected in 10B. Fig.
9 shows that the energy of the lowest Pauli resonance state is deceasing with
increasing of mass of the ”projectile” $A_{2}$. It is interesting tendency, as
the Coulomb repulsing between 6Li and $A_{2}$ is increased with increasing of
mass of the second cluster $A_{2}$. It necessary to underline that the spin-
orbit interaction plays an important role in all cases when both the total
orbital momentum $L$ and total spin $S$ do not equal to zero.

Figure 9: Spectrum of the Pauli resonance states of the negative parity in
8Be, 9Be and 10B created in the state with the total orbital momentum $L$=1.

####  III.3.1 Birth of the Pauli resonance state

To detect the Pauli and shape resonance states we analyzed behavior of phase
shifts as a function of energy. Rapid growth of phase shift was considered as
a signal of a resonance state. There is another way for detecting of resonance
states of both types. This way applicable for any method which involve a
square integrable basis of functions. Unfortunately this method works for
relatively narrow resonance states. The narrow resonance states can be
detected by calculating eigenspectrum of a Hamiltonian with different number
of basis functions. By displaying eigenenergies as a function of the number of
basis functions (we denote them as $N_{O}$) involved in calculations, a
resonance state will display itself as a plateau or/and as a avoid crossing.
Energy of plateau is the energy of a resonance state. Such way of detecting of
resonance states is an essential element of the stabilization method (Ref.
Hazi and Taylor (1970)) and complex scaling method (see definitions of the
method and its recent progress in applications to many-cluster systems in
Refs. Myo  _et al._ (2014); Horiuchi  _et al._ (2012), Myo and Katō (2020)).

In Fig. 10 we show dependence of eigenenergies of the 3/2- state in 7Li as a
functions of the number of oscillator function $N_{0}$ used in calculations.
We gradually change the number of oscillator from 1 to 100. One can see that
it necessary to use at least three oscillator functions to create plateau or,
in other words, to obtain eigenvalue with the energy which is very close to
the energy of resonance state. Such a plateau unambiguously indicate presence
of a narrow resonance state. This result is naturally compatible (consistent)
with the results of phase shift calculations. Besides, the wave functions of
the resonance states obtained with 5, 10 and 100 oscillator functions are very
close to each other in the range of small values of $n$, as it demonstrated in
Fig. 11. It proves that the narrow 3/2- resonance state is formed by
oscillator functions with a very small values of $n$.

Figure 10: Spectrum of the 3/2- states in 7Li as a function of the number of
oscillator functions involved in calculations Figure 11: Convergence of wave
function of the narrow 3/2- resonance state in 7Li in the channel $\alpha+t$

###  III.4 Peculiarities of the Pauli resonance states

Let us consider peculiarities of the wave functions of the Pauli resonance
state. Analysis of wave functions will allow us to reveal (understand) nature
of the Pauli resonances. Wave functions of resonance and nonresonant states
are considered in the oscillator and coordinate representations.

In Fig. 12 we show three wave functions of the 3/2- states in 7Li for
clusterization $\alpha+t$. One of these functions is the wave function of the
ground state (GS), the second function is the Pauli resonance state (PR) with
energy $E$=25.810 MeV and the third function is the wave function of
nonresonant, elastic $\alpha+t$ scattering state (SC) ($E$=10.1 MeV). Main
difference between the Pauli resonance and nonresonant states wave functions
is that the contribution of the oscillator function $\left|0\right\rangle$.
This function gives the largest contribution to the wave function of the Pauli
resonance states, and it has smallest contribution to the wave functions of
the ground and continuous spectrum state.

Figure 12: Wave functions of the ground state (GS), Pauli resonance state (PR)

In Fig. 13 we demonstrate wave functions of these states in coordinate space.
As one should expect, nonresonant wave functions have a node at small
distances ($r$$<$2.5 fm), while resonance wave function has a first node at
relatively large distance ($r\approx$5.5fm). Besides, the resonance function
has a very large amplitude, at list two times larger than the amplitude of the
bound and scattering states.

Fig. 14 shows general picture of contribution of oscillator functions with the
quantum numbers $n=0$ and $n=1$ to wave functions of continuous spectrum
states over large energy range. Fig. 14 confirms also that the oscillator wave
function with $n=0$ contribute mainly to the Pauli resonance state and gives a
small contribution to other states of the $\alpha+t$ continuous spectrum.

