# Planar equilibrium measure problem
in the quadratic fields with a point charge

Sung-Soo Byun  Center for Mathematical Challenges, Korea Institute for
Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea
<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

We consider a two-dimensional equilibrium measure problem under the presence
of quadratic potentials with a point charge and derive the explicit shape of
the associated droplets. This particularly shows that the topology of the
droplets reveals a phase transition: (i) in the post-critical case, the
droplets are doubly connected domain; (ii) in the critical case, they contain
two merging type singular boundary points; (iii) in the pre-critical case,
they consist of two disconnected components. From the random matrix theory
point of view, our results provide the limiting spectral distribution of the
complex and symplectic elliptic Ginibre ensembles conditioned to have zero
eigenvalues, which can also be interpreted as a non-Hermitian extension of the
Marchenko-Pastur law.

###### Key words and phrases:

Planar equilibrium measure problem, two-dimensional Coulomb gases, elliptic
Ginibre ensemble, conditional point process, conformal mapping method, a non-
Hermitian extension of the Marchenko-Pastur law

Sung-Soo Byun was partially supported by Samsung Science and Technology
Foundation (SSTF-BA1401-51) and by the National Research Foundation of Korea
(NRF-2019R1A5A1028324) and by a KIAS Individual Grant (SP083201) via the
Center for Mathematical Challenges at Korea Institute for Advanced Study.

##  1\. Introduction and Main results

In this paper, we study a planar equilibrium problem with logarithmic
interaction under the influence of quadratic potentials with a point charge.
This problem is purely deterministic, but its motivation comes from the random
world, more precisely, from the random matrix theory or the theory of two-
dimensional Coulomb gases in general. To be more concrete, for given points
$\boldsymbol{\zeta}=(\zeta_{j})_{j=1}^{N}\in\mathbb{C}^{N}$ of configurations,
we consider the Hamiltonians

(1.1) $\displaystyle\mathbf{H}_{N}^{\mathbb{C}}(\boldsymbol{\zeta})$
$\displaystyle:=\sum_{1\leq j<k\leq
N}\log\frac{1}{|\zeta_{j}-\zeta_{k}|^{2}}+N\sum_{j=1}^{N}W(\zeta_{j}),$ (1.2)
$\displaystyle\mathbf{H}_{N}^{\mathbb{H}}(\boldsymbol{\zeta})$
$\displaystyle:=\sum_{1\leq j<k\leq
N}\log\frac{1}{|\zeta_{j}-\zeta_{k}|^{2}}+\sum_{1\leq j\leq k\leq
N}\log\frac{1}{|\zeta_{j}-\bar{\zeta}_{k}|^{2}}+2N\sum_{j=1}^{N}W(\zeta_{j}).$

Here $W:\mathbb{C}\to{\mathbb{R}}$ is a suitable function called the external
potential. These are building blocks to define joint probability distributions

(1.3) $d\mathbb{P}_{N,\beta}^{\mathbb{C}}(\boldsymbol{\zeta})\propto
e^{-\frac{\beta}{2}\mathbf{H}_{N}^{\mathbb{C}}(\boldsymbol{\zeta})}\prod_{j=1}^{N}\,dA(\zeta_{j}),\qquad
d\mathbb{P}_{N,\beta}^{\mathbb{H}}(\boldsymbol{\zeta})\propto
e^{-\frac{\beta}{2}\mathbf{H}_{N}^{\mathbb{H}}(\boldsymbol{\zeta})}\prod_{j=1}^{N}\,dA(\zeta_{j}),$

where $dA(\zeta)=d^{2}\zeta/\pi$ is the area measure and $\beta>0$ is the
inverse temperature. Both point processes $\mathbb{P}_{N,\beta}^{\mathbb{C}}$
and $\mathbb{P}_{N,\beta}^{\mathbb{H}}$ represent two-dimensional Coulomb gas
ensembles [38, 55, 62]. In particular, if $\beta=2$, they are also called
determinantal and Pfaffian Coulomb gases respectively due to their special
integrable structures, see [26, 28] for recent reviews on this topic.
Furthermore, they have an interpretation as eigenvalues of non-Hermitian
random matrices with unitary and symplectic symmetry. For instance, if
$W(\zeta)=|\zeta|^{2}$, the ensembles (1.3) corresponds to the eigenvalues of
complex and symplectic Ginibre matrices [40].

One of the fundamental questions regarding such point processes is their
macroscopic/global behaviours as $N\to\infty.$ For the case $\beta=2$, this
can be regarded as a problem determining the limiting spectral distribution of
given random matrices. The classical results in this direction include the
circular law for the Ginibre ensembles. As expected from the structure of the
Hamiltonians (1.1) and (1.2), the macroscopic behaviours of the system can be
effectively described using the logarithmic potential theory [59].

For this purpose, let us briefly recap some basic notions in the logarithmic
potential theory. Given a compactly supported probability measure $\mu$ on
$\mathbb{C}$, the weighted logarithmic energy $I_{W}[\mu]$ associated with the
potential $W$ is given by

(1.4)
$I_{W}[\mu]:=\int_{\mathbb{C}^{2}}\log\frac{1}{|z-w|}\,d\mu(z)\,d\mu(w)+\int_{\mathbb{C}}W\,d\mu.$

For a general potential $W$ satisfying suitable conditions, there exists a
unique probability measure $\mu_{W}$ which minimises $I_{W}[\mu]$. Such a
minimiser $\mu_{W}$ is called the equilibrium measure associated with $W$ and
its support $S_{W}:=\operatorname{supp}(\mu_{W})$ is called the droplet.
Furthermore, if $W$ is $C^{2}$-smooth in a neighbourhood of $S_{W}$, it
follows from Frostman’s theorem that $\mu_{W}$ is absolutely continuous with
respect to $dA$ and takes the form

(1.5) $d\mu_{W}(z)=\Delta W(z)\cdot\mathbbm{1}_{\\{z\in S_{W}\\}}\,dA(z),$

where $\Delta:=\partial\bar{\partial}$ is the quarter of the usual Laplacian.

In relation with the point processes (1.3), it is well known [31, 15, 41] that

(1.6) $\mu_{N,W}:=\frac{1}{N}\sum_{j=1}^{N}\delta_{\zeta_{j}}\to\mu_{W}$

in the weak star sense of measure. From the statistical physics viewpoint,
this convergence is quite natural since the weighted energy $I_{W}$ in (1.4)
corresponds to the continuum limit of the discrete Hamiltonians (1.1) and
(1.2) after proper renormalisations. (In the case of (1.2), it is required to
further assume that $W(\zeta)=W(\bar{\zeta})$.)

Contrary to the density (1.5) of the measure $\mu_{W}$, there is no general
theory on the determination of its support $S_{W}$. (See however [60] for a
general theory on the regularity and [49] on the connectivity of the droplet
associated with Hele-Shaw type potentials.) This leads to the following
natural question.

For a given potential $W$, what is the precise shape of the associated
droplet?

In view of the energy functional (1.4), this is a typical form of an inverse
problem in the potential theory and is called an equilibrium measure problem.
Beyond the case when $W$ is radially symmetric (cf. [59, Section IV.6]), this
problem is highly non-trivial even for some explicit potentials with a simple
form, see [3, 12, 44, 14, 21, 35, 54, 36] for some recent works. Let us also
stress that such a problem is important not only because it provides the
intrinsic macroscopic behaviours of the point processes (1.3) but also because
it plays the role of the first step to perform the Riemann-Hilbert analysis
which gives rise to a more detailed statistical information ($k$-point
functions) of the point processes, see [12, 13, 17, 18, 44, 45, 46, 47, 51,
53, 56] for extensive studies in this direction. In this work, we aim to
contribute to the equilibrium problems associated with the potentials (1.7)
and (1.14) below, which are of particular interest in the context of non-
Hermitian random matrix theory.

###  1.1. Main results

For given parameters $\tau\in[0,1)$ and $c\geq 0$, we consider the potential

(1.7)
$Q(\zeta):=\frac{1}{1-\tau^{2}}\Big{(}|\zeta|^{2}-\tau\operatorname{Re}\zeta^{2}\Big{)}-2c\log|\zeta|.$

When $\beta=2$, the ensembles (1.3) associated with $Q$ correspond to the
distribution of random eigenvalues of the elliptic Ginibre matrices of size
$(c+1)N$ conditioned to have zero eigenvalues with multiplicity $cN$. We
mention that such a model with $c>0$ was also studied in the context of
Quantum Chromodynamics [1].

In (1.7), the logarithmic term can be interpreted as an insertion of a point
charge, see [4, 33, 23, 22, 27] for recent investigations of the models (1.3)
in this situation. Such insertion of a point charge has also been studied in
the theory of planar orthogonal polynomials [12, 13, 17, 51, 52, 53, 16]. On
the other hand, the parameter $\tau\in[0,1)$ captures the non-Hermiticity of
the model. To be more precise, the models (1.3) associated with $Q$
interpolate the complex/symplectic Ginibre ensembles ($\tau=0$) with the
Gaussian Unitary/Symplectic ensembles ($\tau=1$) conditioned to have zero
eigenvalues, see Remark 1.6 for further discussion in relation to our main
results.

For the case $c=0$, the terminology “elliptic” comes from the fact that the
limiting spectrum $S_{\tau,0}$ is given by the ellipse

(1.8)
$S_{\tau,0}:=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x}{1+\tau}\Big{)}^{2}+\Big{(}\frac{y}{1-\tau}\Big{)}^{2}\leq
1\Big{\\}},$

which is known as the elliptic law, see e.g. [34, 37]. We refer to [50, 5, 8,
57, 20, 19] and references therein for more about the recent progress on the
complex elliptic Ginibre ensembles and [43, 6, 24, 25] for their symplectic
counterparts. For the rotationally invariant case when $\tau=0,$ it is easy to
show that the associated droplet $S_{0,c}$ is given by

(1.9) $S_{0,c}:=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:c\leq x^{2}+y^{2}\leq
1+c\Big{\\}},$

see e.g. [59, Section IV.6] and [26, Section 5.2].

The primary goal of this work is to determine the precise shape of the droplet
associated with the potential (1.7) for general $\tau\in[0,1)$ and $c\geq 0$.
For this, we set some notations. Let us write

(1.10) $\tau_{c}:=\frac{1}{1+2c}$

for the critical non-Hermiticity parameter. For $\tau\in(\tau_{c},1)$, we
define

(1.11) $f(z)\equiv
f_{\tau}(z):=\frac{(1+\tau)(1+2c)}{2}\frac{(1-az)(z-a\tau)^{2}}{z(z-a)},\qquad
a=-\frac{1}{\sqrt{\tau(1+2c)}}.$

We are now ready to present our main result.

######  Theorem 1.1.

Let $Q$ be given by (1.7). Then the droplet $S\equiv S_{\tau,c}=S_{Q}$ of the
equilibrium measure

(1.12) $d\mu_{Q}(z)=\frac{1}{1-\tau^{2}}\mathbbm{1}_{S}(z)\,dA(z)$

is given as follows.

  * •

(Post-critical case) If $\tau\in(0,\tau_{c}]$, we have

(1.13)
$S_{\tau,c}=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x}{(1+\tau)\sqrt{1+c}}\Big{)}^{2}+\Big{(}\frac{y}{(1-\tau)\sqrt{1+c}}\Big{)}^{2}\leq
1\,,\,\frac{x^{2}+y^{2}}{(1-\tau^{2})c}\geq 1\Big{\\}}.$

  * •

(Pre-critical case) If $\tau\in[\tau_{c},1)$, the droplet $S_{\tau,c}$ is the
closure of the interior of the real analytic Jordan curves given by the image
of the unit circle with respect to the map $z\mapsto\pm\sqrt{f(z)}$, where $f$
is given by (1.11).

(a) $\tau=1/6<\tau_{c}$

(b) $\tau=1/3=\tau_{c}$

(c) $\tau=1/2>\tau_{c}$

Figure 1.  The droplet $S_{\tau,c}$, where $c=1$ and a Fekete point
configuration with $N=2048$.

Note that if $c=0$ (resp., $\tau=0$), the droplet (1.13) corresponds to (1.8)
(resp., (1.9)). We mention that the post-critical case of Theorem 1.1 is
indeed shown in a more general setup, see (2.1) and Proposition 2.1 below.

######  Remark 1.2 (Phase transition of the droplet).

In Theorem 1.1, we observe that if $c>0$, the topology of the droplet reveals
a phase transition. Namely, for the post-critical case when $\tau<\tau_{c}$,
the droplet is a doubly connected domain, whereas for the pre-critical case
$\tau>\tau_{c}$, it consists of two disconnected components. At criticality
when $\tau=\tau_{c}$, the droplet contains two symmetric double points. We
refer to [12, 14, 3, 36] for further models whose droplets reveal various
phase transitions. Let us also mention that recently, there have been several
works on the models (1.3) with multi-component droplets, see e.g. [13, 30, 9,
10]. In this pre-critical regime, some theta-function oscillations are
expected to appear for various kinds of statistics; cf. [32, 9]. The precise
asymptotic behaviours of the partition function would also be interesting in
connection with the conjecture that these depend on the Euler index of the
droplets, see [42, 29] and [26, Sections 4.1 and 5.3] for further discussion.

######  Remark 1.3 (Fekete points and numerics).

A configuration $\\{\zeta_{j}\\}_{j=1}^{N}$ which makes the Hamiltonians (1.1)
or (1.2) minimal is known as a Fekete configuration. This can be interpreted
as the ensembles (1.3) with low temperature limit $\beta=\infty$, see e.g.
[61, 58, 7, 11] and references therein. Since the droplet is independent of
the inverse temperature $\beta>0$ (excluding the high-temperature regime [2]
when $\beta=O(1/N)$), the Fekete points are useful to numerically observe the
shape of the droplets. In Figures 1 and 2, Fekete configurations associated
with the Hamiltonian (1.1) are also presented, which show good fits with
Theorems 1.1 and 1.4.

