# On the extension for Toeplitz matrices of certain Markov inequalities

K. Castillo  CMUC, Department of Mathematics, University of Coimbra, 3000-143
Coimbra, Portugal<EMAIL_ADDRESS>and  A. Suzuki  CMUC, Department of
Mathematics, University of Coimbra, 3000-143 Coimbra, Portugal<EMAIL_ADDRESS>
###### Abstract.

Starting from a doubly infinite sequence of complex numbers, the aim of this
paper is to extend certain Markov inequalities for the determinant of Hankel
matrices and the zeros of the corresponding orthogonal polynomials on the real
line (A. Markov in Notes of the Imperial Academy of Sciences, St. Petersburg,
74 (Appendix nº2) (1894) 1-30. English translation, by J. Shohat, Duke Math.
J. 7 (1940), 85-96) to the Toeplitz case, where the central role is played by
CD kernels and paraorthogonal polynomials on the unit circle. In particular,
we consider the case in which the starting sequence is a two-sided Pólya
frequency sequence.

###### Key words and phrases:

Markov’s inequalities, Toeplitz matrices, CD kernels, paraorthogonal
polynomials on the unit circle, two-sided Pólya frequency sequences

###### 2010 Mathematics Subject Classification:

30C15, 42C05

##  1\. Introduction

Given an infinite sequence of real numbers $(a_{j})_{j\in\mathbb{N}}$, define
the $(n+1)$-by-$(n+1)$ Hankel matrices

$\displaystyle A_{n+1}=(a_{j+k-2})_{j,k=1}^{n+1}.$

Denote by $A^{(j,k)}$ the submatrix of a matrix $A$ obtained by deleting the
row $j$ and the column $k$. In order to guarantee the existence of a solution
of the Stieltjes moment problem, i.e., that the numbers $a_{j}$ may be
represented as

$a_{j}=\int_{0}^{\infty}x^{j}\ \mathrm{d}\mu(x),$

for some nontrivial Borel measure $\mu$ on $\mathbb{R}$, it is necessary and
sufficient that the matrices $A_{n+1}$ and their $n$-by-$n$ submatrices

$A_{n+1}^{(n+1,1)}=(a_{j+k-1})_{j,k=1}^{n},$

be positive definite for all $n\in\mathbb{N}$. (Recall that a matrix is called
positive definite if it is Hermitian and its leading principal minors,
including the determinant of the matrix itself, are positive.) In addition,
the matrices $A_{n+1}$ and $A_{n+1}^{(n+1,1)}$ are positive definite111In
1915, Pólya connected these conditions with the Laguerre-Pólya class, see [20,
Theorem IV and V, pp.235-236], and, in this case, the sequence
$(a_{j})_{j\in\mathbb{N}}$ is called one-sided Pólya frequency sequence. if
and only if $A_{n+1}$ is totally positive. (Recall that a totally positive
matrix is one in which all the minors are positive definite.) From the above,
the interested reader can easily prove that the matrices $A_{n+1}$ and
$A_{n+1}^{(n+1,1)}$ are positive definite if and only if there exists a
(unique) sequence of monic orthogonal polynomials on the real line (OPRL) with
respect to $\mu$, say $(P_{j})_{j\in\mathbb{N}}$, given by

$P_{0}=1,\quad P_{n+1}(x)=\frac{1}{\det
A_{n+1}}\,\det\,\begin{pmatrix}a_{0}&a_{1}&a_{2}&\cdots&a_{n+1}\\\\[7.0pt]
a_{1}&a_{2}&a_{3}&\cdots&a_{n+2}\\\\[7.0pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[7.0pt]
a_{n}&a_{n+1}&a_{n+2}&\cdots&a_{2n+1}\\\\[7.0pt]
1&x&x^{2}&\dots&x^{n+1}\end{pmatrix},$

whose zeros are all positive.

The above results, and others, were proven, using the analytic theory of
continued fractions, in two articles published in 1894 by two of the most
prominent mathematicians of the 19th century (see [17] and [29]): Markov and
Stieltjes. Stieltjes’ work is his celebrated memoir ‘‘Recherches sur les
fractions continues’’, which contains our modern ‘‘Stieltjes integral’’ and
also gave rise to the terminology ‘‘Problem of Moments’’, whose first
systematic treatment appears in works of Chebyshev, Markov and Stieltjes
himself, and, later, Hamburger, Nevanlinna, Riesz, Hausdorff, Carleman, and
Stone. However, as far as we know, Markov’s work222Read at the meeting of the
Physics and Mathematics Department on October 13, 1893. The article was
printed in March 1894 and at the time was sold for 40 kopecks. On April 1,
1894, the United States Treasury valuation of the ruble was 37.2 cents. There
are 100 kopecks in a ruble. never went beyond his own results333In his work,
implicitly, Markov proved that $A_{n+1}$ and $A^{(n+1,1)}$ are positive
definite if and only if $A_{n+1}$ is totally positive. Note, for instance,
that the proof that appears in [19, Theorem 4.4] is essentially given by
Markov in [17, pp. 18-20].. His work, written in Russian, only began to appear
in literature after being translated into English by Shohat in 1940. In fact,
Markov’s theorems appear in the monographs on the problem of moments by Shohat
and Tamarkin (see [22, p. 114]) and by Krein and Nudel’man (see [15, pp.
210-211, p. 224]), and in the classical book on matrix theory by Gantmacher
(see [9, Chapter XV]). The definitive recognition came in 1987, when
Kolmogorov and Yushkevich collected Markov’s results in their ‘‘Mathematics of
the 19th Century’’ (see [14, p. 52]).