Figure 13: Wave functions of 7Li=$\alpha+t$ in coordinate space of the 3/2-
bound (BS), resonance (PR) and scattering (SC) states.  Figure 14:
Contribution of the oscillator wave functions with $n=0$ and $n=1$ to wave
functions of the continuous spectrum 3/2- states of the $\alpha+t$ channel

Let us now consider a case with two Pauli resonance states. They are detected
in the 1/2+ state of 7Li in the channel $\alpha+t$. Wave functions of two
Pauli resonance states are shown in Fig. 15 in oscillator space and in Fig. 16
in coordinate space. Two oscillator functions with $n=0$ and $n=1$ give the
main contribution to the Pauli resonance functions. As the results, these wave
function in coordinate space describe a very compact two-cluster system, main
part of which are concentrated at the small distances between clusters, namely
$0\leq r\leq 5$ fm. Amplitudes of the wave functions in this small region are
substantially larger than their amplitude at large distances.

Figure 15: Wave functions in oscillator space of two Pauli resonance state in
the 1/2+ state of 7Li Figure 16: Wave functions in coordinate space of two
Pauli resonances in the 1/2+ state in the $\alpha+t$ channel

###  III.5 Overlap

As it was well (widely) recognized that mysteries (spurious) resonance states
appear due to the Pauli principle, it is then expedient to analyze its effects
on the norm kernels. Matrix of norm kernel in general case (for improved and
advanced version of the RGM) is nondiagonal, thus we start the analysis with
3D picture of the matrix. In Fig. 17 we display the overlap matrix
$\left\|\left\langle n|m\right\rangle\right\|$ for the channel $\alpha$+$t$ in
the state $L$=0, $S$=1/2 and $J^{\pi}$=1/2+. One can see that this matrix is a
quasi-diagonal. The largest matrix elements are located on the main diagonal,
and the larger is $m=n$, the close to unity they are. Off-diagonal matrix
elements $\left\langle n|m\right\rangle$ are very small. A few diagonal matrix
element with small values of $n$ are also small due to the Pauli principle.
One may conclude that the Pauli principle has a short-range nature as it
affects a relatively small number of cluster basis functions
$\left|n\right\rangle$ (15) and corresponding matrix elements $\left\langle
n|m\right\rangle$. Note that Fig. 17 demonstrates a typical behavior of matrix
elements of norm kernel in the advanced version of the RGM for all nuclei and
all states considered in this paper.

Figure 17: Matrix of norm kernel for the state $L$=0, $S$=1/2 and
$J^{\pi}$=1/2+ of the channel $\alpha+t$

Fig. 17 prompts us to study only diagonal matrix elements of the norm kernel,
which reflects fully (complete) effects of the Pauli principle. Consequently
in this section we discuss the diagonal matrix elements and eigenvalues of the
norm kernel. In Fig. 18 we compare diagonal matrix elements of the norm
kernel, determined in the standard (S) and advanced (A) version of the RGM,
for 7Li as two-cluster configuration $\alpha+t$. It is worthwhile recalling
that in the standard version, the matrix of the norm kernel is diagonal. This
figure demonstrates general features of the quantities $\left\langle
n|n\right\rangle$ and $\Lambda_{\alpha}$ for all two-cluster systems under
consideration. As was pointed our in previous paragraph, the major part of
diagonal matrix elements are equal unity and only small fraction of them
differ from unity, showing effects of the Pauli principle. It is necessary to
recall that oscillator wave functions with small values of the quantum number
$n$ describe two clusters at smallest relative distance, thus effects of the
Pauli principle for these functions are prominent. One can see that there are
two Pauli forbidden states in the 1/2+ state and one in the 3/2- state within
the standard version. In the advanced versions these basis states, namely,
$\left|0\right\rangle$ and $\left|1\right\rangle$ for 1/2+ and
$\left|0\right\rangle$ for 3/2-, can be considered as almost-forbidden Pauli
states, as the corresponding diagonal matrix elements are very small
($\left\langle n|n\right\rangle<0.1$). Fig. 18 demonstrates important features
of matrix elements: the number of forbidden states in the standard version
coincides with the number of almost-forbidden states in the advanced version.