Notice that the potential (1.7) and the droplet $S_{\tau,c}$ are invariant
under the map $\zeta\mapsto-\zeta$. We now discuss an equivalent formulation
of Theorem 1.1 under the removal of such symmetry. (See [36, Section 1.3] for
a similar discussion in a vector equilibrium problem on a sphere with point
charges.) The motivation for this formulation will be clear in the next
subsection.

For this purpose, we denote

(1.14)
$\widehat{Q}(\zeta):=\frac{2}{1-\tau^{2}}\Big{(}|\zeta|-\tau\operatorname{Re}\zeta\Big{)}-2c\log|\zeta|.$

By definition, the potentials $Q$ in (1.7) and $\widehat{Q}$ in (1.14) are
related as

(1.15) $Q(\zeta)=\frac{1}{2}\widehat{Q}(\zeta^{2}).$

Denoting by $\widehat{S}$ the droplet associated with $\widehat{Q}$, it
follows from [14, Lemma 1] that

(1.16) $S=\\{\zeta\in\mathbb{C}:\zeta^{2}\in\widehat{S}\\}.$

Due to the relation (1.16) and

(1.17) $\Delta\widehat{Q}(\zeta)=\frac{1}{2(1-\tau^{2})}\frac{1}{|\zeta|},$

we have the following equivalent formulation of Theorem 1.1.

######  Theorem 1.4.

Let $\widehat{Q}$ be given by (1.14). Then the droplet
$\widehat{S}\equiv\widehat{S}_{\tau,c}=S_{\widehat{Q}}$ of the equilibrium
measure

(1.18)
$d\mu_{\widehat{Q}}(\zeta)=\frac{1}{2(1-\tau^{2})}\frac{1}{|\zeta|}\mathbbm{1}_{\widehat{S}}(\zeta)\,dA(\zeta)$

is given as follows.

  1. (i)

(Post-critical case) If $\tau\in(0,\tau_{c}]$, we have

(1.19)
$\widehat{S}_{\tau,c}=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x-2\tau(1+c)}{(1+\tau^{2})(1+c)}\Big{)}^{2}+\Big{(}\frac{y}{(1-\tau^{2})(1+c)}\Big{)}^{2}\leq
1\,,\,\frac{x^{2}+y^{2}}{(1-\tau^{2})^{2}c^{2}}\geq 1\Big{\\}}.$

  2. (ii)

(Pre-critical case) If $\tau\in[\tau_{c},1)$, the droplet
$\widehat{S}_{\tau,c}$ is the closure of the interior of the real analytic
Jordan curve given by the image of the unit circle with respect to the
rational map $z\mapsto f(z)$, where $f$ is given by (1.11).

(a) $\tau=1/6<\tau_{c}$

(b) $\tau=1/3=\tau_{c}$

(c) $\tau=1/2>\tau_{c}$

Figure 2.  The droplet $\widehat{S}_{\tau,c}$, where $c=1$ and a Fekete point
configuration with $N=2048$.

######  Remark 1.5 (Joukowsky transform in the critical case).

If $\tau=\tau_{c}$ with (1.10), we have $a=1/a=-1$. Thus in this critical
case, the rational function $f_{\tau}$ in (1.11) is simplified as

(1.20)
$f_{\tau_{c}}(z)=(1+c)\frac{(z+\tau)^{2}}{z}=(1+c)\Big{(}z+2\tau+\frac{\tau^{2}}{z}\Big{)}.$

Note that compared to the general case (1.11), there is one less zero and one
less pole in (1.20). Indeed, in the critical case, the rational map
$f_{\tau_{c}}$ is a (shifted) Joukowsky transform

(1.21)
$f_{\tau_{c}}:\mathbb{D}^{c}\to\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x-2\tau(1+c)}{(1+\tau^{2})(1+c)}\Big{)}^{2}+\Big{(}\frac{y}{(1-\tau^{2})(1+c)}\Big{)}^{2}\geq
1\Big{\\}}.$

In [3], a similar type of Joukowsky transform was used to solve an equilibrium
measure problem. For the models under consideration in the present work, due
to a more complicated form of the rational function (1.11), the required
analysis for the associated equilibrium problem turns out to be more involved.

######  Remark 1.6 (A non-Hermitian extension of the Marchenko-Pastur
distribution).

In the Hermitian limit $\tau\uparrow 1,$ by (1.7) and (1.14), we have

(1.22) $\displaystyle\lim_{\tau\uparrow
1}Q(x+iy)=V(x):=\begin{cases}\displaystyle\frac{x^{2}}{2}-2c\log|x|,&\text{if
}y=0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\infty&\text{otherwise},\end{cases}$ (1.23) $\displaystyle\lim_{\tau\uparrow
1}\widehat{Q}(x+iy)=\widehat{V}(x):=\begin{cases}\displaystyle
x-2c\log|x|,&\text{if }y=0,\,x>0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\infty&\text{otherwise}.\end{cases}$

Then the associated equilibrium measures are given by the well-known
Marchenko-Pastur law (with squared variables) [38, Proposition 3.4.1], i.e.

(1.24) $\displaystyle d\mu_{V}(x)$
$\displaystyle=\frac{1}{2\pi|x|}\,\sqrt{(\lambda_{+}^{2}-x^{2})(x^{2}-\lambda_{-}^{2})}\cdot\mathbbm{1}_{[-\lambda_{+},-\lambda_{-}]\cup[\lambda_{-},\lambda_{+}]}\,dx,$
(1.25) $\displaystyle d\mu_{\widehat{V}}(x)$ $\displaystyle=\frac{1}{2\pi
x}\,\sqrt{(\lambda_{+}^{2}-x)(x-\lambda_{-}^{2})}\cdot\mathbbm{1}_{[\lambda_{-}^{2},\lambda_{+}^{2}]}\,dx,$

where $\lambda_{\pm}:=\sqrt{2c+1}\pm 1$, cf. Remark 2.3. Therefore one can
interpret Theorem 1.1 (resp., Theorem 1.4) as a non-Hermitian generalisation
of the Marchenko-Pastur distribution (1.24) (resp., (1.25)), see [8, Section
2] for more about the geometric meaning with the notion of the statistical
cross-section. We also refer to [3] for another non-Hermitian extension of
(1.24) and (1.25) in the context of the chiral Ginibre ensembles.

######  Remark 1.7 (Inclusion relations of the droplets).

Let us write

(1.26) $\displaystyle
S_{1}=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x}{1+\tau}\Big{)}^{2}+\Big{(}\frac{y}{1-\tau}\Big{)}^{2}\leq
1+c\Big{\\}},\qquad
S_{2}:=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:x^{2}+y^{2}\leq(1-\tau^{2})c\Big{\\}}$

and denote by $\widehat{S}_{j}$ $(j=1,2)$ the image of $S_{j}$ under the map
$z\mapsto z^{2}$. Then it follows from the definition (1.10) that

(1.27) $\tau\in(0,\tau_{c})\qquad\textup{if and only if}\qquad S_{1}^{c}\cap
S_{2}=\emptyset.$

By Theorems 1.1 and 1.4, for general $\tau\in[0,1)$ and $c\geq 0$, one can
observe that

(1.28) $S_{\tau,c}\subseteq
S_{1}\cap(\textup{Int}\,S_{2})^{c},\qquad\widehat{S}_{\tau,c}\subseteq\widehat{S}_{1}\cap(\textup{Int}\,\widehat{S}_{2})^{c}.$

Here, equality in (1.28) holds if and only if in the post-critical case. (This
property holds in a more general setup, see Proposition 2.1.) On the other
hand, in the pre-critical case one can interpret that the particles in
$S_{1}^{c}\cap S_{2}$ are smeared out to $S_{1}\cap S_{2}^{c}$ which makes the
inclusion relations (1.28) strictly hold, see Figure 3.

(a) $S_{\tau,c}$

(b) $\widehat{S}_{\tau,c}$

Figure 3.  The droplets $S_{\tau,c}$ and $\widehat{S}_{\tau,c}$ in the pre-
critical case, where $c=2$ and $\tau=0.7>\tau_{c}$. Here, the dashed lines
display the boundaries of $S_{j}$ and $\widehat{S}_{j}$ ($j=1,2$).

###  1.2. Outline of the proof

Recall that $\mu_{W}$ is a unique minimiser of the energy (1.4). It is well
known that the equilibrium measure $\mu_{W}$ is characterised by the
variational conditions (see [59, p.27])

(1.29)
$\displaystyle\int\log\frac{1}{|\zeta-z|^{2}}\,d\mu_{W}(z)+W(\zeta)=C,\quad\text{q.e.}\quad\text{if
}\zeta\in S_{W};\vskip 3.0pt plus 1.0pt minus 1.0pt$ (1.30)
$\displaystyle\int\log\frac{1}{|\zeta-z|^{2}}\,d\mu_{W}(z)+W(\zeta)\geq
C,\quad\text{q.e.}\quad\text{if }\zeta\notin S_{W}.$

Here, q.e. stands for quasi-everywhere. (Nevertheless, this notion is not
important in the sequel as we will show that for the models we consider the
conditions (1.29) and (1.30) indeed hold everywhere.)

Due to the uniqueness of the equilibrium measure, all we need to show is that
if $W=Q$, then $\mu_{Q}$ in (1.12) satisfies the variational principles (1.29)
and (1.30). Equivalently, by (1.16), it also suffices to show the variational
principles for the equilibrium measure $\mu_{\widehat{Q}}$ in (1.18).

However, it is far from being obvious to obtain the “correct candidate” of the
droplets. Perhaps one may think that at least for the post-critical case, the
shape of the droplet (1.13) is quite natural given the well-known cases (1.8)
and (1.9) as well as the fact that the area of $S_{\tau,c}$ should be
$(1-\tau^{2})\pi$. On the other hand, for the pre-critical case, one can
easily notice that there is some secret behind deriving the explicit formula
of the rational function (1.11). To derive the correct candidate, we use the
conformal mapping method with the help of the Schwarz function, see Appendix
A.

######  Remark 1.8 (Removal of symmetry).

We emphasise that the conformal mapping method does not work for the multi-
component droplet, i.e. the pre-critical case of Theorem 1.1. This is
essentially due to the lack of the Riemann mapping theorem. Nevertheless, one
can observe that once we remove the symmetry $\zeta\mapsto-\zeta$, the droplet
in the pre-critical case of Theorem 1.4 is simply connected. This explains the
reason why we need the idea of removing symmetry.

The rest of this paper is organised as follows.

  * •

In Section 2, we prove Theorems 1.1 and 1.4. In Subsection 2.1, we show the
post-critical case of Theorem 1.1 in a more general setup, see Proposition
2.1. On the other hand, in Subsection 2.2, we show the pre-critical case of
Theorem 1.4. Then by the relation (1.16), these complete the proof of our main
results.

  * •

This article contains two appendices. In Appendix A, we explain the conformal
mapping method to derive the “correct candidate” of the droplets. In Appendix
B, we present a way to solve a one-dimensional equilibrium problem in Remark
2.3, which shares a common feature with the conformal mapping method. These
appendices are made only for instructive purposes and the readers who only
want the proof of the main theorems may stop at the end of Section 2.

##  2\. Proof of main theorem

In this section, we show Theorems 1.1 and 1.4.

###  2.1. Post critical cases

Extending (1.7), we consider the potential

(2.1)
$Q_{p}(\zeta):=\frac{1}{1-\tau^{2}}\Big{(}|\zeta|^{2}-\tau\operatorname{Re}\zeta^{2}\Big{)}-2c\log|\zeta-p|,\qquad
p\in\mathbb{C}.$

For the case $\tau=0$, the shape of the droplet associated with the potential
(2.1) was fully characterised in [12]. (In this case, it suffices to consider
the case $p\geq 0$ due to the rotational invariance.) In particular, it was
shown that if

(2.2) $c<\frac{(1-p^{2})^{2}}{4p^{2}},\qquad\tau=0,\qquad p\geq 0,$

the droplet is given by
$S=\overline{\mathbb{D}(0,\sqrt{1+c})}\setminus\mathbb{D}(p,\sqrt{c}),$ where
$\mathbb{D}(p,R)$ is the disc with centre $p$ and radius $R$, cf. see Remark
A.5 for the other case $c>(1-p^{2})^{2}/(4p^{2})$.

To describe the droplets associated with $Q_{p}$, we denote

(2.3)
$S_{1}:=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x}{1+\tau}\Big{)}^{2}+\Big{(}\frac{y}{1-\tau}\Big{)}^{2}\leq
1+c\Big{\\}}$

and

(2.4)
$S_{2}:=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:(x-\operatorname{Re}p)^{2}+(y-\operatorname{Im}p)^{2}\leq(1-\tau^{2})c\Big{\\}}.$

Then we obtain the following.

######  Proposition 2.1.

Suppose that the parameters $\tau,c\in{\mathbb{R}}$ and $p\in\mathbb{C}$ are
given to satisfy

(2.5) $S_{2}\subset S_{1},$

where $S_{1}$ and $S_{2}$ are given by (2.3) and (2.4). Then the droplet
$S\equiv S_{Q_{p}}$ associated with (2.1) is given by

(2.6) $S=S_{1}\cap(\textup{Int}\,S_{2})^{c}.$

See Figure 4 for the shape of the droplets and numerical simulations of Fekete
point configurations. We remark that with slight modifications, Proposition
2.1 can be further extended to the case with multiple point charges, i.e. the
potential of the form

(2.7)
$\frac{1}{1-\tau^{2}}\Big{(}|\zeta|^{2}-\tau\operatorname{Re}\zeta^{2}\Big{)}-2\sum
c_{j}\log|\zeta-p_{j}|,\qquad p_{j}\in\mathbb{C},\quad c_{j}\geq 0.$

(See Remark A.4 for a related discussion.) Let us also mention that a similar
statement for an equilibrium problem on the sphere was shown in [21]. For a
treatment of a more general case, we refer the reader to [35, 54, 36].