Assume that one element of the sequence $(a_{j})_{j\in\mathbb{N}}$ is a
continuously differentiable real valued function, say $a_{k}=a_{k}(t)$,
depending on a parameter $t$ varying in a real open interval. Suppose also
that $A_{n+1}=A_{n+1}(t)$ and $A_{n+2}^{(n+2,1)}=A_{n+2}^{(n+2,1)}(t)$ are
positive definite for all $t$ (i.e., the zeros of the corresponding OPRL are
all positive), and

$(-1)^{k}\frac{\mathrm{d}a_{k}}{\mathrm{dt}}(t)<0.$

Roughly speaking, under these hypotheses, Markov proved in [17], as he himself
wrote, ‘‘two remarkable theorems’’444две замечательные теоремы. (see [17, p.
21]) regarding the monotonicity of the zeros $(x_{j}(t))_{j=1}^{n+1}$ of
$P_{n+1}=P_{n+1}(\cdot;t)$ and the determinants $\det A_{n+1}(t)$:

$\displaystyle\frac{\mathrm{d}x_{j}}{\mathrm{dt}}(t)>0,\quad\frac{\mathrm{d}\det
A_{n+1}}{\mathrm{dt}}(t)<0.$

These results are directly related to the sign of the coefficients of certain
real polynomials obtained from the OPRL ([17, pp. 15-16]). This is the
technical reason why Markov (implicitly) considered OPRL with only positive
zeros, which easily allowed to control the sign of their coefficients after
having factored $x_{j}(t)$. While all the above is ‘‘easy’’ for real
polynomials with real zeros, it can be a much more delicate question if we are
dealing with complex polynomials. The aim of this paper is to extend as far as
possible the above Markov results to doubly infinite sequences of complex
numbers and the corresponding Toeplitz matrices. In this context, the
orthogonal polynomials will be ‘‘replaced’’ by the CD kernels and
paraorthogonal polynomials on the unit circle (POPUC), which once again
reaffirms that these polynomials are the correct analogue of the OPRL. In
particular, we consider the case in which the starting sequence is a two-sided
Pólya frequency sequence. It is worth pointing out that POPUC have received
significant attention over the last years (see for instance [30, 3, 13, 27,
26, 23, 24, 7, 18, 5, 6] and references therein) and, of course, Pólya
frequency sequences and Toeplitz matrices appear in quite a large number of
applications. Section 2 contains our the main results and some corollaries. In
Section 3 we prove the theorems of Section 2.

##  2\. Basic definitions and main results

Denote by $\mathbb{S}^{1}_{r}(c)$ the boundary of the open disk
$\mathbb{D}_{r}(c)$ of radius $r>0$ with center $c$. Since the unit disk with
center at the origin plays a distinguished role, we use the notation
$\mathbb{D}=\mathbb{D}_{1}(0)$ and $\mathbb{S}^{1}=\mathbb{S}^{1}_{1}(0)$. Let
us introduce the notation $C_{r}(c)=\mathbb{D}_{r}(c)\cap\mathbb{S}^{1}$,
$I_{r}(c)=\mathbb{D}_{r}(c)\cap\mathbb{R}$, and
$\mathbb{S}^{1}_{\mp}=\\{z\in\mathbb{S}^{1}\,:\,\Re(z)\lessgtr 0\\}$. A
(doubly infinite) sequence $(c_{j})_{j\in\mathbb{Z}}$ of complex numbers is
called a two-sided Pólya frequency sequence if the four-way infinite matrix

$\begin{pmatrix}\ddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\iddots\\\\[7.0pt]
\cdots&c_{2}&c_{1}&c_{0}&c_{-1}&c_{-2}&c_{-3}&c_{-4}&\cdots\\\\[7.0pt]
\cdots&c_{3}&c_{2}&c_{1}&c_{0}&c_{-1}&c_{-2}&c_{-3}&\cdots\\\\[7.0pt]
\cdots&c_{4}&c_{3}&c_{2}&c_{1}&c_{0}&c_{-1}&c_{-2}&\cdots\\\\[7.0pt]
\iddots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

is totally positive (see, for instance, [8, Definition 1] and [12, Chapter
8]). Define the sequence of (monic) orthogonal polynomials on the unit circle
(OPUC) from an arbitrary sequence of complex numbers
$(c_{j})_{j\in\mathbb{Z}}$ as follows:

(1) $\displaystyle Q_{0}=1,\quad Q_{n}(z)=\frac{1}{\det
C_{n}}\,\det\,\begin{pmatrix}c_{0}&\overline{c}_{1}&\overline{c}_{2}&\cdots&\overline{c}_{n}\\\\[7.0pt]
c_{1}&c_{0}&\overline{c}_{1}&\cdots&\overline{c}_{n-1}\\\\[7.0pt]
\vdots&\vdots&\vdots&\ddots&\vdots\\\\[7.0pt]
c_{n-1}&c_{n-2}&c_{n-3}&\cdots&\overline{c}_{1}\\\\[7.0pt]
1&z&z^{2}&\dots&z^{n}\end{pmatrix},$

where $C_{n}=(c_{l-j})_{j,l=1}^{n}$, $\overline{c}_{j}=c_{-j}$, with $\det
C_{n}>0$. Define the normalized OPUC by $q_{n}(z)=\kappa_{n}\,z^{n}+\cdots$,
where $\kappa_{n}=\|Q_{n}\|^{-1}$. Of course, here and subsequently, whenever
we consider a sequence of OPUC, we are implicitly assuming that $C_{n+1}$ is
positive definite for all $n\in\mathbb{N}$. The CD kernel is defined (see [28,
(2.13.1)]) for $w,z\in\mathbb{C}$ by

(2) $\displaystyle K_{n}(w,z)=\sum_{j=0}^{n}\overline{q_{j}(w)}q_{j}(z).$

Recall that a monic POPUC of degree $n+1$ is defined from a sequence of
complex numbers $(c_{j})_{j\in\mathbb{Z}}$ as follows (see [28, p. $115$]):

(3) $\displaystyle
P_{n+1}(z,b)=P_{n+1}(z)=zQ_{n}(z)-\overline{b}\,Q^{*}_{n}(z),\quad
b\in\mathbb{S}^{1},$

where $Q_{n}^{*}(z)=z^{n}\,\overline{Q_{n}(1/\overline{z})}$. Recall that a
Hurwitz polynomial is a polynomial whose zeros are located in the open left
half-plane (see [9, p. 195]).