Figure 18: Comparison of diagonal matrix elements of the norm kernel,
determined in the standard (S) and advanced (A) version of the RGM for 1/2+
and 3/2- states of 7Li=$\alpha+t$

The diagonal matrix elements $\left\langle n|n\right\rangle$ and eigenvalues
$\Lambda_{\alpha}$ of the norm kernel for the $0^{+}$ and $1^{-}$ states of
two-cluster system 6Li+$n$ are shown in Fig. 20. The almost forbidden states
are found for the states $L^{\pi}=0^{+}$ and $S$=1/2 and $S$=3/2. Comparing
Figs. 19 and 20 we see that the larger are interacting clusters, the larger is
the region of diagonal matrix elements $\left\langle n|n\right\rangle$ which
are affected of the Pauli principle.

In Fig. 19 we display the diagonal matrix elements $\left\langle
n|n\right\rangle$ and eigenvalues $\Lambda_{\alpha}$ of the norm kernel for
the $0^{+}$ states of cluster system 6Li+$d.$ Diagonal matrix element also
show that there are a few almost forbidden states when $\left\langle
n|n\right\rangle$ are close to zero. One may observe a set of the super-
allowed Pauli states ($\left\langle n|n\right\rangle>1$) for the total spin
$S$=1 and $S$=2. There are similarities between eigenvalues and the diagonal
matrix elements of the norm kernel. The eigenvalues $\Lambda_{\alpha}$ reveals
a few almost forbidden states, two states for $S$=1 and one state for the
total spin $S$=0 and $S$=2. Similar to the diagonal matrix elements, the
eigenvalues for $S$=0 and $S$=2 possess the super allowed states.

Figure 19: Diagonal matrix elements (the upper panel) and eigenvalues (the
lower panel) of the norm kernel for the $L^{\pi}$= $0^{+}$ states and
different values of the total spin $S$ in two-cluster system 6Li+$d$ Figure
20: Diagonal matrix elements (the upper panel) and eigenvalues (the lower
panel) of the norm kernel for two-cluster system 6Li+$n$.

Diagonal matrix elements $\left\langle n|n\right\rangle$ of the norm kernel
and its eigenvalues $\Lambda_{\alpha}$ for the channel 6Li+$\alpha$ are
displayed in Fig. 21. Two almost forbidden states are demonstrated by both
diagonal matrix elements and eigenvalues. They are observed in two states:
$L$=0, $S$=1, $J^{\pi}$=1+ and $L$=1, $S$=1, $J^{\pi}$=1-.

Figure 21: Diagonal matrix elements (upper panel) and eigenvalues (lower
panel) of the norm kernel for the channel 6Li+$\alpha$ in the states $L$=0,
$S$=1, $J^{\pi}$=1+ and $L$=1, $S$=1, $J^{\pi}$=1-

Finishing this subsections, we conclude that the number of almost forbidden
states coincides with the number of almost forbidden eigenstates. Almost
forbidden states $\left|n\right\rangle$ obey the restriction $\left\langle
n|n\right\rangle<0.3$, while almost forbidden eigenstates have
$\Lambda_{\alpha}<0.2$. Comparing results demonstrated in Figs. 17, 18, 19, 20
with results of Tables 2 and 3 we came to the conclusion that the number of
almost forbidden states equals of the number of the Pauli resonance states.

##  IV Method REV

Let us consider main ideas of the REV method formulated in Ref. Kruglanski and
Baye (1992). The author of Ref. Kruglanski and Baye (1992) paid attention that
a set of new eigenstates of the norm kernel appeared in the case when
different oscillator lengths were used for an alpha particle and 16O. These
eigenstates have very small values comparatively to eigenstates with the
common oscillator length. For example, the smallest eigenvalue obtained for
the total orbital momentum $L^{\pi}$=0+ with the common oscillator length is
equal to 0.229, while there are four eigenstates with eigenvalues less than
0.03. Similar picture was also observed for the state $L^{\pi}$=1-. The lowest
eigenvalue obtained with the common oscillator length equals 0.344, for
different oscillator lengths $b_{\alpha}$=1.395 fm and $b_{O}$=1.776 fm, four
eigenstates emerged with eigenvalues less than 0.04.