(a) $p=\frac{2}{21}\sqrt{14}\,i$

(b) $p=\frac{2}{7}\sqrt{14}$

(c) $p=\frac{3}{5}+\frac{1}{5}i$

Figure 4.  The droplet $S$ in Proposition 2.1, where $\tau=1/3$ and $c=1/7$.
Here, a Fekete point configuration with $N=2048$ is also displayed.

######  Remark 2.2.

If $p=0$, the condition (2.5) corresponds to

(2.8) $\tau<\frac{1}{1+2c}=\tau_{c}.$

Therefore Proposition 2.1 for the special value $p=0$ gives Theorem 1.1 (i).
As a consequence, by (1.16), Theorem 1.4 (i) also follows. We also mention
that if $\tau=0$ and $p>0$, the condition (2.5) coincides with (2.2).

######  Remark 2.3 (Equilibrium measure in the Hermitian limit).

Before moving on to the planar equilibrium problem for (2.1), we first discuss
the one-dimensional problem arising in the Hermitian limit. For
$p\in{\mathbb{R}}$, the Hermitian limit $\tau\uparrow 1$ of the potential
$Q_{p}$ is given by

(2.9) $\lim_{\tau\uparrow
1}Q_{p}(x+iy)=V_{p}(x):=\begin{cases}\displaystyle\frac{x^{2}}{2}-2c\log|x-p|,&\text{if
}y=0,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
+\infty&\text{otherwise}.\end{cases}$

Then one can show that the associated equilibrium measure
$\mu_{V}\equiv\mu_{V_{p}}$ is given by

(2.10)
$\frac{d\mu_{V}(x)}{dx}=\frac{\sqrt{-\prod_{j=1}^{4}(x-\lambda_{j})}}{2\pi|x-p|}\cdot\mathbbm{1}_{[\lambda_{1},\lambda_{2}]\cup[\lambda_{3},\lambda_{4}]}(x),$

where

(2.11)
$\displaystyle\lambda_{1}=\frac{p-2-\sqrt{(p+2)^{2}+8c}}{2},\qquad\lambda_{2}=\frac{p+2-\sqrt{(p-2)^{2}+8c}}{2},$
(2.12)
$\displaystyle\lambda_{3}=\frac{p-2+\sqrt{(p+2)^{2}+8c}}{2},\qquad\lambda_{4}=\frac{p+2+\sqrt{(p-2)^{2}+8c}}{2}.$

We remark that when $p=0$, it recovers (1.24). See Figure 5 for the graphs of
the equilibrium measure $\mu_{V_{p}}$. The equilibrium measure (2.10) follows
from the standard method using the Stieltjes transform and the Sokhotski-
Plemelj inversion formula. For reader’s convenience, we provide a proof of
(2.10) in Appendix B.

(a) $p=0$

(b) $p=1$

(c) $p=4$

Figure 5.  Graphs of the equilibrium measure $\mu_{V_{p}}$, where $c=1$.

In the rest of this subsection, we prove Proposition 2.1. First, let us show
the following elementary lemmas.

######  Lemma 2.4.

For $a,b>0$, let

$K:=\Big{\\{}(x,y)\in{\mathbb{R}}^{2}:\Big{(}\frac{x}{a}\Big{)}^{2}+\Big{(}\frac{y}{b}\Big{)}^{2}\leq
1\Big{\\}}.$

Then we have

(2.13)
$\int_{K}\frac{1}{\zeta-z}\,dA(z)=\begin{cases}\displaystyle\bar{\zeta}-\frac{a-b}{a+b}\,\zeta&\text{if
}\zeta\in K,\vskip 3.0pt plus 1.0pt minus 1.0pt\\\
\displaystyle\frac{2ab}{a^{2}-b^{2}}\Big{(}\zeta-\sqrt{\zeta^{2}-a^{2}+b^{2}}\Big{)}&\text{otherwise.}\end{cases}$

In particular, for $\zeta\in K$, there exists a constant
$c_{0}\in{\mathbb{R}}$ such that

(2.14)
$\int_{K}\log|\zeta-z|^{2}\,dA(z)=|\zeta|^{2}-\frac{a-b}{a+b}\operatorname{Re}\zeta^{2}+c_{0}.$

######  Remark 2.5.

The Cauchy transform in (2.13) is useful to explicitly compute the moments of
the equilibrium measure. Namely, by definition, we have

$\int_{K}\frac{1}{\zeta-z}\,dA(z)=\frac{1}{\zeta}\sum_{k=0}^{\infty}\frac{1}{\zeta^{k}}\int_{K}z^{k}\,dA(z),\qquad\zeta\to\infty.$

On the other hand, we have

$\displaystyle\zeta-\sqrt{\zeta^{2}-a^{2}+b^{2}}=\frac{a^{2}-b^{2}}{\zeta}\sum_{k=0}^{\infty}\binom{1/2}{k+1}\frac{(b^{2}-a^{2})^{k}}{\zeta^{2k}},\qquad\zeta\to\infty.$

Combining the above equations with (2.13), we obtain that for any non-negative
integer $k,$

(2.15) $\frac{1}{ab}\int_{K}z^{2k}\,dA(z)=2\binom{1/2}{k+1}(b^{2}-a^{2})^{k}.$

###### Proof of Lemma 2.4.

Recall that $\mathbb{D}$ is the unit disc with centre the origin. Then the
Joukowsky transform $f:\bar{\mathbb{D}}^{c}\to K^{c}$ is given by

(2.16) $f(z)=\frac{a+b}{2}\,z+\frac{a-b}{2}\,\frac{1}{z}.$

By applying Green’s formula, we have

(2.17) $\int_{K}\frac{1}{\zeta-z}\,dA(z)=\frac{1}{2\pi i}\int_{\partial
K}\frac{\bar{z}}{\zeta-z}\,dz+\bar{\zeta}\cdot\mathbbm{1}_{\\{\zeta\in K\\}}.$

Furthermore, by the change of variable $z=f(w)$, it follows that

(2.18) $\displaystyle\int_{\partial K}\frac{\bar{z}}{\zeta-z}\,dz$
$\displaystyle=\int_{\partial\mathbb{D}}\frac{\overline{f(1/\bar{w})}}{\zeta-f(w)}f^{\prime}(w)\,dw=\int_{\partial\mathbb{D}}g_{\zeta}(w)\,dw,$

where $g_{\zeta}$ is the rational function given by

(2.19)
$g_{\zeta}(w):=\frac{1}{\zeta-f(w)}\Big{(}\frac{a+b}{2}\frac{1}{w}+\frac{a-b}{2}w\Big{)}\Big{(}\frac{a+b}{2}-\frac{a-b}{2}\frac{1}{w^{2}}\Big{)}.$

Observe that

$\zeta=f(w)\qquad\textrm{if and only if}\qquad
w=w_{\zeta}^{\pm}:=\frac{\zeta\pm\sqrt{\zeta^{2}-a^{2}+b^{2}}}{a+b},$

i.e. the points $w_{\zeta}^{\pm}$ are solutions to the quadratic equation

$(a+b)w^{2}-2\zeta\,w+(a-b)=0.$

Here, the branch of the square root is chosen such that

$w_{\zeta}^{-}\to 0\qquad\zeta\to\infty.$

By above observation, the function $g_{\zeta}$ has poles only at

$0,\qquad w_{\zeta}^{+},\qquad w_{\zeta}^{-}.$

Moreover note that

$\zeta\in K\qquad\textrm{if and only if}\qquad w_{\zeta}^{\pm}\in\mathbb{D}.$

Notice that if $\zeta\in K^{c}$, then $w_{\zeta}^{-}\in\mathbb{D}$ and
$w_{\zeta}^{+}\in\mathbb{D}^{c}$.

Using the residue calculus, we have

(2.20)
$\underset{w=0}{\textrm{Res}}\,\Big{[}g_{\zeta}(w)\Big{]}=\frac{a+b}{a-b}\,\zeta.$

On the other hand, we have

(2.21)
$\underset{w=w_{\zeta}^{\pm}}{\textrm{Res}}\,\Big{[}g_{\zeta}(w)\Big{]}=-\overline{f(1/\bar{w}_{\zeta}^{\pm})}=-\frac{a+b}{2}\frac{1}{w_{\zeta}^{\pm}}-\frac{a-b}{2}w_{\zeta}^{\pm}.$

In particular,

(2.22)
$\underset{w=w_{\zeta}^{+}}{\textrm{Res}}\,\Big{[}g_{\zeta}(w)\Big{]}+\underset{w=w_{\zeta}^{-}}{\textrm{Res}}\,\Big{[}g_{\zeta}(w)\Big{]}=-2\frac{a^{2}+b^{2}}{a^{2}-b^{2}}\,\zeta.$

Combining all of the above, we obtain the desired identity (2.13). The second
assertion immediately follows from (2.13) and the real-valuedness of
$\zeta\mapsto\int_{K}\log|\zeta-z|^{2}\,dA(z)$. ∎

######  Lemma 2.6.

For $R>0$ and $p\in\mathbb{C}$ we have

(2.23) $\int_{\mathbb{D}(p,R)}\log|\zeta-z|\,dA(z)=\begin{cases}\displaystyle
R^{2}\log|\zeta-p|&\text{if }\zeta\notin\mathbb{D}(p,R),\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ \displaystyle R^{2}\log
R-\frac{R^{2}}{2}+\frac{|\zeta-p|^{2}}{2}&\text{otherwise}.\end{cases}$

###### Proof.

First, recall the well-known Jensen’s formula: for $r>0$,

(2.24) $\frac{1}{2\pi}\int_{0}^{2\pi}\log|\zeta-
re^{i\theta}|\,d\theta=\begin{cases}\log r&\text{if }r>|\zeta|,\vskip 3.0pt
plus 1.0pt minus 1.0pt\\\ \log|\zeta|&\text{otherwise}.\end{cases}$

By the change of variables, we have

$\displaystyle\int_{\mathbb{D}(p,R)}\log|\zeta-z|\,dA(z)$
$\displaystyle=\int_{\mathbb{D}(0,R)}\log|\zeta-
p-z|\,dA(z)=\frac{1}{\pi}\int_{0}^{R}r\int_{0}^{2\pi}\log|\zeta-p-
re^{i\theta}|\,d\theta\,dr.$

Suppose that $\zeta\notin\mathbb{D}(p,R)$. Then by applying (2.24), we have

$\displaystyle\frac{1}{\pi}\int_{0}^{R}r\int_{0}^{2\pi}\log|\zeta-p-
re^{i\theta}|\,d\theta\,dr$
$\displaystyle=2\int_{0}^{R}r\,\log|\zeta-p|\,dr=R^{2}\log|\zeta-p|.$

On the other hand if $\zeta\in\mathbb{D}(p,R)$, we have

$\displaystyle\frac{1}{\pi}\int_{0}^{R}r\int_{0}^{2\pi}\log|\zeta-p-
re^{i\theta}|\,d\theta\,dr$
$\displaystyle=2\int_{0}^{|\zeta-p|}r\,\log|\zeta-p|\,dr+2\int_{|\zeta-p|}^{R}r\,\log
r\,dr$ $\displaystyle=R^{2}\log R-\frac{R^{2}}{2}+\frac{|\zeta-p|^{2}}{2},$

which completes the proof. ∎

We are now ready to complete the proof of Proposition 2.1.

###### Proof of Proposition 2.1.

Note that by (1.5), the equilibrium measure $\mu$ associated with $Q_{p}$ is
of the form

(2.25) $d\mu(z):=\Delta
Q_{p}(z)\cdot\mathbbm{1}_{S}(z)\,dA(z)=\frac{1}{1-\tau^{2}}\cdot\mathbbm{1}_{S}(z)\,dA(z).$

Due to the assumption (2.5), we have

$\displaystyle\int\log\frac{1}{|\zeta-z|^{2}}\,d\mu(z)=\frac{1}{1-\tau^{2}}\Big{(}\int_{S_{1}}\log\frac{1}{|\zeta-z|^{2}}\,dA(z)-\int_{S_{2}}\log\frac{1}{|\zeta-z|^{2}}\,dA(z)\Big{)}.$

Note that by Lemma 2.4, there exists a constant $c_{0}$ such that

(2.26)
$\int_{S_{1}}\log\frac{1}{|\zeta-z|^{2}}\,dA(z)=-|\zeta|^{2}+\tau\operatorname{Re}\zeta^{2}-c_{0}.$

On the other hand, by Lemma 2.6, we have

(2.27)
$\int_{S_{2}}\log\frac{1}{|\zeta-z|^{2}}\,dA(z)=-2(1-\tau^{2})c\,\log|\zeta-p|.$

Combining (2.26), (2.27) and (2.1), we obtain

(2.28)
$\int\log\frac{1}{|\zeta-z|^{2}}\,d\mu(z)=-Q_{p}(\zeta)-\frac{c_{0}}{1-\tau^{2}},$

which leads to (1.29).