The next theorem is the analogue of Markov’s result for determinants of Hankel
matrices.

######  Theorem 2.1.

Fix $k\in\mathbb{Z}$ and let $c_{k}=c_{k}(t)$ be a continuously differentiable
complex valued function depending on a parameter $t$ varying in a real open
interval containing the origin. Let
$(c_{j})_{\begin{subarray}{c}j\in\mathbb{Z},j\not=\pm k\end{subarray}}$ be a
sequence of complex numbers. Define, from the elements of the sequence
$(c_{j})_{j\in\mathbb{Z}}$, the $(n+1)$-by-$(n+1)$ parametric Toeplitz matrix

$\displaystyle
C_{n+1}(t)=(c_{l-j})_{j,l=1}^{n+1},\quad\overline{c}_{j}=c_{-j}.$

If $(c_{j})_{j\in\mathbb{Z}}$ is a two-sided Pólya frequency sequence for all
$t$, then $\det C_{n+1}(t)$ is a strictly increasing function of $t$, provided
that $(-1)^{k}c_{k}(t)$ is a strictly increasing function of $t$.

When $(c_{j})_{j\in\mathbb{Z}}$ is not a two-sided Pólya frequency sequence,
the conclusion of Theorem 2.1 remains valid whenever a certain CD kernel is a
Hurwitz polynomial.

######  Theorem 2.2.

Assume the hypotheses and notation of Theorem 2.1 with all the elements of the
sequence $(c_{j})_{j\in\mathbb{Z}}$ taking real values. If the CD kernel
$K_{n}(1,-z)=K_{n}(1,-z;t)$ is a Hurwitz polynomial for all $t$, then $\det
C_{n+1}(t)$ is a strictly increasing function of $t$, provided that
$(-1)^{k}c_{k}(t)$ is a strictly increasing function of $t$.

The next theorem is much more general that Markov’s result for zeros of OPRL.

######  Theorem 2.3.

Assume the hypotheses and notation of Theorem 2.1. Let
$P_{n+1}(z,b)=P_{n+1}(z,b;t)=P_{n+1}(z;t)$ be defined as in (3) with
$Q_{n}(z)=Q_{n}(z;t)$ and $b=b(t)\in\mathbb{S}^{1}$ for all $t$. Let
$K_{n}(w,z)=K_{n}(w,z;t)$ be defined as in (2) for all $t$. Assume that $b(t)$
is a continuously differentiable complex valued function. If
$P_{n+1}(\zeta_{0};0)=0$, then there exist $\epsilon>0$ and $\delta>0$, and a
unique function, $\zeta:I_{\epsilon}(0)\to C_{\delta}(\zeta_{0})$,
differentiable on $I_{\epsilon}(0)$, with $\zeta(0)=\zeta_{0}$, and such that
$P_{n+1}(\zeta(t);t)=0$ for each $t\in I_{\epsilon}(0)$. Set
$\zeta(t)=e^{i\varphi(t)}$ for all $t$. If
$P_{n+1}(e^{i\varphi(t)};t)=P_{n+1}(e^{-i\varphi(t)};t)=0$ for each $t\in
I_{\epsilon}(0)$, then

$\displaystyle\operatorname{sgn}\,\left(\frac{\mathrm{d}\varphi}{\mathrm{dt}}(t)\right)=-\operatorname{sgn}(\Im(e^{i\varphi(t)}))\,\operatorname{sgn}\,\left(\Re\left(\overline{b_{k}(t)}\,\frac{\mathrm{d}c_{k}}{\mathrm{dt}}(t)\right)\right),$

where

$\displaystyle b_{k}(t)$ $\displaystyle=-\int
e^{-ik\theta}\,\frac{\displaystyle\sin\left(\frac{\theta-\varphi(t)}{2}\right)}{\displaystyle\sin\left(\frac{\theta+\varphi(t)}{2}\right)}\,\left|K_{n}(e^{i\varphi(t)},e^{i\theta};t)\right|^{2}\frac{\mathrm{d}\theta}{2\pi}$
$\displaystyle=-\frac{e^{i\varphi(t)}}{\overline{Q^{2}_{n}(e^{i\varphi(t)};t)}}\int
e^{-ik\theta}K_{n}(e^{i\varphi(t)},e^{i\theta};t)K_{n}(e^{i\theta},e^{-i\varphi(t)};t)\frac{\mathrm{d}\theta}{2\pi}.$

Furthermore, under the assumption $P_{n+1}(e^{-i\varphi(t)};t)=0$ replaced by
$P_{n+1}(e^{i\theta_{0}};t)=0$,

$\displaystyle\operatorname{sgn}\,\left(\frac{\mathrm{d}\varphi}{\mathrm{dt}}(t)\right)=\operatorname{sgn}\,\left(\Re\left(\overline{b_{k}(t)}\,\frac{\mathrm{d}c_{k}}{\mathrm{dt}}(t)\right)\right),$

where

$b_{k}(t)=-\frac{i\,e^{-i\theta_{0}}(e^{i\varphi(t)}-e^{i\theta_{0}})}{Q_{n}(e^{i\theta_{0}};t)\,\overline{Q_{n}(e^{i\varphi(t)};t)}}\int
e^{-ik\theta}K_{n}(e^{i\varphi(t)},e^{i\theta};t)K_{n}(e^{i\theta},e^{i\theta_{0}};t)\frac{\mathrm{d}\theta}{2\pi}.$

Depending on the location of the zeros of the corresponding POPUC, we can give
more concrete results on zeros of POPUC from the previous theorem. Below we
consider a series of examples in the form of corollaries.