It was suggested in Ref. Kruglanski and Baye (1992) to eliminate such
eigenvalues and to use smaller set of norm kernel eigenstates. Thus in the
case of different oscillator lengths, all eigenstates with eigenvalue smaller
than eigenvalues with common oscillator length were treated as the Pauli
forbidden states. Actually, the border between the Pauli allowed and Pauli
forbidden state the Pauli allowed and Pauli forbidden states in system
$\alpha+^{16}$O was selected to be 0.1. Having applied such restrictions, all
Pauli resonance states disappeared.

We decided to use this method to eliminate the Pauli resonance states which
appear in light nuclei within the advanced resonating group method. Analysis
of the eigenvalues of the norm kernel carried out in Section III.5 indicates
that we have to redetermine the border between the Pauli allowed and Pauli
forbidden states.

Efficiency of the REV method will be demonstrated in Section V.1.

##  V Method ROF

We suggest another method to struggle with the Pauli resonance states in light
nuclei. This method relay on properties of matrix elements of the norm kernel.
By analyzing properties of the matrix elements, our attention was attracted by
behavior of diagonal matrix elements $\left\langle n|n\right\rangle$. In many
cases the matrix element $\left\langle 0|0\right\rangle$ and some times matrix
element $\left\langle 1|1\right\rangle$ are very small with respect to other
diagonal matrix elements. The analysis also showed (revealed) that matrix
elements of corresponding row ($\left\langle 0|n\right\rangle$, $\left\langle
1|n\right\rangle$) and columns ($\left\langle n|0\right\rangle$, $\left\langle
n|1\right\rangle$) are also very small. Besides, it was shown above (Section
III.4) that oscillator functions with $n=0$ and sometimes with $n=1$ dominate
in the wave function of the Pauli resonance states. Thus we suggest to omit
those part of the matrix $\left\|\left\langle
n|\widetilde{n}\right\rangle\right\|$ whose diagonal matrix elements are very
small. We also suggest a criteria for smallness of the diagonal matrix
elements. Let us introduce minimal value of the diagonal matrix elements
$O_{\min}$ which will mark a border between the Pauli forbidden (or almost
forbidden) and Pauli allowed states. Within our method, all diagonal matrix
elements which are smaller than $O_{\min}$ will be omitted with their
correspondent rows and columns.

Analysis of the diagonal matrix elements of the norm kernel leads us to the
conclusion that in many two-cluster cases, considered above, $O_{\min}$ can be
set to 0.2. This can be seen in Figs. 19, 20, 21. Such a value can be also
used both for the case of one or two Pauli resonance states.

It is important to notice, that from mathematical point of view almost
forbidden basis states or eigenstates are allowed states and should not create
any problems. The same is true also from computational point of view, as the
smallest eigenvalues are much larger than the smallest numerical value
(numerical zero) in modern computers. Indeed, almost forbidden states do not
create any problem for bound states and their parameters, such as root-mean-
square mass and proton radii and so on. Presence of almost forbidden states
affects (distorts) only continuous spectrum states. In this respect, both
methods suggest redetermination of essentially allowed Pauli states. Both
methods determine border between almost forbidden and allowed states. This
border is marked by $\Lambda_{\min}$ and $O_{\min}$ in the REF and ROF
methods, respectively. In general case, one can use $\Lambda_{\min}$ and
$O_{\min}$ as variational parameters to control the number of eliminated basis
states $\left|n\right\rangle$ or eigenstates $\left|\alpha\right\rangle$ and
their effects on scattering parameters. Naturally, the main aim of such
procedure is to eliminate the Pauli resonance state(s) and to cause minimal
effects on bound states and shape resonance states.

###  V.1 Demonstration of the REV and ROF methods

Having analyzed the diagonal matrix elements and eigenvalues of the overlap
matrix, we deduced $O_{\min}$ and $\Lambda_{\min}$ for all nuclei and for
those states $J^{\pi}$ which have the Pauli resonance states. These quantities
are displayed in Table 4.