Next, we show the variational inequality (1.30). Note that if $\zeta\in
S_{2}$, it immediately follows from Lemma 2.6. Thus it is enough to verify
(1.30) for the case $\zeta\in S_{1}^{c}$. Let

(2.29) $H_{p}(\zeta):=\int\log\frac{1}{|\zeta-z|^{2}}\,d\mu(z)+Q_{p}(\zeta).$

Suppose that the variational inequality (1.30) does not hold. Then since
$H_{p}(\zeta)\to\infty$ as $\zeta\to\infty$, there exists $\zeta_{*}\in
S_{1}^{c}$ such that

(2.30) $\partial_{\zeta}H_{p}(\zeta)|_{\zeta=\zeta_{*}}=0.$

On the other hand, by Lemmas 2.4 and 2.6, if $\zeta\in S_{1}^{c}$, the Cauchy
transform of the measure $\mu$ is computed as

(2.31)
$\int\frac{d\mu(z)}{\zeta-z}=\frac{1}{2\tau}\Big{(}\zeta-\sqrt{\zeta^{2}-4\tau(1+c)}\Big{)}-\frac{c}{\zeta-p}.$

Together with (2.1), this gives rise to

(2.32)
$\displaystyle\partial_{\zeta}Q_{p}(\zeta)-\int\frac{d\mu(z)}{\zeta-z}=\frac{1}{1-\tau^{2}}\Big{(}\bar{\zeta}-\tau\zeta\Big{)}-\frac{1}{2\tau}\Big{(}\zeta-\sqrt{\zeta^{2}-4\tau(1+c)}\Big{)}.$

Then it follows that the condition $\partial_{\zeta}H_{p}(\zeta)=0$ is
equivalent to

(2.33)
$(1+\tau^{2})|\zeta|^{2}-\tau(\zeta^{2}+\bar{\zeta}^{2})=(1-\tau^{2})^{2}(1+c).$

Therefore, by (2.3), one can notice that $\partial_{\zeta}H_{p}(\zeta)=0$ if
and only if $\zeta\in\partial S_{1}$. This yields a contradiction with the
assumption $\zeta_{*}\in S_{1}^{c}.$ Therefore we conclude that the
variational inequality (1.30) holds for $\zeta\in S^{c}$, which completes the
proof. ∎

######  Remark 2.7.

Let us denote by

(2.34) $m_{k}:=\int z^{k}\,d\mu(z)$

the $k$-th moment of the equilibrium measure. Notice that the Cauchy transform
of $\mu$ satisfies the asymptotic expansion

(2.35)
$\int\frac{d\mu(z)}{\zeta-z}=\frac{1}{\zeta}\sum_{k=0}^{\infty}\frac{m_{k}}{\zeta^{k}},\qquad\zeta\to\infty.$

Using this property and (2.31), after straightforward computations, one can
verify that the equilibrium measure $\mu$ in Proposition 2.1 has the moments

(2.36)
$m_{2k}=2\frac{(2k-1)!}{(k-1)!(k+1)!}\tau^{k}(1+c)^{k+1}-c\,p^{2k},\qquad
m_{2k+1}=-c\,p^{2k+1}.$

Notice in particular that if $p=0$, all odd moments vanish.

###  2.2. Pre-critical case

In this subsection, we show Theorem 1.4 (ii). Then by (1.16), Theorem 1.1 (ii)
follows.

###### Proof of Theorem 1.4 (ii).

Recall that $\widehat{Q}$ is given by (1.14) and that all we need to show is
the variational principles (1.29) and (1.30) for $W=\widehat{Q}.$ For this,
similar to above, let

(2.37)
$H(\zeta):=\int\log\frac{1}{|\zeta-z|^{2}}\,d\widehat{\mu}(z)+\widehat{Q}(\zeta),$

where $\widehat{\mu}$ is the equilibrium measure associated with
$\widehat{Q}$. Then

(2.38)
$\partial_{\zeta}H(\zeta)=\partial_{\zeta}\widehat{Q}(\zeta)-C(\zeta)=\frac{1}{1-\tau^{2}}\Big{(}\sqrt{\frac{\bar{\zeta}}{\zeta}}-\tau\Big{)}-\frac{c}{\zeta}-C(\zeta),$

where $C(\zeta)$ is the Cauchy transform of $\widehat{\mu}$ given by

(2.39)
$C(\zeta)=\frac{1}{2(1-\tau)^{2}}\int_{\widehat{S}}\frac{1}{\zeta-z}\frac{1}{|z|}\,dA(z).$

Here, we have used (1.5).

Applying Green’s formula, we have

(2.40) $\displaystyle\begin{split}(1-\tau^{2})C(\zeta)&=\frac{1}{2\pi
i}\int_{\partial\widehat{S}}\frac{1}{\zeta-z}\sqrt{\frac{\bar{z}}{z}}\,dz+\sqrt{\frac{\bar{\zeta}}{\zeta}}\cdot\mathbbm{1}_{\\{\zeta\in\operatorname{Int}(\widehat{S})\\}}.\end{split}$

Recall that $f$ is given by (1.11). Let

(2.41)
$\displaystyle\begin{split}g(w)&:=\sqrt{\frac{\overline{f(1/\bar{w})}}{f(w)}}\,f^{\prime}(w).\end{split}$

Since $f^{\prime}(a\tau)=0,$ the function $g(w)$ has poles only at $0,1/a,a$.
We also write

(2.42) $h_{\zeta}(w):=\frac{g(w)}{\zeta-f(w)}.$

Using the change of variable $z=f(w)$,

(2.43) $\displaystyle\begin{split}\frac{1}{2\pi
i}\int_{\partial\widehat{S}}\frac{1}{\zeta-z}\sqrt{\frac{\bar{z}}{z}}\,dz&=\frac{1}{2\pi
i}\int_{\partial\mathbb{D}}\frac{1}{\zeta-f(w)}\sqrt{\frac{\overline{f(1/\bar{w})}}{f(w)}}\,f^{\prime}(w)\,dw=\frac{1}{2\pi
i}\int_{\partial\mathbb{D}}h_{\zeta}(w)\,dw.\end{split}$

By the residue calculus, we have

(2.44)
$\underset{w=0}{\textrm{Res}}\,\Big{[}h_{\zeta}(w)\Big{]}=\frac{1}{\tau},\qquad\underset{w=a}{\textrm{Res}}\,\Big{[}h_{\zeta}(w)\Big{]}=0.$

Note that $\zeta=f(w)$ is equivalent to

(2.45) $d(1-aw)(w-a\tau)^{2}=w(w-a)\zeta,\qquad d=\frac{(1+\tau)(1+2c)}{2},$

which can be rewritten as a cubic equation

(2.46)
$adw^{3}-(d+2a^{2}d\tau-\zeta)w^{2}+a(2d\tau+a^{2}\tau^{2}d-\zeta)w-a^{2}\tau^{2}d=0.$

For given $\zeta\in\mathbb{C}$, there exist $w_{\zeta}^{(j)}$ $(j=1,2,3)$ such
that $f(w_{\zeta}^{(j)})=\zeta.$ Note that by (2.46), we have

(2.47) $w_{\zeta}^{(1)}w_{\zeta}^{(2)}w_{\zeta}^{(3)}=a\tau^{2}\in(-1,0).$

Furthermore, since $f$ is a conformal map from $\mathbb{D}^{c}$ onto
$\widehat{S}^{c}$, we have the following:

  1. (1)

If $\zeta\in\operatorname{Int}(\widehat{S})$, then all $w_{\zeta}^{(j)}$’s are
in $\mathbb{D}$;

  2. (2)

If $\zeta\in\widehat{S}^{c}$, then two of $w_{\zeta}^{(j)}$’s are in
$\mathbb{D}$.

By the residue calculus using (1.11) and (2.41), for each $j,$

(2.48)
$\displaystyle\begin{split}\underset{w=w_{\zeta}^{(j)}}{\textrm{Res}}\,\Big{[}h_{\zeta}(w)\Big{]}&=-\frac{g(w_{\zeta}^{(j)})}{f^{\prime}(w_{\zeta}^{(j)})}=-\frac{(w^{(j)}_{\zeta}-a)(1-a\tau
w^{(j)}_{\zeta})}{(w^{(j)}_{\zeta}-a\tau)(1-aw^{(j)}_{\zeta})}\\\
&=\frac{d}{\zeta}\frac{(a\tau
w_{\zeta}^{(j)}-1)(w_{\zeta}^{(j)}-a\tau)}{w_{\zeta}^{(j)}}=\frac{d}{\zeta}\Big{(}a\tau
w_{\zeta}^{(j)}-(a^{2}\tau^{2}+1)+\frac{a\tau}{w_{\zeta}^{(j)}}\Big{)},\end{split}$

where we have used (2.45). On the other hand, it follows from (2.46) that

(2.49)
$\sum_{j=1}^{3}w_{\zeta}^{(j)}=\frac{d+2a^{2}d\tau-\zeta}{ad},\qquad\sum_{j=1}^{3}\frac{1}{w_{\zeta}^{(j)}}=\frac{-\zeta+2d\tau+a^{2}\tau^{2}d}{a\tau^{2}d}.$

These relations give rise to

(2.50) $\displaystyle\begin{split}d\sum_{j=1}^{3}\Big{(}a\tau
w_{\zeta}^{(j)}-(a^{2}\tau^{2}+1)+\frac{a\tau}{w_{\zeta}^{(j)}}\Big{)}&=d\tau+2a^{2}d\tau^{2}-\tau\zeta-\frac{\zeta}{\tau}+2d+a^{2}\tau
d-3d(a^{2}\tau^{2}+1)\\\
&=-\Big{(}\tau+\frac{1}{\tau}\Big{)}\zeta-c(1-\tau^{2}).\end{split}$

Combining all of the above, we have shown that if
$\zeta\in\operatorname{Int}(\widehat{S})$,

(2.51)
$\sum_{j=1}^{3}\underset{w=w_{\zeta}^{(j)}}{\textrm{Res}}\,\Big{[}h_{\zeta}(w)\Big{]}=-\Big{(}\tau+\frac{1}{\tau}\Big{)}-\frac{c(1-\tau^{2})}{\zeta}.$

Therefore if $\zeta\in\operatorname{Int}(\widehat{S})$, we obtain

(2.52)
$\displaystyle\begin{split}(1-\tau^{2})C(\zeta)&=\sqrt{\frac{\bar{\zeta}}{\zeta}}-\frac{1}{\tau}-c(1-\tau^{2})=(1-\tau^{2})\partial_{\zeta}\widehat{Q}(\zeta).\end{split}$

Then by (2.40), the variational equality (1.29) follows.

Now it remains to show the variational inequality (1.30). Note that by
definition, $H(\zeta)\to\infty$ as $\zeta\to\infty$. Suppose that the
variational inequality (1.30) does not hold. Then there exists
$\zeta_{*}\in\widehat{S}^{c}$ such that

(2.53)
$\partial_{\zeta}H(\zeta)|_{\zeta=\zeta_{*}}=\partial\widehat{Q}(\zeta_{*})-C(\zeta_{*})=0.$

Recall that if $\zeta\in\widehat{S}^{c}$, then only one of
$w_{\zeta}^{(j)}$’s, say $w_{\zeta}$, is in $\mathbb{D}^{c}$. By combining the
above computations, we have that for $\zeta\in\widehat{S}^{c}$,

(2.54)
$\displaystyle\begin{split}(1-\tau^{2})\Big{(}\partial_{\zeta}\widehat{Q}(\zeta)-C(\zeta)\Big{)}=\sqrt{\frac{\bar{\zeta}}{\zeta}}-\underset{w=w_{\zeta}}{\textrm{Res}}\,\Big{[}h_{\zeta}(w)\Big{]}=\sqrt{\frac{\bar{\zeta}}{\zeta}}-\frac{d}{\zeta}\Big{(}a\tau
w_{\zeta}-(a^{2}\tau^{2}+1)+\frac{a\tau}{w_{\zeta}}\Big{)}.\end{split}$

Therefore the identity (2.53) holds if and only if

(2.55) $|\zeta_{*}|=d\Big{(}a\tau
w_{\zeta_{*}}-(a^{2}\tau^{2}+1)+\frac{a\tau}{w_{\zeta_{*}}}\Big{)}.$

Note that by (1.11),

$\displaystyle-\frac{d}{f(x)}\Big{(}a\tau
x-(a^{2}\tau^{2}+1)+\frac{a\tau}{x}\Big{)}$
$\displaystyle=-\frac{1}{f(x)}\frac{(1+\tau)(1+2c)}{2}\frac{(a\tau
x-1)(x-a\tau)}{x}=\frac{(a\tau x-1)(x-a)}{(ax-1)(x-a\tau)}.$

Therefore if $x<1/(a\tau)$,

$\displaystyle d\Big{(}a\tau
x-(a^{2}\tau^{2}+1)+\frac{a\tau}{x}\Big{)}<\tau|f(x)|<|f(x)|.$

From this, we notice that (2.55) does not hold for
$w_{\zeta_{*}}\in{\mathbb{R}}$. Furthermore, this implies that the right-hand
side of (2.55) is real-valued if and only if
$w_{\zeta_{*}}\in\partial\mathbb{D}$, equivalently,
$\zeta_{*}\in\partial\widehat{S}.$ This contradicts with the assumption that
$\zeta_{*}\in\widehat{S}^{c}$. Now the proof is complete. ∎

##  Appendix A Conformal mapping method: the pre-critical case

In this appendix, we present the conformal mapping method, which is helpful to
derive the candidate of the droplet given in terms of the rational function
(1.11).

######  Proposition A.1.

Let $\tau\in(\tau_{c},1)$. Suppose that $\widehat{S}$ in (1.18) is simply
connected. Let $f$ be a unique conformal map
$(\bar{\mathbb{D}}^{c},\infty)\to(\widehat{S}^{c},\infty)$, which satisfies

(A.1) $f(z)=r_{1}\,z+r_{2}+O\Big{(}\frac{1}{z}\Big{)},\qquad z\to\infty.$

Then the following holds.