######  Corollary 2.1.

Assume the hypotheses and notation of Theorem 2.3 with all the elements of the
sequence $(c_{j})_{j\in\mathbb{Z}}$ taking real values and $b=1$. If the CD
kernel $K_{n}(1,-z)=K_{n}(1,-z;t)$ is a Hurwitz polynomial for all $t$, then
$\zeta(t)$ moves clockwise $($respectively, counterclockwise$)$ along
$\mathbb{S}^{1}_{+}$ as $t$ increases on $I_{\epsilon}(0)$ provided
$\Im(\zeta_{0})>0$ $($respectively, $\Im(\zeta_{0})<0$$)$ and
$(-1)^{k}c_{k}(t)$ is a strictly decreasing function of $t$.

###### Proof.

By [28, Proposition 2.14.3 (iii)], the zeros of
$P_{n+1}(\cdot,1;t)=P_{n+1}(\cdot;t)$ are $1$ plus the zeros of
$K_{n}(1,\cdot;t)$. Since all the zeros of $P_{n+1}(\cdot;t)$ are on
$\mathbb{S}^{1}$ (see [28, Theorem 2.14.4]), they are indeed on
$\mathbb{S}^{1}_{+}$. By hypothesis, all the coefficients of
$P_{n+1}(\cdot;t)$ are real, and so its nonreal zeros can be arranged in
conjugate pairs. Assume that $\Im(\zeta_{0})\not=0$ and write

(4) $\displaystyle
R(z;t)=\dfrac{P_{n+1}(z;t)}{(z-\zeta(t))(z-\overline{\zeta(t)})}=\displaystyle\sum_{j=0}^{n-1}f_{j}(t)z^{j}.$

By hypothesis, $(-1)^{n-1}R(-z;t)$ is a monic Hurwitz polynomial, and so its
coefficients are all positive (see [21, Definition 11.4.1 and Proposition
11.4.2]). (Since the zeros of $P_{n+1}(\cdot;t)$ are simple (see [28, Theorem
2.14.4]) and lie on $\mathbb{S}_{+}^{1}$, and $(-1)^{n-1}R(-z;t)$ is a monic
Hurwitz polynomial, we must have $n$ even.) Hence

$\operatorname{sgn}\,(f_{j}(t))=(-1)^{j+1}.$

Note that

$\displaystyle\left|R(e^{i\theta};t)\right|^{2}$
$\displaystyle=\left(\displaystyle\sum_{j=0}^{n-1}f_{j}(t)e^{ij\theta}\right)\left(\displaystyle\sum_{j=0}^{n-1}\overline{f_{j}(t)}e^{-ij\theta}\right)$
$\displaystyle=\left(\displaystyle\sum_{j=0}^{n-1}f_{j}(t)e^{ij\theta}\right)\left(\displaystyle\sum_{j=-n+1}^{0}f_{-j}(t)e^{ij\theta}\right)=\sum_{j=-n+1}^{n-1}g_{j}(t)e^{ij\theta}$

where

$g_{j}(t)=\sum_{\begin{subarray}{c}0\leq j_{1},\,j_{2}\leq n-1\\\\[3.0pt]
j_{1}-j_{2}=j\end{subarray}}f_{j_{1}}(t)f_{j_{2}}(t).$

Since
$\operatorname{sgn}\,(f_{j_{1}}(t)f_{j_{2}}(t))=(-1)^{j_{1}+1}(-1)^{j_{2}+1}=(-1)^{j}$,
we obtain $\operatorname{sgn}\,(g_{j}(t))=(-1)^{j}$. Write

$\displaystyle\sum_{j=-n}^{n}d_{j}(t)e^{ij\theta}=(e^{-i\theta}-2\Re(\zeta(t))+e^{i\theta})\left|R(e^{i\theta};t)\right|^{2}.$

A similar argument to the above implies that
$\operatorname{sgn}\,(d_{k}(t))=(-1)^{k+1}$. By [28, Proposition 2.14.3
(i)-(ii)], we have that the zeros of $P_{n+1}(\cdot;t)$ are the $n+1$
solutions of the equation

(5) $\displaystyle z\,\frac{Q_{n}(z;t)}{Q^{*}_{n}(z;t)}=\overline{b(t)}.$

Since $P_{n+1}(\zeta(t);t)=P_{n+1}(\overline{\zeta(t)};t)=0$, the equation (5)
yields

$\overline{b(\zeta(t))}=\zeta(t)\,\frac{Q_{n}(\zeta(t);t)}{Q^{*}_{n}(\zeta(t);t)}=\overline{b(t)}=\overline{\zeta(t)}\,\frac{Q_{n}(\overline{\zeta(t)};t)}{Q^{*}_{n}(\overline{\zeta(t)};t)}=\overline{b(\overline{\zeta(t)})},$

and so
$P_{n+1}(z,b(\zeta(t));t)=P_{n+1}(z,b(\overline{\zeta(t)});t)=P_{n+1}(z,b;t)$.
Therefore, by [28, Proposition 2.14.3 (iii)], we get

$\frac{1}{k^{2}_{n}\,\overline{Q_{n}(\zeta(t);t)}}K_{n}(\zeta(t),e^{i\theta};t)=\frac{P_{n+1}(e^{i\theta},b;t)}{e^{i\theta}-\zeta(t)}.$