Table 4: Actual values of $O_{\min}$ and $\Lambda_{\min}$, and the number of forbidden states $N_{f.s.}$, which are used to eliminate the Pauli resonance states Nucleus | Clusterization | $L$ | $S$ | $J^{\pi}$ | $O_{\min}$ | $\Lambda_{\min}$ | $N_{f.s}$
---|---|---|---|---|---|---|---
6Li |  4He+$d$ | 0 | 1 | $1^{+}$ | 0.2 | 0.2 | 1
|  | 1 | 1 | $2^{-}$ | 0.2 | 0.2 | 1
|  3H+3He | 0 | 1 | $1^{+}$ | 0.1 | 0.1 | 1
|  | 1 | 1 | $2^{-}$ | 0.1 | 0.1 | 1
7Li |  4He+3H | 1 | 1/2 | $3/2^{-}$ | 0.1 | 0.1 | 1
|  | 1 | 1/2 | $1/2^{-}$ | 0.1 | 0.1 | 1
|  | 0 | 1/2 | $1/2^{+}$ | 0.1 | 0.1 | 2
|  | 2 | 1/2 | $3/2^{+}$ | 0.1 | 0.1 | 1
|  6Li+$n$ | 0 | 1/2 | $1/2^{+}$ | 0.3 | 0.3 | 1
8Be |  6Li+$d$ | 0 | 0 | $0^{+}$ | 0.2 | 0.2 | 1
|  | 0 | 1 | $1^{+}$ | 0.1 | 0.1 | 1
|  | 1 | 0 | $1^{-}$ | 0.2 | 0.2 | 1
|  | 1 | 1 | $2^{-}$ | 0.3 | 0.3 | 1
|  | 0 | 2 | $2^{+}$ | 0.1 | 0.1 | 1
9Be |  6Li+$t$ | 0 | 1/2 | $1/2^{+}$ | 0.2 | 0.1 | 2
|  | 1 | 1/2 | $1/2^{-}$ | 0.1 | 0.1 | 1
|  | 0 | 3/2 | $3/2^{+}$ | 0.2 | 0.1 | 2
|  | 1 | 3/2 | $5/2^{-}$ | 0.1 | 0.1 | 1
|  | 1 | 3/2 | ,$5/2^{-}$ | 0.2 | 0.2 | 1
10B |  6Li+4He | 1 | 1 | $0^{-}$ | 0.3 | 0.2 | 2
|  | 1 | 1 | $1^{-}$ | 0.3 | 0.2 | 2
|  | 2 | 1 | $1^{+}$ | 0.2 | 0.2 | 2
|  | 2 | 1 | $2^{+}$ | 0.2 | 0.2 | 1
|  | 2 | 1 | $3^{+}$ | 0.2 | 0.2 | 1

In Fig. 22 we demonstrate efficiency of the REV and ROF methods for the
$\alpha+t$ scattering in the 1/2+ state. Here OA stands for ordinary algorithm
of obtaining phase shifts within the advanced version of the RGM. The phase
shift in this approach exhibits two Pauli resonance states, parameter of which
are shown in Table 2. As we can see both methods remove the Pauli resonance
states. They also yield the phase shifts which are close to the standard
version at low energy region 0$\leq E<$6 MeV. There is very small difference
of phase shifts obtained with the REV and ROF methods. We used minimal values
of $\Lambda_{\min}=O_{\min}=0.2$. This restriction eliminated two functions in
both methods.

Figure 22: Phase shifts of the elastic $\alpha+t$ scattering in 1/2+ state as
a function of energy determined in three different approaches.

The similar picture is observed for the $\alpha+d$ scattering in the 2- state,
see Fig. 23. Only one Pauli resonance state is generated in this case. Both
REV and ROF methods remove that Pauli resonance state and produce the phase
shifts with very small differences. In this case we also used minimal values
of $\Lambda_{\min}=O_{\min}=0.2$. This restriction eliminated only one
function in both methods.

Figure 23: Phase shifts of the elastic $\alpha+d$ scattering in the 2- state

Phase shifts of the elastic 6Li+$d$ scattering obtained within three different
approaches are shown in Fig. 24. As one can see, in this case we observe both
low-energy shape and high-energy Pauli resonance states. The REV and ROF
methods eliminating one eigenfunction and one oscillator function,
respectively, remove the Pauli resonance state. They also slightly change
parameters of the shape resonance. In the original approach (OA) parameters of
the shape resonance are $E$=0.153 MeV and $\Gamma$=0.013 MeV, while in the REV
method they are $E$=0.374 MeV and $\Gamma$=0.485 MeV and in the ROF method we
obtain $E$=0.352 MeV and $\Gamma$=0.371 MeV. Note that the REV and ROF give
almost identical phase shifts of the 6Li+$d$ scattering. It means that the
eliminated eigenfunction of the norm kernel and the eliminated oscillator
function are close to each other.