  1. (i)

The conformal map $f$ is a rational function of the form

(A.2) $f(z)=r_{1}z+r_{2}+\frac{r_{3}}{z}+\frac{r_{4}}{z-a},\qquad a\in(-1,0),$

which satisfies

(A.3) $f(1/a)=\frac{r_{1}}{a}+r_{2}+r_{3}a+\frac{ar_{4}}{1-a^{2}}=0.$

  2. (ii)

The parameters $r_{j}$ $(j=1,\dots,4)$ are given by

(A.4)
$\displaystyle\begin{split}&r_{1}=\frac{1+\tau}{2}\sqrt{\frac{1+2c}{\tau}},\qquad\quad
r_{2}=\frac{1+\tau}{2\tau}(\tau(1+2c)+2\tau-1),\\\
&r_{3}=\frac{1+\tau}{2}\tau\sqrt{\tau(1+2c)},\qquad
r_{4}=\frac{(1-\tau)^{2}(1+\tau)(1-(1+2c)\tau)}{2\tau\sqrt{\tau(1+2c)}}\end{split}$

and

(A.5) $a=-\frac{1}{\sqrt{\tau(1+2c)}}.$

Note that the rational function $f$ with the choice of parameters
(LABEL:r1_r2_r3_r4) corresponds to (1.11). Therefore Proposition A.1 gives
rise to Theorem 1.4 (ii) under the assumption that $\widehat{S}$ is simply
connected. However, there is no general theory characterising the connectivity
of the droplet. (Nevertheless, we refer the reader to [49, 48] for sharp
connectivity bounds of the droplets associated with a class of potentials.)
Thus we need to directly verify the variational principles as in Subsection
2.2.

###### Proof of Proposition A.1 (i).

By differentiating the variational equality (1.29), we have

(A.6)
$\partial_{\zeta}\widehat{Q}(\zeta)=C(\zeta):=\int\frac{d\widehat{\mu}(z)}{\zeta-z},\qquad\zeta\in\widehat{S}.$

Using (1.14), this can be rewritten as

(A.7)
$\bar{\zeta}=\zeta\Big{[}(1-\tau^{2})\Big{(}C(\zeta)+\frac{c}{\zeta}\Big{)}+\tau\Big{]}^{2}.$

Therefore the Schwarz function $F$ associated with the droplet $\widehat{S}$
exists. Furthermore, it is expressed in terms of $C$ as

(A.8)
$F(\zeta)=\zeta\Big{[}(1-\tau^{2})\Big{(}C(\zeta)+\frac{c}{\zeta}\Big{)}+\tau\Big{]}^{2}.$

Note that for $z\in\partial\mathbb{D},$

(A.9)
$\overline{f(1/\bar{z})}=\overline{f(z)}=f(z)\Big{[}(1-\tau^{2})\Big{(}C(f(z))+\frac{c}{f(z)}\Big{)}+\tau\Big{]}^{2}.$

Using this, we define $f:\bar{\mathbb{D}}\backslash\\{0\\}\to\mathbb{C}$ by
analytic continuation as

(A.10)
$f(z):=\overline{f(1/\bar{z})\Big{[}(1-\tau^{2})\Big{(}C(f(1/\bar{z}))+\frac{c}{f(1/\bar{z})}\Big{)}+\tau\Big{]}^{2}}.$

Therefore $f$ has simple poles only at $0,\infty$ and the point
$a\in{\mathbb{R}}$ such that $f(1/a)=0$, which leads to (A.2). ∎

Next, we need to specify the constants $r_{j}$ and $a.$ For this, we shall
find interrelations among the parameters.

######  Lemma A.2.

We have

(A.11) $r_{3}=r_{1}\tau^{2}$

and

(A.12) $r_{4}=a(1-\tau^{2})\Big{(}r_{2}-2\tau(1+c)\Big{)}.$

Furthermore, we have

(A.13)
$r_{2}=r_{1}\frac{1+a^{2}\tau^{2}}{1-a^{2}\tau^{2}}\frac{a^{2}-1}{a}+\frac{2a^{2}(1-\tau^{2})\tau(1+c)}{1-a^{2}\tau^{2}}.$

###### Proof.

Note that

(A.14)
$\overline{f(1/\bar{z})}=\frac{r_{1}}{z}+r_{2}+r_{3}z+\frac{r_{4}z}{1-az}.$

Therefore, we have

(A.15)
$\frac{1}{\overline{f(1/\bar{z})}}=\frac{1}{r_{1}}\,z-\frac{r_{2}}{r_{1}^{2}}\,z^{2}+\frac{r_{2}^{2}-r_{1}r_{3}-r_{1}r_{4}}{r_{1}^{3}}\,z^{3}+O(z^{4}),\qquad
z\to 0.$

Since the Cauchy transform $C$ satisfies the asymptotic behaviour

(A.16) $C(\zeta)=\frac{1}{\zeta}+O(\frac{1}{\zeta^{2}}),\qquad\zeta\to\infty,$

we have

(A.17) $\overline{C(f(1/\bar{z}))}=\frac{1}{r_{1}}\,z+O(z^{2}),\qquad z\to 0.$

Combining these equations with (A.10), we obtain

(A.18) $\displaystyle
f(z)=\frac{r_{1}\tau^{2}}{z}+\Big{(}r_{2}\tau^{2}+2\tau(1-\tau^{2})(1+c)\Big{)}+O(z),\qquad
z\to 0.$

On the other hand, by using (A.2), we have

(A.19) $f(z)=\frac{r_{3}}{z}+\Big{(}r_{2}-\frac{r_{4}}{a}\Big{)}+O(z),\qquad
z\to 0.$

Then by comparing the coefficients in (A.18) and (A.19), we obtain (A.11) and
(A.12).

Note that by (A.3), we have

(A.20) $r_{4}=\frac{a^{2}-1}{a}\Big{(}\frac{r_{1}}{a}+r_{2}+r_{3}a\Big{)}.$

Then by (A.11), we have

(A.21)
$r_{4}=\frac{a^{2}-1}{a}\Big{(}\frac{r_{1}}{a}+r_{2}+r_{1}a\tau^{2}\Big{)}=r_{1}\frac{(1+a^{2}\tau^{2})(a^{2}-1)}{a^{2}}+r_{2}\frac{a^{2}-1}{a}.$

Combining this identity with (A.12), we obtain

(A.22)
$a(1-\tau^{2})r_{2}-2a(1-\tau^{2})\tau(1+c)=r_{1}\frac{(1+a^{2}\tau^{2})(a^{2}-1)}{a^{2}}+r_{2}\frac{a^{2}-1}{a},$

which leads to (A.13). ∎

######  Lemma A.3.

We have

(A.23)
$\Big{(}(2-a^{2}+a^{4}\tau^{2})r_{1}+ar_{2}\Big{)}\Big{(}r_{2}-2\tau(1+c)\Big{)}=(1-\tau^{2})c^{2}a(a^{2}-1).$

###### Proof.

Using (A.3), we have

(A.24)
$\frac{1}{\overline{f(1/\bar{z})}}=\frac{a^{2}(a^{2}-1)}{(2-a^{2})r_{1}+ar_{2}+a^{4}r_{3}}\,\frac{1}{z-a}+O(1),\qquad
z\to a.$

Then by (A.10) and (A.11), we obtain

(A.25)
$r_{4}=\frac{(1-\tau^{2})^{2}c^{2}\,a^{2}(a^{2}-1)}{(2-a^{2})r_{1}+ar_{2}+a^{4}r_{3}}=\frac{(1-\tau^{2})^{2}c^{2}\,a^{2}(1-a^{2})^{2}}{r_{1}(a^{2}\tau^{2}-1)(1-a^{2})^{2}+r_{4}a^{2}}.$

Now lemma follows from (A.12). ∎

###### Proof of Proposition A.1 (ii).

Since $\widehat{\mu}$ is a probability measure, we have

(A.26)
$1=\int_{\widehat{S}}\frac{1}{2(1-\tau^{2})}\frac{1}{|z|}\,dA(z)=\frac{1}{2\pi
i}\int_{\partial\widehat{S}}\frac{1}{1-\tau^{2}}\sqrt{\frac{\bar{z}}{z}}\,dz,$

where we have used Green’s formula for the second identity. Using the change
of variable $z=f(w)$, where $f$ is of the form (A.2), this can be rewritten as

(A.27) $\frac{1}{2\pi
i}\int_{\partial\mathbb{D}}\sqrt{\overline{f(1/\bar{w})}f(w)}\,\frac{f^{\prime}(w)}{f(w)}\,dw=1-\tau^{2}.$

By Lemma A.2 and (A.2), we have

(A.28) $\displaystyle f(z)$
$\displaystyle=\frac{1-az}{z(z-a)}\Big{(}-\frac{r_{1}}{a}z^{2}+\Big{(}\frac{a^{2}-1}{a^{2}}r_{1}-\frac{r_{2}}{a}\Big{)}z-a\tau^{2}r_{1}\Big{)},$
(A.29) $\displaystyle\overline{f(1/\bar{z})}$
$\displaystyle=\frac{z-a}{z(1-az)}\Big{(}-a\tau^{2}r_{1}z^{2}+\Big{(}\frac{a^{2}-1}{a^{2}}r_{1}-\frac{r_{2}}{a}\Big{)}z-\frac{r_{1}}{a}\Big{)}.$

Note here that by construction, two zeros of $f$ other than $1/a$ are
contained in the unit disc. Using these together with straightforward residue
calculus, we obtain

(A.30)
$\textup{Res}_{w=0}\Big{[}\sqrt{\overline{f(1/\bar{w})}f(w)}\,\frac{f^{\prime}(w)}{f(w)}\Big{]}=(1+c)(1-\tau^{2})$

and

(A.31)
$\textup{Res}_{w=a}\Big{[}\sqrt{\overline{f(1/\bar{w})}f(w)}\,\frac{f^{\prime}(w)}{f(w)}\Big{]}=-\frac{1}{a}\Big{[}\Big{(}\frac{1+a^{2}\tau^{2}}{a}r_{1}+r_{2}\Big{)}\Big{(}\frac{a^{4}\tau^{2}-a^{2}+2}{a}r_{1}+r_{2}\Big{)}\Big{]}^{1/2}.$

Furthermore, it follows from Lemma A.3 that

(A.32)
$\textup{Res}_{w=a}\Big{[}\sqrt{\overline{f(1/\bar{w})}f(w)}\,\frac{f^{\prime}(w)}{f(w)}\Big{]}=-c(1-\tau^{2}).$

Combining (A.27), (A.30) and (A.32), one can notice that the function $f$ has
a double zero, which implies that

(A.33) $\frac{a^{2}-1}{a^{2}}r_{1}-\frac{r_{2}}{a}=2r_{1}\tau.$

By solving the system of equations given in Lemmas A.2, A.3 and (A.33), the
desired result follows. ∎

######  Remark A.4 (The use of higher moments of the equilibrium measure).

In a more complicated case, for instance for the case with multiple point
charges such as (2.7), the mass-one condition (A.26) may not be enough to
characterise the parameters. In this case, one can further use the higher
order asymptotic expansions appearing in the above lemmas, which involve the
$k$-th moments of the equilibrium measure; cf. (2.35). Thus in principle, one
can always find enough (algebraic) interrelations to characterise the
parameters appearing in the conformal map.

######  Remark A.5.

For the case $\tau=0$ and $p>0$, it was shown in [12] that if

$c>\frac{(1-p^{2})^{2}}{4p^{2}},$

the droplet associated with (2.1) is a simply connected domain whose boundary
is given by the image of the conformal map

$f(z)=R\,z-\frac{\kappa}{z-q}-\frac{\kappa}{q},\qquad
R=\frac{1+p^{2}q^{2}}{2pq},\qquad\kappa=\frac{(1-q^{2})(1-p^{2}q^{2})}{2pq}.$

Here, $q$ is given by the unique solution of $P(q^{2})=0$, where

$P(x):=x^{3}-\Big{(}\frac{p^{2}+4c+2}{2p^{2}}\Big{)}x^{2}+\frac{1}{2p^{4}}$

such that $0<q<1$ and $\kappa>0.$

Beyond the case $\tau=0$, the conformal mapping method described above also
works for the potential (2.1) with general $\tau\in[0,1),c\in{\mathbb{R}}$ and
$p\in\mathbb{C}$ under the assumption that the associated droplet is simply
connected. Under this assumption, one can show that the boundary of the
droplet is given by the image of the rational conformal map $f$ of the form

(A.34) $f(z)=R_{1}\,z+R_{2}+\frac{R_{3}}{z}+\frac{R_{4}}{z-q},\qquad
q\in\mathbb{D},$

which satisfies $f(1/q)=0$. Furthermore, following the strategy above, one can
characterise the coefficients $R_{j}$ ($j=1,\dots,4$) of this rational map as
well as the position of the pole $q\in\mathbb{D}$.

However, as previously mentioned, it is far from being obvious to characterise
a condition for which the droplet is simply connected. Nevertheless, since the
radius of curvature of the ellipse (2.3) at the point $(1+\tau)\sqrt{1+c}$ is
given by

$\frac{(1-\tau)^{2}}{1+\tau}\,\sqrt{1+c},$

one can expect that if

(A.35)
$p>\max\Big{\\{}\frac{4\tau}{1+\tau}\sqrt{1+c}\,,\,(1+\tau)\sqrt{1+c}-\sqrt{1-\tau^{2}}\sqrt{c}\Big{\\}}$

then the droplet is a simply connected domain.

##  Appendix B One-dimensional equilibrium measure problem in the Hermitian
limit

In this appendix, we present a proof of (2.10). Let us write

(B.1) $V(z)\equiv V_{p}(z)=\frac{z^{2}}{2}-2c\log|z-p|.$

Recall that $\mu_{V}\equiv\mu_{V_{p}}$ is the equilibrium measure associated
with $V_{p}(x)$ ($x\in{\mathbb{R}}$).