Hence

(6) $\displaystyle\displaystyle\sum_{j=-n}^{n}d_{j}(t)e^{ij\theta}$
$\displaystyle=\frac{e^{i\theta}\left|P_{n+1}(e^{i\theta};t)\right|^{2}}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}$
$\displaystyle=-\left|\frac{K_{n}(\zeta(t),e^{i\theta};t)}{k_{n}^{2}\,Q_{n}(\zeta(t);t)}\right|^{2}\,\frac{\displaystyle\sin\left(\frac{\theta-\varphi(t)}{2}\right)}{\displaystyle\sin\left(\frac{\theta+\varphi(t)}{2}\right)}.$

Multiplying both sides of the above equation by $e^{-ik\theta}$ and then
integrating on $\mathbb{S}^{1}$, we get

(7) $\displaystyle
d_{k}(t)=\frac{1}{\displaystyle\left|k_{n}^{2}\,Q_{n}(\zeta(t);t)\right|^{2}}\,b_{k}(t),$

and the result follows from Theorem 2.3. ∎

######  Corollary 2.2.

Assume the hypotheses and notation of Corollary 2.1, under the assumption
$b=1$ and the arc $\mathbb{S}^{1}_{+}$ replaced by $b=-1$ and
$\mathbb{S}^{1}_{-}$, respectively. Then $\zeta(t)$ moves clockwise
$($respectively, counterclockwise$)$ along $\mathbb{S}_{-}^{1}$ as $t$
increases on $I_{\epsilon}(0)$, provided $\Im(\zeta_{0})>0$ $($respectively,
$\Im(\zeta_{0})<0$$)$ and $c_{k}(t)$ is a strictly increasing function of $t$.

###### Proof.

The proof follows as the proof of Corollary 2.1. ∎

######  Corollary 2.3.

Assume the hypotheses and notation of Theorem 2.3 with $b=1$ and $n$ even.
Suppose that $(c_{j})_{j\in\mathbb{Z}}$ is a two-sided Pólya frequency
sequence for all $t$. If $\zeta(t)$ is the closest zero to $z=-1$ with
$\Im(\zeta_{0})>0$, then $\zeta(t)$ moves clockwise along $\mathbb{S}^{1}$ as
$t$ increases on $I_{\epsilon}(0)$, provided $(-1)^{k}\,c_{k}(t)$ is a
strictly decreasing function of $t$.

###### Proof.

Set

$Q_{j}(z;t)=\sum_{l=0}^{j}a_{j,l}(t)z^{l},\quad\alpha_{j}(t)=(-1)^{j}\frac{\det\,C_{j+2}^{(j+2,1)}(t)}{\det\,C_{j+1}(t)}.$

We claim that $\operatorname{sgn}\,(a_{j,l}(t))=(-1)^{j-l}$. Indeed, by
Szegő’s recursion (see [25, Theorem 1.5.2 and Theorem 1.5.11]), we have
$Q_{1}(z;t)=z-\alpha_{0}(t)$, and so $a_{1,0}(t)<0$ and $a_{1,1}(t)>0$. The
proof of our claim is by induction on $j$. Assume that $Q_{j-1}(\cdot;t)$ is
such that $\operatorname{sgn}\,(a_{j-1,l}(t))=(-1)^{j-l-1}$. By Szegő’s
recursion, we get

$\displaystyle Q_{j}(z;t)$
$\displaystyle=zQ_{j-1}(z;t)-\overline{\alpha}_{j-1}(t)Q_{j-1}^{*}(z;t)$
$\displaystyle=z^{j}+\sum_{l=1}^{j-1}(a_{j-1,l-1}(t)-\alpha_{j}(t)a_{j-1,j-l-1}(t))z^{l}-\alpha_{j-1}(t),$

and, hence,

$a_{j,l}=(1-\delta_{l,0})a_{j-1,l-1}(t)-(1-\delta_{l,j})\alpha_{j-1}(t)a_{j-1,j-l-1}(t),$

from which our claim follows. From (3), it is easily seen that

$P_{n+1}(z,1;t)=\sum_{j=0}^{n+1}r_{j}(t)z^{j},\quad
r_{j}(t)=(1-\delta_{j,0})a_{n,j-1}(t)-(1-\delta_{j,n+1})a_{n,n-j}(t),$

and so $\operatorname{sgn}(r_{j}(t))=(-1)^{j+1}$. Consequently, all the
coefficients of $-P_{n+1}(-z,1;t)$ are positive. By [2, Theorem 1.1](see also
[4, Theorem 2.4.5])555In particular, this interesting result gives a positive
answer to a conjecture posed of B. Conroy. Borwein and Erdelyi wrote in their
book that the proof of this result is “surprisingly complicated.”, it follows
that all the coefficients of

$\frac{-P_{n+1}(-z,1;t)}{(z+\zeta(t))(z+\overline{\zeta(t)})}$

are positive and, hence, $\operatorname{sgn}\,(f_{j}(t))=(-1)^{j+1}$, where
$f_{j}(t)$ is defined in (4). Note that $d_{j}(t)$, defined in (6), is given
by

$d_{j}(t)=\sum_{\begin{subarray}{c}0\leq j_{1}\leq n+1,\,0\leq j_{2}\leq
n-1\\\\[3.0pt] -j_{1}+j_{2}=j-1\end{subarray}}r_{j_{1}}(t)f_{j_{2}}(t).$

Since
$\operatorname{sgn}\,(r_{j_{1}}(t)f_{j_{2}}(t))=(-1)^{-j_{1}+1}(-1)^{j_{2}+1}=(-1)^{j+1}$,
we have $\operatorname{sgn}\,(d_{j}(t))=(-1)^{j+1}$, and the result follows as
the proof of Corollary 2.1.

∎

######  Remark 2.1.

In Corollary 2.3 we may replace the assumption that $(c_{j})_{j\in\mathbb{Z}}$
is a two-sided Pólya frequency sequence for all $t$ by the assumption that
$C_{j+2}^{(j+2,1)}(t)$ is a P-matrix $($see [11, p. 11]$)$ for all $t$ and
$j\in\mathbb{N}$.