Figure 24: Phase shifts of the elastic 6Li+$d$ scattering in the state $L=0$,
$S=0$ and $J^{\pi}=0^{+}$

We found several cases when the REV and ROF methods give noticeable different
phase shifts. One of such examples is shown in Fig. 25 where phase shifts of
the 6Li+$\alpha$ scattering in the state $L=1$, $S=1$ and $J^{\pi}=1^{-}$ are
drawn. Note that almost the same results are observed for the state
$J^{\pi}=0^{-}$ and $J^{\pi}=2^{-}$ generated by the coupling of the total
orbital momentum $L=1$ with the total spin $S=1$. Two Pauli resonance states
were removed by eliminating two eigenfunctions of the norm kernel obeying the
restriction $\Lambda_{\alpha}\leq$0.2, and two oscillator functions with the
restriction $\left\langle n|n\right\rangle\leq$0.3. Noticeable deviation of
the phase shifts obtained in the REV and ROF methods is seen at the energy
region $E>$3 MeV. Such deviation can be explained by structure of the
eigenfunctions and their relation to oscillator functions. If an eigenfunction
is mainly represented by one oscillator function, one may expect close results
of both methods. If eigenfunction is spread over large number of oscillator
functions, results obtained with these two method would be different. To prove
this statement, we show in Fig. 27 eigenfunctions
$\left\|U_{n}^{\alpha}\right\|$of the norm kernel as a function of $n$ for two
different cases with two Pauli resonance states. We selected cases for elastic
6Li+$\alpha$ scattering with quantum numbers $L=S$=1, $J^{\pi}$=1-and $L=0,$
$S$=1, $J^{\pi}$=1+. Phase shifts for them are shown in Figs 25 and 26. Fig.
27 demonstrates that for the $J^{\pi}$=1- state, a large number of oscillator
functions participate in formation of eigenfunctions $U_{n}^{1}$ and
$U_{n}^{2}$. While for the $J^{\pi}$=1+ state, lowest oscillator functions
with $n=0$ and $n=1$ totally dominates in corresponding eigenfunctions
$U_{n}^{1}$ and $U_{n}^{2}$. Similar dominance of oscillator function with the
quantum number $n=0$ in the eigenfunction $U_{n}^{1}$ are observed in all
cases, when phase shifts obtained with the REV and ROF are coincide.

Figure 25: Phase shifts of the 6Li+$\alpha$ scattering in the state $L=1$,
$S=1$, $J^{\pi}=1^{-}$ obtained with three different approaches. Figure 26:
Phase shifts of the elastic 6Li+$\alpha$ scattering with Jπ=1+ caculated
within three approximations Figure 27: Eigenfunctions $U_{n}^{\alpha}$ of the
norm kernel as a function of $n$ for $J^{\pi}$=1- and $J^{\pi}$=1+ states in
10B, considered as two-cluster system 6Li+$\alpha$

In Table 5 we demonstrate effects of eliminated eigenfunctions and oscillator
functions on parameters of bound and resonance states. These results are
obtained for the 1+ states in 10B. By increasing $\Lambda_{\min}$ ($O_{\min})$
from zero to a certain value, indicated in the second column of Table 5, we
manage to eliminate one two and three eigenfunctions (oscillator functions).
In the fourth column of Table 28, we demonstrate how eliminated eigenfunctions
and oscillator functions affect energy of the $1^{+}$ bound state of 10B. In
Fig. 28 we show effects of eliminated functions on the 6Li+$\alpha$ phase
shift. By eliminating one eigenfunction or one oscillator function, we remove
the lowest Pauli resonance state and change position (lower down) of the
second resonance on approximately 6.5 MeV. However, energy of the ground state
is slightly changed after removing one function. When we remove two
eigenfunctions or two oscillator functions, both Pauli resonance states are
disappeared. Two removed eigenfunctions increase energy of the bound state on
0.9 MeV, while two removed oscillator functions increase the energy on
$\approx$1.3 MeV.