We define

(B.2)
$R(z):=\Big{(}\frac{V^{\prime}(z)}{2}\Big{)}^{2}-\int_{{\mathbb{R}}}\frac{V^{\prime}(z)-V^{\prime}(s)}{z-s}\,d\mu_{V}(s).$

By applying Schiffer variations (see e.g. [35, Section 3]), we have

(B.3)
$R(z)=\Big{(}\int\frac{d\mu_{V}(s)}{z-s}-\frac{V^{\prime}(z)}{2}\Big{)}^{2},\qquad
z\in\mathbb{C}\setminus\operatorname{supp}(\mu_{V}).$

Combining the asymptotic behaviour

$\int\frac{d\mu_{V}(s)}{z-s}\sim\frac{1}{z},\qquad z\to\infty,$

with (B.3), we obtain

(B.4)
$R(z)=\frac{1}{4}z^{2}-(c+1)-\frac{cp}{z}+O\Big{(}\frac{1}{z^{2}}\Big{)},\qquad
z\to\infty.$

On the other hand, since

$\displaystyle
V^{\prime}(z)=z-\frac{2c}{z-p},\qquad\frac{V^{\prime}(z)-V^{\prime}(s)}{z-s}=1+\frac{2c}{z-p}\frac{1}{s-p},$

we have

(B.5) $\displaystyle
R(z)=\frac{1}{4}\Big{(}z-\frac{2c}{z-p}\Big{)}^{2}-1-\frac{2c}{z-p}\int_{{\mathbb{R}}}\frac{d\mu_{V}(s)}{s-p}.$

Thus we obtain

(B.6) $R(z)=\frac{c^{2}}{(z-p)^{2}}+O\Big{(}\frac{1}{z-p}\Big{)},\qquad z\to
p.$

In the expression (B.5), one can observe that $R$ is a rational function with
a double pole at $z=p$. Therefore it is of the form

(B.7) $R(z)=\frac{1}{4}z^{2}+\frac{Az^{2}+Bz+C}{(z-p)^{2}}$

for some constants $A,B$ and $C$. As in the previous subsection, we need to
specify these parameters.

By direct computations, we have

(B.8)
$R(z)=\frac{1}{4}z^{2}+A+\frac{2Ap+B}{z}+O\Big{(}\frac{1}{z^{2}}\Big{)},\qquad
z\to\infty,$

and

(B.9) $R(z)=\frac{Ap^{2}+Bp+C}{(z-p)^{2}}+O\Big{(}\frac{1}{z-p}\Big{)},\qquad
z\to p.$

By comparing coefficients in (B.4) and (B.8), we have

(B.10) $A=-c-1,\qquad-cp=2Ap+B.$

Similarly, by (B.6) and (B.9),

(B.11) $Ap^{2}+Bp+C=c^{2}.$

By solving these algebraic equations, we obtain

(B.12) $B=p(c+2),\qquad C=c^{2}-p^{2}.$

Combining all of the above with (B.7), we have shown that

(B.13)
$\displaystyle\begin{split}R(z)&=\frac{1}{4}z^{2}+\frac{-(c+1)z^{2}+p(c+2)z+(c^{2}-p^{2})}{(z-p)^{2}}\\\
&=\frac{((z-p)(z-2)-2c)((z-p)(z+2)-2c)}{4(z-p)^{2}}=\frac{\prod_{j=1}^{4}(z-\lambda_{j})}{4(z-p)^{2}},\end{split}$

where $\lambda_{j}$’s are given by (2.11) and (2.12). Therefore by (B.3), the
Stieltjes transform of $\mu_{V}$ is given by

(B.14)
$\displaystyle\begin{split}\int\frac{d\mu_{V}(s)}{z-s}&=\frac{V^{\prime}(z)}{2}-R(z)^{1/2}=\frac{z}{2}-\frac{c}{z-p}-\frac{1}{2}\sqrt{\frac{\prod_{j=1}^{4}(z-\lambda_{j})}{(z-p)^{2}}}.\end{split}$

Letting $z=x+i{\varepsilon}\to x\in{\mathbb{R}}$, we find

$\lim_{{\varepsilon}\to
0+}\operatorname{Im}\int\frac{d\mu_{V}(s)}{(x+i{\varepsilon})-s}=\begin{cases}\displaystyle\frac{\sqrt{-\prod_{j=1}^{4}(x-\lambda_{j})}}{2|x-p|}&\text{if
}x\in[\lambda_{1},\lambda_{2}]\cup[\lambda_{3},\lambda_{4}],\vskip 3.0pt plus
1.0pt minus 1.0pt\\\ 0&\text{otherwise}.\end{cases}$

Now the desired identity (2.10) follows from the Sokhotski-Plemelj inversion
formula, see e.g. [39, Section I.4.2].

### Acknowledgements

The author is greatly indebted to Yongwoo Lee for the figures and numerical
simulations.

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# Response of Energy Levels and B(E2)s to Variations of
Single Particle Energies

Castaly Fan1, Praveen C. Srivastava2, and Larry Zamick1

1Department of Physics and Astronomy, Rutgers University,
Piscataway, New Jersey 08854, USA
2Department of Physics, Indian Institute of Technology
Roorkee, Roorkee 247667, India

([-57])

###### Abstract

We perform shell model calculations for 48Cr using GXPF1 interaction by
varying single particle energies. First we keep the splitting of the pair A
($p_{3/2}-f_{7/2}$) constant and likewise pair B ($p_{1/2}-f_{5/2}$). We then
shift the pair B energy by an amount $\Delta$ relative to the first pair. We
then study the dependence of the energies of the even J states in 48Cr. We
further study the effects on B(E2)’s $J$ to $J-2$. We then do a similar
analysis with pair C ($f_{5/2}-f_{7/2}$) constant and we shift pair D
($p_{1/2}-p_{3/2}$) by an amount $\Delta$ . In the latter case the spin orbit
splittings are held constant. Again we study the effects on energies and
B(E2)s. We also discuss the possibility of scaling.

##  1 Introduction

In this work we examine the effects of varying single particle energies in a
shell model calculation using the GXFP1A interaction [1]. We will focus on the
yrast spectrum of 48Cr $J=0,2,4,...,16$. We perform calculations with the
GXFP1A interaction [1]. There have been many publications using these
interactions in the PF shell with the shell model program ANTOINE [3], notably
the works of E. Caurier et al. [4][5] and more recently of V. Kumar et al.
[6]. Other works related to 48Cr include those of K. Hara et al. [7],
F.Brandolini et al. [8], E.Caurier et al. [9], Z.C.Gao et al. [10] and R.A.
Herrera et al. [11].

##  2 Arrangement of Tables and Figures

The original single particle energies are given in the second column of Table
1. In the third column we show the corresponding values for the FPD6
interaction [2]. The orbits are 0$f_{7/2},1p_{3/2},0f_{5/2}$ and $1p_{1/2}$.
We will then alter the single particle energies as indicated in Tables 1, 1,
1, and 1. We first perform calculations with what we call CASE-1.

Table 1: Single particle energies (MeV) of GXPF1A and FPD6. Orbit | GXPF1A | FPD6
---|---|---
$0f_{7/2}$ | 0 (-8.6240) | 0 (-8.3876)
$1p_{3/2}$ | 2.9447 | 1.8942
$0f_{5/2}$ | 7.2411 | 6.4910
$1p_{1/2}$ | 4.4870 | 3.9093

In Table 1 we keep the splitting of the pair A ($p_{3/2}-f_{7/2}$) constant
and likewise pair B ($p_{1/2}-f_{5/2}$). We then shift the pair B energy by a
positive amount $\Delta$ relative to the first pair. We then study the
dependence of the energies of the even $J$ states in 48Cr. We further study
the effects on B(E2)’s $J$ to $J-2$. In Table 1 we extend the calculation to
negative $\Delta$.

Next we perform calculations which what we call CASE-2. In Tables 1 and 1 do a
similar analysis as was done in CASE-1, but now with pair C
($f_{5/2}-f_{7/2}$) constant and we pair D ($p_{1/2}-p_{3/2}$) shifted by an
amount $\Delta$. In the latter case the spin orbit splittings are held
constant. Again we study the effects on energies and B(E2)s.

[H] Energy spectra of 48Cr using GXPF1A interaction. Here we have keep the
single-particle energies of $0f_{7/2}$ and $1p_{3/2}$ as the original one, and
changed the single-particle energies of $0f_{5/2}$ and $1p_{1/2}$ moved up by
original plus $\Delta$. Energy GXPF1A $\Delta$ = 1 2 3 4 5 6 7 8 9 10 20 40 60
80 100 $0^{+}$ 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.000 0.000 0.000 0.000 0.000 0.000 $2^{+}$ 0.788 0.767 0.750 0.737 0.727
0.718 0.711 0.705 0.700 0.696 0.692 0.669 0.654 0.646 0.644 0.642 $4^{+}$
1.717 1.687 1.664 1.646 1.632 1.620 1.610 1.601 1.594 1.588 1.582 1.547 1.519
1.508 1.501 1.498 $6^{+}$ 3.229 3.152 3.090 3.039 2.998 2.963 2.934 2.908
2.886 2.867 2.849 2.747 2.671 2.639 2.622 2.610 $8^{+}$ 4.753 4.649 4.553
4.478 4.415 4.361 4.314 4.274 4.239 4.208 4.179 4.001 3.881 3.827 3.796 3.778
$10^{+}$ 6.429 6.238 6.080 5.952 5.846 5.758 5.684 5.621 5.565 5.517 5.474
5.219 5.033 4.957 4.915 4.889 $12^{+}$ 7.722 7.479 7.296 7.153 7.037 6.941
6.860 6.791 6.731 6.679 6.632 6.356 6.153 6.070 6.024 5.996 $14^{+}$ 9.701
9.432 9.227 9.063 8.929 8.818 8.724 8.693 8.572 8.511 8.456 8.129 7.887 7.788
7.733 7.699 $16^{+}$ 12.805 12.411 12.115 11.845 11.699 11.546 11.417 11.308
11.213 11.130 11.057 10.623 10.305 10.156 10.105 10.061 B(E2: J $\rightarrow$
J-2 ) ($e^{2}fm^{4}$) $2^{+}$ 249 237 228 222 217 212 209 206 204 202 200 191
184 180 179 178 $4^{+}$ 336 321 310 302 296 291 287 283 280 278 276 262 252
248 245 244 $6^{+}$ 336 317 304 296 290 285 280 277 274 271 270 256 247 244
241 240 $8^{+}$ 306 288 276 268 262 257 253 250 248 245 244 232 223 222 218
217 $10^{+}$ 212 195 192 186 186 185 185 184 184 184 183 181 180 179 179 178
$12^{+}$ 162 163 162 162 162 161 161 160 160 160 159 157 156 155 154 154
$14^{+}$ 126 125 124 123 123 123 123 122 122 121 121 120 118 118 118 117
$16^{+}$ 62 65 66 67 68 68 68 68 68 68 68 68 68 67 67 67 Q$(eb)$ $2^{+}$ -0.30
-0.29 -0.29 -0.28 -0.28 -0.27 -0.27 -0.27 -0.27 -0.27 -0.26 -0.26 -0.25 -0.24
-0.25 -0.25 $4^{+}$ -0.40 -0.39 -0.38 -0.38 -0.36 -0.36 -0.36 -0.35 -0.35
-0.34 -0.34 -0.33 -0.31 -0.31 -0.30 -0.29 $6^{+}$ -0.40 -0.38 -0.37 -0.36
-0.35 -0.34 -0.33 -0.33 -0.32 -0.32 -0.32 -0.29 -0.28 -0.28 -0.27 -0.27
$8^{+}$ -0.41 -0.38 -0.36 -0.34 -0.33 -0.32 -0.31 -0.30 -0.29 -0.28 -0.28
-0.26 -0.22 -0.24 -0.22 -0.22 $10^{+}$ -0.21 -0.17 -0.16 -0.15 -0.13 -0.13
-0.12 -0.12 -0.12 -0.11 -0.11 -0.10 -0.09 -0.09 -0.09 -0.09 $12^{+}$ -0.03
-0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02 -0.02
-0.02 -0.02 $14^{+}$ -0.05 -0.04 -0.03 -0.03 -0.03 -0.03 -0.03 -0.03 -0.02
-0.02 -0.02 -0.02 -0.01 -0.01 -0.01 -0.01 $16^{+}$ -0.09 -0.08 -0.07 -0.07
-0.06 -0.06 -0.06 -0.06 -0.05 -0.05 -0.05 -0.04 -0.04 -0.04 -0.04 -0.04

[H] Energy spectra of 48Cr using GXPF1A interaction. Here we have keep the
single-particle energies of $0f_{7/2}$ and $1p_{3/2}$ as the original one, and
changed the single-particle energies of $0f_{5/2}$ and $1p_{1/2}$ moved up by
original minus $\Delta$.