######  Remark 2.2.

Although it is implicitly above, we emphasize that $b_{k}(t)$ in Theorem 2.3
is, depending on which case we are considering, the coefficient of order $k$
of one of the following Laurent polynomials:

$k_{n}^{4}\,\left|Q_{n}(\zeta(t);t)\right|^{2}\,\frac{z\left|P_{n+1}(z;t)\right|^{2}}{(z-\zeta(t))(z-\overline{\zeta(t)})}=\cdots+b_{k}(t)z^{k}+\cdots$

or

$i\,k_{n}^{4}\left(\zeta(t)-e^{i\theta_{0}}\right)\,\frac{z\left|P_{n+1}(z;t)\right|^{2}}{(z-\zeta(t))\left(z-e^{i\theta_{0}}\right)}=\cdots+b_{k}(t)z^{k}+\cdots.$

In other words, in view of Theorem 2.3, if we ‘‘perturb’’ the moment $c_{k}$
in the sequence $(c_{j})_{j\in\mathbb{Z}}$, all information about the
monotonicity of zeros of the corresponding POPUC depends exclusively on the
coefficient of order $k$ of one of the above polynomials and on $c_{k}$
itself.

We end this section with an example which gives readers a final taste of the
complexity of the problems we are dealing with.

######  Example 2.1.

Let $r$ be a continuously differentiable real function depending on a
parameter varying in a real open interval containing the origin. Fix
$k\in\mathbb{Z}$ and let $c_{\pm k}=c_{\pm k}(t)=(-1)^{k}r(t)^{{k^{2}}/2}$.
Set $q=r(0)\in(0,1)$ and define $c_{j}=(-1)^{j}q^{{j^{2}}/2}$ for all
$j\in\mathbb{Z}\setminus\\{-k,k\\}$. For $t=0$, $(c_{j})_{j\in\mathbb{Z}}$ is
the sequence of moments of the rotated Rogers-Szegő polynomials $($see [25, p.
87]$)$, and so $C_{n+1}(0)$ is positive definite for all $n\in\mathbb{N}$,
where $C_{n+1}(t)=(c_{l-j})_{j,l=1}^{n+1}$. Consequently, for $t$ small
enough, $C_{n+1}(t)$ is positive definite and the corresponding OPUC have
positive coefficients. Let $Q_{n}=Q_{n}(\cdot;t)$ be the OPUC defined as in
(1) and let $P_{n+1}(\cdot,-1)=P_{n+1}(\cdot,-1;t)$ be the POPUC defined as in
(3) with $b=-1$. Let $\zeta(t)$ denote one of the closest zeros to $z=1$ of
$P_{n}(\cdot,-1;t)$. Since the coefficients of $P_{n}(\cdot,-1;t)$ are
positive, by [2, Theorem 1.1], all the coefficients of the Laurent polynomial

$z\overline{P_{n+1}(z,-1;t)}\frac{P_{n+1}(z,-1;t)}{(z-\zeta(t))(z-\overline{\zeta(t)})}$

are positive for $n$ even. Thus, from the proof of Corollary 2.1 $($see also
Remark 2.2$)$, it is easily seen, for $n$ even, that $\zeta(t)$ moves
clockwise $($respectively, counterclockwise$)$ on $\mathbb{S}^{1}$ as $t$
strictly increases, provided $\Im(\zeta(0))>0$ $($respectively,
$\Im(\zeta(0))<0$$)$ and $c_{k}(t)$ is a strictly increasing function of $t$.
We conjecture that this is true for all the zeros in the upper half plane.

Figure 1. The zeros of $P_{n+1}(z;-1)$ for $n$ odd.

Note that when $n$ is odd, $P_{n+1}(z;-1)$ has a zero in $z=1$. In Figure 1 we
illustrate this situation, where the black dots represent the zeros of
$P_{n+1}(\cdot,-1)$. From [2, Theorem 1.1], we know that since the
coefficients of $P_{n+1}(\cdot,-1)$ are positive, the coefficients of the
monic polynomial whose zeros are all the black points to the left of the
vertical dashed line are also positive. However, in this case, the information
about the monotonicity of $\zeta(t)$ has to be extracted from a certain
polynomial with positive coefficients multiplied by the linear factor $z-1$,
and controlling the sign of a given coefficient of such a polynomial is not an
easy task.

##  3\. Proof of the main results

We now proceed to prove our main results.

###### Proof of Theorem 2.1.

By Jacobi’s formula,

$\displaystyle\frac{\mathrm{d}\det C_{n+1}}{\mathrm{dt}}(t)$
$\displaystyle={\rm tr}\left({\rm adj}\,C_{n+1}(t)\frac{\mathrm{d}\det
C_{n+1}}{\mathrm{dt}}(t)\right)$
$\displaystyle=\displaystyle\sum_{j,l=1}^{n+1}(-1)^{l+j}\det{C}_{n+1}^{(l,j)}(t)\frac{\mathrm{d}c_{l-j}}{\mathrm{dt}}(t)$
(8) $\displaystyle=\sum\limits_{\begin{subarray}{c}j,l=1\\\
|l-j|=k\end{subarray}}^{n+1}(-1)^{k}\det{C}_{n+1}^{(l,j)}(t)\frac{\mathrm{d}c_{k}}{\mathrm{dt}}(t),$

and, since $(c_{j})_{j\in\mathbb{Z}}$ is a two-sided Pólya frequency sequence
for all $t$, the result follows. ∎

###### Proof of Theorem 2.2.

By Carathéodory-Toeplitz’s theorem (see [25, p. 27]), there exists, for each
$t$, a nontrivial positive Borel measure $\sigma$ on $\mathbb{S}^{1}$
parametrized by $z=e^{i\theta}$ such that