Table 5: Evolution of bound and resonance states in 10B. Energies and widths are in MeV Method | $\Lambda_{\min}/O_{\min}$ | $N_{f}$ | $E_{GS}$ | $E$ | $\Gamma$ | $E$ | $\Gamma$
---|---|---|---|---|---|---|---
OA | 0.0 | 100 | -2.477 | 13.427 | 0.056 | 41.144 | 2.751
REV | 0.1 | 99 | -2.280 | - | - | 34.539 | 3.503
ROF | 0.1 | 99 | -2.264 | - | - | 34.826 | 3.684
REV | 0.2 | 98 | -1.527 | - | - | - | -
ROF | 0.2 | 98 | -1.183 | - | - | - | -
REV | 0.7 | 97 | -0.132 | - | - | - | -
ROF | 0.7 | 97 | 0.526 | - | - | - | -
Figure 28: Phase shifts of the elastic 6Li+$\alpha$ scattering obtained with
different values of $\Lambda_{\min}$ (the upper part) and $O_{\min}$ (the
lower part)

As we pointed out above, oscillator functions with small values of the quantum
number $n$ and eigenfunctions with small value of index $\alpha$ describe the
most compact two-cluster configurations. It is interesting to analyze effects
of their deletion on energies of bound states and shape resonances, if they
appear. For this aim we collected in Table 6 energies of bound and resonance
states

Table 6: Effects of removed eigenfunctions and oscillator functions on energy of bound and shape resonance states Nucleus | Channel | $J^{\pi}$ | Parameter | OA | REV | ROF
---|---|---|---|---|---|---
7Li | $\alpha+t$ | 3/2- | $N_{f}$ | 100 | 99 | 99
|  |  |  $E_{GS}$, MeV | -1.127 | -1.105 | -1.064
8Be |  6Li$+d$ | 0+ | $N_{f}$ | 100 | 99 | 99
|  |  |  $E_{GS}$, MeV | -18.971 | -15.281 | -14.426
|  |  |  $E$, MeV | 0.153 | 0.374 | 0.352
|  |  |  $\Gamma$, MeV | 0.013 | 0.485 | 0.371
|  | 2- | $N_{f}$ | 100 | 99 | 99
|  |  |  $E$, MeV | 0.800 | 0.825 | 0.805
|  |  |  $\Gamma$, MeV | 0.738 | 0.957 | 0.762
10B |  6Li$+\alpha$ | 1+ | $N_{f}$ | 100 | 98 | 98
|  |  |  $E_{GS}$, MeV | -2.477 | -1.527 | -1.183

####  V.1.1 Preliminary conclusions

At the end of this section we made preliminary conclusions concerning the REV
and ROF methods. In all cases, presented above, both methods completely remove
all detected Pauli resonance states. In many cases, both methods give close
results for phase shifts. In some cases, phase shifts are somewhat (slightly)
different. Such a difference as we demonstrated, appear when eigenfunctions of
the norm kernel are spread over a large number of oscillator functions. In
other words, removed eigenfunctions and removed oscillator functions are quit
different. When results of both method coincide, removed eigenfunctions are
presented mainly by removed oscillator functions.

We demonstrated that the ROF method, formulated in this paper, is an
alternative method to the one suggested by Kruglanski and Baye. Advantage of
the ROF is that it does no require an orthogonalization procedure of matrices
of norm kernel and then transformation of matrix of Hamiltonian to new
representation. This procedure is time consuming when a large number of basis
functions are involved. We also demonstrated that oscillator representation is
appropriate tool for studying effects of the Pauli principle on kinematic
(matrix of norm kernel) and dynamics (matrix of a Hamiltonian) of two- and
many-cluster systems.