Energy |  $\Delta$ = -1 | -2 | -3 | -4 | -5 | -6 | -7 | -8 | -9 | -10 | -20 | -40 | -60 | -80 | -100
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0^{+}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
$2^{+}$ | 0.816 | 0.845 | 0.865 | 0.872 | 0.878 | 0.882 | 0.876 | 0.857 | 0.842 | 0.910 | 0.490 | 0.385 | 0.359 | 0.349 | 0.340
$4^{+}$ | 1.757 | 1.808 | 1.861 | 1.915 | 1.968 | 1.995 | 1.973 | 1.890 | 1.760 | 1.710 | 1.424 | 1.134 | 1.061 | 1.025 | 1.007
$6^{+}$ | 3.316 | 3.376 | 3.352 | 3.359 | 3.433 | 3.496 | 3.494 | 3.410 | 3.234 | 1.864 | 2.650 | 2.153 | 2.036 | 1.9165 | 1.931
$8^{+}$ | 4.877 | 4.970 | 4.987 | 5.056 | 5.177 | 5.261 | 5.260 | 5.147 | 4.626 | 4.058 | 2.966 | 2.433 | 2.297 | 2.237 | 2.202
$10^{+}$ | 6.621 | 6.663 | 6.633 | 6.760 | 6.965 | 7.102 | 7.114 | 6.998 | 6.439 | 5.770 | 7.675 | 6.865 | 6.651 | 6.552 | 6.495
$12^{+}$ | 8.060 | 8.539 | 9.099 | 9.397 | 9.469 | 9.449 | 9.373 | 9.131 | 8.195 | 7.487 | 15.588 | 35.059 | 54.929 | 74.871 | 94.838
$14^{+}$ | 10.060 | 10.547 | 11.133 | 11.585 | 11.648 | 11.589 | 11.512 | 11.397 | 10.959 | 10.434 | 26.249 | 65.601 | 105.436 | 145.363 | 185.319
$16^{+}$ | 13.361 | 14.034 | 14.169 | 13.720 | 13.424 | 13.263 | 13.134 | 12.965 | 12.778 | 12.854 | 28.411 | 67.637 | 107.439 | 147.349 | 187.297
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$2^{+}$ | 268 | 295 | 326 | 348 | 359 | 362 | 360 | 356 | 343 | 306 | 141 | 128 | 124 | 123 | 122
$4^{+}$ | 362 | 400 | 449 | 484 | 499 | 501 | 492 | 457 | 388 | 242 | 182 | 165 | 161 | 158 | 156
$6^{+}$ | 369 | 421 | 448 | 466 | 491 | 511 | 516 | 501 | 429 | 172 | 151 | 137 | 114 | 135 | 130
$8^{+}$ | 344 | 418 | 487 | 523 | 552 | 568 | 571 | 540 | 205 | 199 | 8 | 8 | 21 | 6 | 7
$10^{+}$ | 249 | 317 | 382 | 428 | 476 | 511 | 517 | 400 | 223 | 191 | 0.12 | 0.06 | 0.06 | 0.05 | 0.05
$12^{+}$ | 155 | 141 | 236 | 360 | 412 | 440 | 446 | 15 | 198 | 173 | 0.008 | 0.009 | 0.009 | 0.009 | 0.009
$14^{+}$ | 131 | 147 | 212 | 324 | 390 | 403 | 389 | 0.45 | 6 | 2 | 0.028 | 0.02 | 0.02 | 0.02 | 0.02
$16^{+}$ | 56 | 24 | 98 | 114 | 209 | 245 | 255 | 246 | 75 | 46 | 0.23 | 0.73 | 0.96 | 0.94 | 1.093
Q$(eb)$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$2^{+}$ | -0.32 | -0.32 | -0.31 | -0.29 | -0.27 | -0.26 | -0.26 | -0.26 | -0.26 | -0.18 | +0.24 | +0.22 | +0.22 | +0.23 | +0.22
$4^{+}$ | -0.42 | -0.43 | -0.43 | -0.41 | -0.41 | -0.41 | -0.42 | -0.45 | -0.49 | -0.51 | +0.30 | +0.29 | +0.28 | +0.28 | +0.28
$6^{+}$ | -0.42 | -0.42 | -0.37 | -0.34 | -0.34 | -0.35 | -0.37 | -0.41 | -0.54 | -0.61 | +0.29 | +0.26 | +0.13 | +0.27 | +0.24
$8^{+}$ | -0.43 | -0.43 | -0.40 | -0.39 | -0.38 | -0.38 | -0.38 | -0.46 | -0.70 | -0.68 | -0.40 | -0.38 | -0.38 | -0.38 | -0.38
$10^{+}$ | -0.27 | -0.37 | -0.41 | -0.42 | -0.42 | -0.42 | -0.41 | -0.42 | -0.74 | -0.72 | -0.01 | -0.005 | -0.003 | -0.002 | -0.0019
$12^{+}$ | -0.05 | -0.10 | -0.30 | -0.42 | -0.42 | -0.41 | -0.40 | -0.75 | -0.74 | -0.72 | -0.48 | -0.46 | -0.45 | -0.45 | -0.45
$14^{+}$ | -0.08 | -0.16 | -0.33 | -0.44 | -0.43 | -0.42 | -0.41 | -0.40 | -0.49 | -0.48 | -0.42 | -0.39 | -0.39 | -0.39 | -0.39
$16^{+}$ | -0.01 | -0.32 | -0.36 | -0.43 | -0.44 | -0.44 | -0.43 | -0.43 | -0.44 | -0.49 | -0.51 | -0.48 | -0.48 | -0.47 | -0.47

[H] Energy spectra of 48Cr using GXPF1A interaction. Here we have keep the
single-particle energies of $0f_{7/2}$ and $0f_{5/2}$ as the original one, and
changed the single-particle energies of $1p_{3/2}$ and $1p_{1/2}$ moved up by
original plus $\Delta$.

Energy |  $\Delta$ = 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 20 | 40 | 60 | 80 | 100
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0^{+}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
$2^{+}$ | 0.918 | 1.008 | 1.068 | 1.107 | 1.133 | 1.152 | 1.165 | 1.175 | 1.182 | 1.187 | 1.205 | 1.207 | 1.206 | 1.206 | 1.205
$4^{+}$ | 1.803 | 1.864 | 1.907 | 1.937 | 1.960 | 1.976 | 1.989 | 1.999 | 2.007 | 2.014 | 2.041 | 2.051 | 2.053 | 2.054 | 2.054
$6^{+}$ | 3.293 | 3.312 | 3.311 | 3.303 | 3.293 | 3.283 | 3.274 | 3.265 | 3.257 | 3.250 | 3.205 | 3.171 | 3.157 | 3.150 | 3.145
$8^{+}$ | 4.729 | 4.699 | 4.671 | 4.647 | 4.625 | 4.607 | 4.591 | 4.578 | 4.565 | 4.555 | 4.491 | 4.444 | 4.425 | 4.416 | 4.409
$10^{+}$ | 6.198 | 6.040 | 5.940 | 5.864 | 5.807 | 5.762 | 5.725 | 5.696 | 5.670 | 5.649 | 6.352 | 5.455 | 5.425 | 5.409 | 5.399
$12^{+}$ | 7.396 | 7.108 | 6.948 | 6.833 | 6.747 | 6.680 | 6.626 | 6.582 | 6.544 | 6.513 | 6.346 | 6.241 | 6.201 | 6.179 | 6.167
$14^{+}$ | 9.244 | 8.954 | 8.757 | 8.614 | 8.507 | 8.423 | 8.355 | 8.300 | 8.253 | 8.213 | 8.002 | 7.869 | 7.819 | 7.792 | 7.776
$16^{+}$ | 12.280 | 11.939 | 11.702 | 11.529 | 11.396 | 11.292 | 11.207 | 11.137 | 11.078 | 11.027 | 10.754 | 10.578 | 10.511 | 10.475 | 10.453
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$2^{+}$ | 213 | 189 | 173 | 161 | 153 | 147 | 142 | 138 | 135 | 133 | 121 | 115 | 112 | 111 | 111
$4^{+}$ | 276 | 234 | 205 | 186 | 171 | 161 | 153 | 148 | 143 | 139 | 122 | 114 | 110 | 109 | 109
$6^{+}$ | 268 | 224 | 196 | 178 | 166 | 156 | 150 | 145 | 140 | 137 | 122 | 115 | 111 | 111 | 111
$8^{+}$ | 255 | 223 | 203 | 190 | 180 | 173 | 168 | 164 | 160 | 157 | 143 | 135 | 132 | 131 | 131
$10^{+}$ | 184 | 170 | 161 | 155 | 150 | 147 | 143 | 140 | 138 | 135 | 127 | 121 | 119 | 118 | 117
$12^{+}$ | 153 | 145 | 139 | 134 | 131 | 127 | 125 | 123 | 121 | 119 | 112 | 107 | 105 | 103 | 103
$14^{+}$ | 120 | 114 | 110 | 107 | 104 | 102 | 100 | 99 | 98 | 97 | 90 | 87 | 85 | 85 | 84
$16^{+}$ | 60 | 59 | 57 | 56 | 55 | 55 | 54 | 54 | 53 | 53 | 50 | 48 | 48 | 47 | 47
Q$(eb)$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$2^{+}$ | -0.26 | -0.22 | -0.18 | -0.16 | -0.13 | -0.12 | -0.10 | -0.09 | -0.08 | -0.08 | -0.04 | -0.01 | -0.0091 | -0.0068 | -0.0047
$4^{+}$ | -0.36 | -0.34 | -0.32 | -0.30 | -0.28 | -0.27 | -0.27 | -0.26 | -0.25 | -0.25 | -0.22 | -0.19 | -0.19 | -0.18 | -0.18
$6^{+}$ | -0.32 | -0.26 | -0.22 | -0.19 | -0.17 | -0.15 | -0.14 | -0.13 | -0.12 | -0.12 | -0.08 | -0.06 | -0.05 | -0.05 | -0.05
$8^{+}$ | -0.33 | -0.27 | -0.23 | -0.20 | -0.18 | -0.17 | -0.15 | -0.15 | -0.13 | -0.13 | -0.09 | -0.08 | -0.07 | -0.070 | -0.06
$10^{+}$ | -0.15 | -0.12 | -0.10 | -0.09 | -0.08 | -0.08 | -0.07 | -0.07 | -0.06 | -0.06 | +0.01 | -0.04 | -0.04 | -0.041 | -0.04
$12^{+}$ | -0.02 | -0.02 | -0.02 | -0.02 | -0.01 | -0.01 | -0.01 | -0.01 | -0.01 | -0.01 | -0.01 | -0.009 | -0.009 | -0.008 | -0.008
$14^{+}$ | -0.04 | -0.04 | -0.03 | -0.03 | -0.03 | -0.03 | -0.03 | -0.03 | -0.03 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02 | -0.02
$16^{+}$ | -0.06 | -0.05 | -0.04 | -0.03 | -0.03 | -0.03 | -0.03 | -0.03 | -0.02 | -0.02 | -0.02 | -0.02 | -0.01 | -0.01 | -0.01

[H] Energy spectra of 48Cr using GXPF1A interaction. Here we have keep the
single-particle energies of $0f_{7/2}$ and $0f_{5/2}$ as the original one, and
changed the single-particle energies of $0p_{3/2}$ and $1p_{1/2}$ moved up by
original minus $\Delta$.

Energy |  $\Delta$ = -1 | -2 | -3 | -4 | -5 | -6 | -7 | -8 | -9 | -10 | -20 | -40 | -60 | -80 | -100
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$0^{+}$ | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000
$2^{+}$ | 0.638 | 0.532 | 0.540 | 0.795 | 1.232 | 1.294 | 1.317 | 1.333 | 1.344 | 1.352 | 1.367 | 1.361 | 1.358 | 1.355 | 1.353
$4^{+}$ | 1.612 | 1.539 | 1.584 | 1.881 | 2.738 | 3.838 | 4.334 | 4.446 | 4.501 | 4.529 | 4.552 | 4.520 | 4.505 | 4.497 | 4.491
$6^{+}$ | 3.103 | 2.991 | 3.061 | 3.577 | 5.074 | 7.084 | 8.712 | 9.843 | 10.860 | 11.862 | 21.830 | 41.799 | 61.787 | 81.781 | 101.778
$8^{+}$ | 4.753 | 4.745 | 4.863 | 5.389 | 6.785 | 8.699 | 10.701 | 12.712 | 14.683 | 15.949 | 25.960 | 45.866 | 65.829 | 85.809 | 105.797
$10^{+}$ | 6.728 | 6.848 | 6.935 | 7.594 | 9.491 | 11.692 | 13.547 | 15.467 | 17.414 | 19.376 | 39.219 | 79.143 | 119.119 | 159.105 | 199.098
$12^{+}$ | 8.341 | 8.942 | 8.984 | 9.706 | 11.763 | 14.550 | 17.441 | 20.075 | 22.152 | 24.139 | 43.927 | 83.780 | 123.727 | 163.699 | 203.682
$14^{+}$ | 10.459 | 11.392 | 11.312 | 12.011 | 14.324 | 17.686 | 21.251 | 24.414 | 27.380 | 30.332 | 60.070 | 119.925 | 179.874 | 239.848 | 299.832
$16^{+}$ | 13.655 | 14.610 | 14.605 | 15.296 | 17.590 | 20.955 | 24.656 | 28.478 | 32.358 | 36.270 | 75.918 | 155.754 | 235.698 | 315.670 | 395.653
B(E2: J $\rightarrow$ J-2 ) ($e^{2}fm^{4}$) |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$2^{+}$ | 302 | 359 | 388 | 363 | 257 | 190 | 170 | 158 | 151 | 145 | 123 | 112 | 109 | 107 | 106
$4^{+}$ | 417 | 496 | 523 | 413 | 65 | 13 | 222 | 188 | 167 | 154 | 116 | 104 | 100 | 99 | 98
$6^{+}$ | 425 | 500 | 508 | 425 | 273 | 189 | 22 | 18 | 44 | 51 | 54 | 54 | 54 | 55 | 54
$8^{+}$ | 385 | 454 | 453 | 368 | 234 | 118 | 8 | 0.04 | 0.60 | 3 | 2 | 2 | 2 | 2 | 1
$10^{+}$ | 293 | 391 | 392 | 319 | 195 | 3 | 2 | 2 | 4 | 3 | 0.45 | 0.32 | 0.29 | 0.28 | 0.27
$12^{+}$ | 171 | 316 | 341 | 325 | 282 | 24 | 21 | 18 | 13 | 11 | 10 | 10 | 10 | 9 | 9
$14^{+}$ | 139 | 232 | 267 | 261 | 210 | 153 | 136 | 56 | 22 | 15 | 8 | 7 | 7 | 7 | 7
$16^{+}$ | 66 | 132 | 160 | 159 | 156 | 154 | 125 | 43 | 23 | 17 | 9 | 8 | 7 | 7 | 7
Q$(eb)$ |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
$2^{+}$ | -0.35 | -0.39 | -0.40 | -0.35 | +0.13 | +0.25 | +0.26 | +0.25 | +0.25 | +0.25 | +0.23 | +0.22 | +0.22 | +0.21 | +0.21
$4^{+}$ | -0.45 | -0.50 | -0.53 | -0.55 | -0.53 | -0.50 | +0.29 | +0.32 | +0.32 | +0.32 | +0.30 | +0.28 | +0.28 | +0.28 | +0.27
$6^{+}$ | -0.49 | -0.56 | -0.61 | -0.63 | -0.61 | -0.49 | -0.14 | +0.01 | +0.07 | +0.09 | +0.12 | +0.13 | +0.13 | +0.13 | +0.13
$8^{+}$ | -0.51 | -0.60 | -0.67 | -0.70 | -0.69 | -0.67 | -0.65 | -0.63 | -0.54 | -0.13 | -0.09 | -0.09 | -0.09 | -0.09 | -0.08
$10^{+}$ | -0.38 | -0.61 | -0.69 | -0.69 | -0.66 | -0.35 | -0.34 | -0.33 | -0.32 | -0.32 | -0.29 | -0.27 | -0.27 | -0.26 | -0.26
$12^{+}$ | -0.08 | -0.59 | -0.69 | -0.69 | -0.67 | -0.63 | -0.58 | -0.37 | -0.28 | -0.26 | -0.24 | -0.24 | -0.24 | -0.23 | -0.24
$14^{+}$ | -0.08 | -0.57 | -0.70 | -0.68 | -0.66 | -0.64 | -0.58 | -0.48 | -0.46 | -0.45 | -0.41 | -0.40 | -0.39 | -0.39 | -0.39
$16^{+}$ | -0.14 | -0.59 | -0.71 | -0.70 | -0.68 | -0.66 | -0.65 | -0.64 | -0.63 | -0.62 | -0.59 | -0.58 | -0.57 | -0.57 | -0.57

Table 2: Ratio E(J)Δ/E(J) with GXPF1A for $\Delta$=1,10, and 20.