$c_{j}(\mathrm{d}\sigma)=c_{j}=\int e^{-ij\theta}\mathrm{d}\sigma(\theta;t)$

if and only if the matrices $C_{n+1}(t)$ are positive definite for all
$n\in\mathbb{N}$. By Heine’s formula (see [25, Theorem 1.5.11]),
$(Q_{n}(\cdot;t))_{n\in\mathbb{N}}$ defined in (1) is the corresponding
sequence of (monic) OPUC, and, consequently,
$(P_{n}(\cdot,1;t))_{n\in\mathbb{N}}$, defined as in (3) with $b=1$, is a
sequence of POPUC associated with the sequence
$(Q_{n}(\cdot;t))_{n\in\mathbb{N}}$ (see [28, p. 115]). To shorten notation,
we omit the dependency on $t$. By [28, Proposition 2.14.3 (iii)], the zeros of
$P_{n+1}$ are $1$ plus the zeros of $K_{n}(1,\cdot)$. Since all the zeros of
$P_{n+1}$ are on $\mathbb{S}^{1}$ (see [28, Theorem 2.14.4]), they are indeed
on $\mathbb{S}^{1}_{+}$. Denote the zeros of $P_{n+1}$ by
$\zeta_{1},\dots,\zeta_{n+1}$. By the mechanical quadrature on
$\mathbb{S}^{1}$ (see [25, Theorem 2.2.12]), there exist positive weights
$\lambda_{n}(\zeta_{1}),\dots,\lambda_{n}(\zeta_{n+1})$ such that

$c_{j}=c_{j}(\mathrm{d}\sigma)=\int
e^{ij\theta}\mathrm{d}\lambda^{n+1}(\theta),$

for each $j\in\\{0,\pm 1,\cdots,\pm n\\}$, $\lambda^{n+1}(\theta)$ being a
measure on $\mathbb{S}^{1}$ given by

$\lambda^{n+1}(\theta)=\sum_{m=1}^{n+1}\lambda_{n}(\zeta_{m})\delta_{\zeta_{m}},$

where $\delta_{\zeta_{m}}$ is a point of mass at $\zeta_{m}$. Writing
$\lambda_{m}=\lambda_{n}(\zeta_{m})$, we get

$c_{j}=\displaystyle\sum_{m=1}^{n+1}\lambda_{m}\overline{\zeta}_{m}^{\,j}.$

Hence, an arbitrary minor of order $n$ of $C_{n+1}$ can be written as

(9)
$D=\det\left(\displaystyle\sum_{m=1}^{n+1}\lambda_{m}\overline{\zeta}_{m}^{\,\alpha_{j}-\beta_{l}}\right)_{j,l=1}^{n},$

where $\alpha_{1}<\cdots<\alpha_{n}\leq n$,
$\alpha_{1},\dots,\alpha_{n}\in\mathbb{N}$, and
$\beta_{1}<\cdots<\beta_{n}\leq n$, $\beta_{1},\dots,\beta_{n}\in\mathbb{N}$.
We can rewrite (9) as

$\displaystyle
D=\det\left(\begin{pmatrix}\lambda_{1}\overline{\zeta}_{1}^{\,\alpha_{1}}&\cdots&\lambda_{n+1}\overline{\zeta}_{n+1}^{\,\alpha_{1}}\\\\[7.0pt]
\vdots&&\vdots\\\\[7.0pt]
\lambda_{1}\overline{\zeta}_{1}^{\,\alpha_{n}}&\cdots&\lambda_{n+1}\overline{\zeta}_{n+1}^{\,\alpha_{n}}\end{pmatrix}\begin{pmatrix}\zeta_{1}^{\beta_{1}}&\cdots&\zeta_{1}^{\beta_{n}}\\\\[7.0pt]
\vdots&&\vdots\\\\[7.0pt]
\zeta_{n+1}^{\beta_{1}}&\cdots&\zeta_{n+1}^{\beta_{n}}\end{pmatrix}\right).$

Let $z_{1},\dots,z_{n}\in\\{\zeta_{1},\dots,\zeta_{n+1}\\}$ be pairwise
distinct and let $\gamma_{1}<\cdots<\gamma_{n}\leq n$,
$\gamma_{1},\dots,\gamma_{n}\in\mathbb{N}$. Writing the generalized
Vandermonde matrix as

(10)
$V_{(\gamma_{1},\dots,\gamma_{n})}(z_{1},\dots,z_{n})=(z_{j}^{\gamma_{l}})_{j,l=1}^{n},$

we have, by Cauchy-Binet’s formula,

(11) $\displaystyle D=$ $\displaystyle\lambda_{1}\cdots\lambda_{n}\det
V_{(\alpha_{1},\dots,\alpha_{n})}(\overline{\zeta}_{1},\dots,\overline{\zeta}_{n})\det
V_{(\beta_{1},\dots,\beta_{n})}(\zeta_{1},\dots,{\zeta}_{n})$
$\displaystyle+\cdots+\lambda_{2}\cdots\lambda_{n+1}\det
V_{(\alpha_{1},\dots,\alpha_{n})}(\overline{\zeta}_{2},\dots,\overline{\zeta}_{n+1})\det
V_{(\beta_{1},\dots,\beta_{n})}({\zeta}_{2},\dots,{\zeta}_{n+1}).$

We claim that each product of determinants on the right-hand side of (11) has
positive real part. Indeed, considering the product

(12) $\det
V_{(\alpha_{1},\dots,\alpha_{n})}(\overline{\zeta}_{1},\dots,\overline{\zeta}_{n})\det
V_{(\beta_{1},\dots,\beta_{n})}(\zeta_{1},\dots,{\zeta}_{n}),$

we write

(13)
$R(z)=\frac{P_{n+1}(z)}{z-\zeta_{n+1}}=\displaystyle\sum_{j=0}^{n}d_{j}z^{j}.$