##  VI Conclusions

Properties of Pauli resonance states in two-body continuum of the light nuclei
6Li, 7Li, 7Be, 9Be and 10B have investigated within the advanced version of
the resonating group method. The advanced version employs a three-cluster
configuration which allows to consider in general case three two-body (binary)
channels. One of constituents of a binary channel is considered as a two-
cluster subsystem, which provides us with more correct description of the
nuclei having distinct two-cluster structure and a small separation energy.
Wave functions of two-cluster subsystem are obtained by solving appropriate
Schrödinger equation. The advanced version we have employed make use of the
square-integrable bases - Gaussian and oscillator bases. Gaussian basis is
used to describe relative motion of two clusters in two-cluster subsystem and
is very efficient in obtaining wave functions of bound states with minimal
number of basis functions. Oscillator basis is used to study interaction of
the third cluster with two-cluster subsystem. It allows us to implement proper
boundary condition for discrete and continuous spectrum states. It was
demonstrated that oscillator basis is suitable tool to study effects of the
Pauli principle and to reveal nature of the Pauli resonance states.

It was demonstrated that the advanced form of two-cluster subsystem is the
origin of the Pauli resonance states. More precisely, an advanced form of wave
function of two-cluster subsystem is responsible for appearance of the Pauli
resonance states.

It has been shown that the Pauli resonance states appear at the relatively
high energy $E>$10 MeV. Some of these resonance states are very narrow
resonance states, however, major part of them are broad resonance states. The
most populated area of resonance states lies in the interval 16$<E<$21 MeV.
Two dense area of widths of resonance states are located in intervals 0.008
$<\Gamma<$0.22 MeV and 0.9$<\Gamma<$1.2 MeV.

It was found that the oscillator functions with minimal value of quantum
number $n$ (the number of radial oscillator quanta) dominates in resonance
wave functions. These basis functions yields very small values of the diagonal
matrix elements $\left\langle n|n\right\rangle$ of the norm kernel. It was
also demonstrated that the very narrow Pauli resonance states can be detected
by using a very small number of oscillator functions: from three to five
functions.

Role of Pauli super-allowed and almost-forbidden states has thoroughly
studied. New method for eliminating the Pauli resonance states is formulated
and its efficiency is demonstrated for nuclei under consideration.

We have established that the Pauli principle predetermine appearance of the
Pauli resonance states by creating almost forbidden states, however energies
and widths of the Pauli resonance states are totally driven (formatted) by
nucleon-nucleon forces.

We found that the number of Pauli resonance states for the given $J^{\pi}$
state, discovered within the advanced version of the RGM, coincides with the
number of the Pauli forbidden states determined in the standard version of the
RGM.

One of the main conclusions of the present paper is that one needs to find
proper definition of the Pauli forbidden and Pauli allowed (fully or
partially) states. Standard or formal definition for Pauli forbidden states is
that the eigenvalues for them should be equal zero $\Lambda_{\alpha}=0$. Then,
the Pauli allowed states should have $\Lambda_{\alpha}>0$. However, the
carried out analysis leads us to the conclusion that for light nuclei with
two-body clusterization the border between forbidden and allowed states is
$\Lambda_{\min}=0.2$. It was also shown that oscillator functions
$\left|n\right\rangle$ which generates (yield) diagonal matrix elements of the
norm kernel $\left\langle n|n\right\rangle\leq O_{\min}=0.2$, can be
considered as the Pauli forbidden states. By removing of the Pauli forbidden
states, one eliminates the Pauli resonance states and causes minor effects on
energy of bound states and energy and width of the shape resonance states, if
they exist. We have not found universal values of $\Lambda_{\min}$ and
$O_{\min}$ for all light nuclei , which have been considered.

As for perspective of this work. In the present paper we have restricted
ourselves to a single-channel approximation to reveal the Pauli resonance
states and find main factors responsible for formation of such states. In the
future paper we are planning to consider appearance of the Pauli resonance
states in many-channel systems and how the REF and ROF can help to eliminate
them. Many-channel cases are specially interesting since small eigenvalues of
the norm kernel can appear due to strong overlap of basis functions belonging
to different channels. This strong coupling is not directly related to the
Pauli principle. This makes the problem more attractive and challenging.

###### Acknowledgements.

We would like to thank K. Katō for stimulating discussions and encouraging
support. This work was supported in part by the Program of Fundamental
Research of the Physics and Astronomy Department of the National Academy of
Sciences of Ukraine (Project No. 0122U000889) and by the Ministry of Education
and Science of the Republic of Kazakhstan, Research Grant IRN: AP 09259876.
V.V. is grateful to the Simons foundation for financial support.

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