$J/\Delta$ | 1 | 10 | 20
---|---|---|---
2 | .973 | .878 | .849
4 | .983 | .921 | .901
6 | .976 | .882 | .851
8 | .978 | .879 | .842
10 | .977 | .851 | .812
12 | .968 | .859 | .823
14 | .972 | .872 | .838
16 | .969 | .863 | .829
Q(2${}^{+})$ | .967 | .867 | .867
$\sqrt{B(E2)}$ | .975 | 0.896 | .876

.

Table 3: Comparison of original energies with normalized ones for $\Delta=$20 using GXPF1A. Renormalization factor = 1.2. $J$ | Original Spectrum |  $\Delta$=20 Renormalized
---|---|---
0 | 0.000 | 0.000
2 | 0.788 | 0.802
4 | 1.717 | 1.854
6 | 3.279 | 3.294?
8 | 4.752 | 4.801
10 | 6.420 | 6.268?
12 | 7.722 | 7.627
14 | 9.701 | 9.755
16 | 12.805 | 12.748

We now discuss the arrangements of the figures. There are 4 sets, each with 4
subsets:

###  2.1 SET 1

### CASE - 1

Pair B ($f_{5/2}-p_{1/2}$) shifted by an amount $\Delta$ relative to Pair A
($p_{3/2}-f_{7/2}$).

Figure 1: E2, E4, E6, E8.

Figure 2: E10, E12, E14, E16.

Figure 3: B(E2) 2, 4, 6, 8.

Figure 4: B(E2) 10, 12, 14, 16.

###  2.2 SET 2

### CASE - 2

Pair D ($p_{1/2}-p_{3/2}$) shifted by an amount $\Delta$ relative to pair D
($f_{5/2}-f_{7/2}$).

Figure 5: E2, E4, E6, E8.

Figure 6: E10, E12, E14, E16.

Figure 7: B(E2) 2, 4, 6, 8.

Figure 8: B(E2) 10, 12, 14, 16.

###  2.3 SET 3

### CASE - 1

Global view: Pair B ($f_{5/2}-p_{1/2}$) shifted by an amount $\Delta$ relative
to Pair A ($p_{3/2}-f_{7/2}$).

Figure 9: E2, E4, E6, E8.

Figure 10: E10, E12, E14, E16.

Figure 11: B(E2) 2, 4, 6, 8.

Figure 12: B(E2) 10, 12, 14, 16.

###  2.4 SET 4

### CASE - 2

Global view: Pair D ($p_{1/2}-p_{3/2}$) shifted by an amount $\Delta$ relative
to pair D ($f_{5/2}-f_{7/2}$).

Figure 13: E2, E4, E6, E8.

Figure 14: E10, E12, E14, E16.

Figure 15: B(E2) 2, 4, 6, 8.

Figure 16: B(E2) 10, 12, 14, 16.

##  3 Overview of All the Results

A casual view of all the figures gives on the impassion that all the curves
look very similar, be it energy levels or B(E2)’s, be it CASE - 1 or CASE - 2.
There are gradual changes with increasing positive $\Delta$ in most cases. For
$\Delta$ becoming more negative things change smoothly for a while but then
there are sudden turn overs. This is undoubtedly due to the fact that there
will be crossovers of single particle energies.

For CASE - 1 we note with GXPF1A the 0$f_{5/2}$ is 4.4870 MeV above 0$f_{7/2}$
so with $\Delta$ lowers that -4.4870 we expect strange things to happen. In
the first 4 figures we indeed see down turnovers at about that value of
$\Delta$.

For CASE - 2 we note that 0 1$p_{3/2}$ is 2.9447 MeV above 0$f_{7/2}$ so we we
expect some sudden changes for $\Delta$ about -3 MeV. In contrast to CASE - 1
here we see sudden upturns as we further decrease $\Delta$. For the B(E2)’s
there is a downturn for CASE - 2.

We will give a more detailed discussion in the next subsection. We will focus
on positive $\Delta$.

###  3.1 SET 1. CASE - 1

In Fig 2 where we shift pair B ($p_{1/2}-f_{5/2}$) relative to pair A we note
that the excitation energies E2, E4, E6, and E8 slowly come down was we go
from $\Delta$ = 0 to 20. However the overall spectrum seems not to change very
much.

In Fig 2 we have E10, E12, E16 and E16 also coming down with increasing
$\Delta$ but at a faster rate than for the lower J’s in Fig 1. Again, the
overall spectrum seems not to change very much.

In Fig 4 The B(E2)s also come down with an increase in $\Delta$ but at a
faster rate than the energies. This is probably due to the fact that when
$\Delta$ increases there is less configuration mixing and hence less
collectivity.

In Fig 4 We show a striking difference in the B(E2) behavior for
$J=10,12,14,16$ when compared with the case in Fig 4 for $J=$2, 4, 6, and 8.
Now the curves for positive $\Delta$ are much flatter-much less variation with
increasing $\Delta$. A flat curve suggests configuration mixing , which is on
the decease as $\Delta$ is increasing , is not so important. The higher the
spin the more important are the single particle orbitals with high J e.g.
$f_{7/2}$.

There have been previous discussion of the band structure of 48Cr and the fact
the the higher J states do not belong to the same ”band” as the lower ones.
The general consensus is that for the higher states there is an alignment of
some of the nucleons along the rotation axis. This shows up more dramatically
in the B(E2)s rather than the energies.

###  3.2 SET 2. CASE - 2

As shown in Fig 6 when we shift ($p_{1/2}-p_{3/2}$) to more positive values we
find that the excitation energies go slightly up although the entire spectrum
does not change very much. This is in contrast to the case of Section1 where
the energies went slightly down. AN exception is $J=8^{+}$which is
surprisingly flat.

In Fig 6 we go to higher spin and on the whole the energies decrease with
increasing $\Delta$. This is another indication that perhaps the high J states
do not belong to the same band as the lower ones. For the first tile we see a
crossover near $\Delta$ = 20 with $J=12^{+}$starting to come down below
$J=10^{+}$.

In Fig 8 we come back to B(E2)s. They decree with increasing $\Delta$ . This
is not so different from The qualitative behavior in Fig 4. This is again due
to the fact that for increasing $\Delta$ there is less configuration mixing
and hence less collectivity.

In Fig 8. We look at B(E2)s for $J=$10, 12, 14 and 16. There is a decrease
with increasing $\Delta$ but not as severe as for $J=$2, 4, and 6. For the
$J=16^{+}$ to 14+transition the curve is very flat. For such a high spin you
need high $J$ single orbits to construct the state, so the reducing of
contributions from $p_{3/2}$ and $p_{1/2}$ is not so important.

###  3.3 Global view SET 3. CASE - 1

Here we will put more emphasis on negative $\Delta$. In Fig 10 we see large
increase in the excitation energies as we say decrease $\Delta$ form +5 to -5.
We could argue that at $\Delta$ =-5 we have a lot of single particle orbitals
close to together and this increases the pairing so that J=0 drops down a lot
relative to $J=$ 2, 4, 6, and 8. For $\Delta$ even more negative the low $J$
states are dominated by the $f_{5/2}$ and $p_{1/2}$ orbitals. As one makes
$\Delta$ more negative configuration for ($f_{7/2},p_{3/2}$) becomes less
important so the curve flattens out.

In Fig 10 ($J=10,12,14,16$) the behavior for negative $\Delta$ is completely
different than for low $J$. Except for $J=10^{+}$, the other energy levels got
up in a linear fashion with increasing negative $\Delta$. This is not
difficult to understand. On needs the high $J$ $f_{7/2}$ orbital to construct
these states and moving say $f_{5/2}$ below $f_{7/2}$ is equivalent to putting
$f_{7/2}$ above $f_{5/2}$.

For the B(E2)s in Figs 12 for $J=$ 2, 4, 6 and 8 the flatness is due to the
reduced configuration mixing due to the wide separation of the $f_{7/2}$ form
$f_{5/2}$.

###  3.4 Global view SET 4. CASE - 2

In this case the lowest orbits have the lowest spins i.e. $p_{3/2}$ and
$p_{1/2}$. It is even harder to make low lying high spin states in this case.
It is therefore not surprising to see in Fig 14 that even lower $J$ state
energies i.e. $J=$6 and 8 have a linear rise with increasing negative
$\Delta$.

In more detail with he $p_{3/2}$ and $p_{1/2}$ orbitals the maximum spin we
can have for 4 protons is $J_{p}$ = 2. via the configuration $p_{3/2}$
$p_{3/2}$ $p_{3/2}$ $p_{1/2}$, which is equivalent to $p_{3/2}^{-1}p_{1/2}$.
The same is true for 4 neutrons. So with $J_{p}$ = 2, $J_{n}$ = 2 we can only
make states up to $J=4$. This explains why only $J=0,2,4$ remain at low
wineries as we increase negative $\Delta$.

And of course in Fig 14 we see for the same reason that all high $J$ state
excitation energies (10, 12, 14, and 16) rise linearly with increasing
negative $\Delta$.

The flattening of the B(E2) curves in Fig 16 and Fig 16 is due to the
reduction in configuration mixing as the space between SET 3. and SET 4.
single particle energies widens. The enhancement near $\Delta$ =-5 is due to
the fact that all single particle orbitals are close to each other so there is
an enhancement of collectivity due to the increase of configuration mixing.

##  4 Closing Remarks - Scaling Behavior

In Table 2 we show the ratio E(J)$\Delta$/E(J) for $\Delta$=1, 10 and 20. Note
that although there are some fluctuations the ratios are similar. If the
ratios for a given $\Delta$ were all the same we would have perfect scaling.
In that idealized situation we would get identical spectra for any finite
$\Delta$ with that of $\Delta$=0 by multiplying the entire matrix for that
$\Delta$ for that by a constant. Thus, is a phenomenological approach. If we
limited ourselves to fitting the spectra of the 16 yrast states of 48Cr, we
would have an infinite number of choices of combinations of 2 body matrix
elements and single particle energies which would yield the same results. In
truth as seen in Table 2 the ratios are not exactly the same but they are
close enough to the idealized situation so that a large range of choices would
lead to equally good results for these spectra. Of course if we expanded the
data i.e. included other states. the result would be different. In Table 3 we
compare the original spectrum of GXPF1A with that for $\Delta$=20 multiplied
by a renormalization factor 1.2. This renormalization factor multiplies the
entire matrix including the $\Delta$=20 single particle energies. We see that
the spectra are reasonably close- it would be hard to prefer one to the other.
However the single particle energies are vastly different. Originally the
0$f_{5/2}$ and 1$p_{1/2}$ are 7.241 and 4.487 MeV above 0$f_{7/2}$. Now they
are 27.241 and 24.487 MeV above the 0$f_{7/2}$ orbit.

When one makes truncation in the PF shell by dropping orbits it is more
natural to drop the spin orbit partners 0$f_{5/2}$ and 1$p_{1/2}$ than it is
to drop the 2 p-shell orbits. This was done by Zamick et al. [13] in the
context of quadrupole moments and B(E2)’s. They studied the effects of
dropping spin-orbit partners 0$f_{5/2}$ and 1 $p_{1/2}$ on these
electromagnetic properties. In the present context this is equivalent to
setting $\Delta$to infinity. To a large extent the results of the truncated
calculations could be put into line with the full calculations by enlarging
the effective charges in the former when the FPD6 interaction is used. The
ratio full to truncated for Q(2+) 2 , Q(4+) 2 ,B(E2, 2+ $\rightarrow$ 0+) and
B(E2 , 4+ $\rightarrow$ 2+) were all very close to 1.4. The possibility of
scaling behavior is intriguing and will be further investigated in the near
future.

## ACKNOWLEDGEMENTS

P C Srivastava acknowledges a research grant from SERB (India),
CRG/2019/000556 and Kalam cluster at Physics Department, IIT-Roorkee. C. Fan
acknowledges supports from Aresty Research Center.

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