By Heineman’s theorem (see [10, Theorem I]), we have

(14) $\dfrac{\det
V_{(0,1,\dots,n-r-1,n-r+1,\dots,n)}(\zeta_{1},\dots,\zeta_{n})}{\det
V_{(0,1,\dots,n-1)}(\zeta_{1},\dots,\zeta_{n})}=(-1)^{r}d_{n-r},$

for $r=0,1,\dots,n$. Hence,

(15) $\displaystyle\det
V_{(\alpha_{1},\dots,\alpha_{n})}(\overline{\zeta}_{1},\dots,\overline{\zeta}_{n})\det
V_{(\beta_{1},\dots,\beta_{n})}(\zeta_{1},\dots,\zeta_{n})$
$\displaystyle\quad=(-1)^{j_{1}+j_{2}}\overline{d}_{n-j_{1}}\,d_{n-j_{2}}\left|\det
V_{(0,1,\dots,n-1)}(\zeta_{1},\dots,\zeta_{n})\right|^{2},$

for some $j_{1},j_{2}\in\\{0,1,\dots,n\\}$. Since $b=1$ and all the elements
of the sequence $(c_{j})_{j\in\mathbb{Z}}$ are real, the coefficients of
$P_{n+1}$ are also real. Consequently, the nonreal zeros of $P_{n+1}$ occur in
conjugate pairs. If $\zeta_{n+1}=1$, then666Note that $(-1)^{n}\,R(-z)$ is a
monic Hurwitz polynomial.

$\operatorname{sgn}\,(d_{n-j})=(-1)^{j},$

for each $j=0,1,\dots,n$. Consequently, $(-1)^{j}d_{n-j}>0$ and, by (15),

(16) $\det
V_{(\alpha_{1},\dots,\alpha_{n})}(\overline{\zeta}_{1},\dots,\overline{\zeta}_{n})\det
V_{(\beta_{1},\dots,\beta_{n})}(\zeta_{1},\dots,\zeta_{n})>0.$

On the other hand, if $\zeta_{n+1}\neq 1$, then we may write

$\displaystyle\dfrac{P_{n+1}(z)}{z-\zeta_{n+1}}$
$\displaystyle=(z-\overline{\zeta}_{n+1})\,\displaystyle\sum_{l=0}^{n-1}f_{l}z^{l}$
$\displaystyle=z^{n}+\left(\displaystyle\sum_{l=1}^{n-1}(f_{l-1}-f_{l}\overline{\zeta}_{n+1})z^{l}\right)-f_{0}\overline{\zeta}_{n+1},\quad
f_{l}\in\mathbb{R},\quad f_{n-1}=1.$

Hence,

$d_{j}=(1-\delta_{j,0})f_{j-1}-(1-\delta_{j,n})f_{j}\overline{\zeta}_{n+1},$

for each $j=0,1,\dots,n$. Using a similar argument as in the previous case, we
obtain $\operatorname{sgn}(f_{n-j})=(-1)^{j+1}$ for each $j=0,1,\dots,n$, and
so

(17) $\displaystyle(-1)^{j}\Re(d_{n-j})$ $\displaystyle>0,$
$\displaystyle(-1)^{j}\operatorname{sgn}(\Im(d_{n-j}))$
$\displaystyle=(\delta_{n-j,n}-1)\operatorname{sgn}\,(\Im(\zeta_{n+1})).$

From (15), we have

(18) $\displaystyle\operatorname{sgn}(\Re(\det
V_{(\alpha_{1},\dots,\alpha_{n})}(\overline{\zeta}_{1},\dots,\overline{\zeta}_{n})\det
V_{(\beta_{1},\dots,\beta_{n})}(\zeta_{1},\dots,\zeta_{n})))$
$\displaystyle\quad=\operatorname{sgn}(\Re((-1)^{j_{1}}\overline{d}_{n-j_{1}}(-1)^{j_{2}}d_{n-j_{2}})).$

Taking into account (17), we obtain

(19)
$\displaystyle\Re((-1)^{j_{1}}\overline{d}_{n-j_{1}}(-1)^{j_{2}}d_{n-j_{2}})$
$\displaystyle\quad=(-1)^{j_{1}}\Re(d_{n-j_{1}})(-1)^{j_{2}}\Re(d_{n-j_{2}})+(-1)^{j_{1}}\Im(d_{n-j_{1}})(-1)^{j_{2}}\Im(d_{n-j_{2}})$
$\displaystyle\quad>0.$

Consequently, the right-hand side of (18) is positive. Hence, from (16) and
(18), the real part of (12) is positive. The positivity of the real part of
the other products of determinants on right-hand side of (18) can be obtained
in the same way and our claim follows. Thus, since $D$ is real, we conclude
that it is positive and the result follows from (3). ∎

###### Proof of Theorem 2.3.

By definition, the coefficients of $P_{n+1}(\cdot;t)$ are rational functions
of the elements of the sequence $(c_{j})_{j\in\mathbb{Z}}$ and $b(t)$, and so
they are differentiable functions for each $t$. Since
$P_{n+1}(\zeta_{0};0)=0$, by simplicity of the zeros of $P_{n+1}(\cdot;t)$
(see [28, Theorem 2.14.4]), we have

$\left.\frac{\partial P_{n+1}}{\partial
z}(z;t)\right|_{z=\zeta_{0},t=0}\not=0,$

and the first part of the theorem follows from the implicit function theorem
(see [1, Theorem 2.4] and its remark). Moreover,

(20)
$\frac{\mathrm{d}\zeta}{\mathrm{dt}}(t)=-\frac{\displaystyle\frac{\partial
P_{n+1}}{\partial t}(\zeta(t);t)}{\displaystyle\frac{\partial
P_{n+1}}{\partial z}(\zeta(t);t)}$

for each $t\in I_{\epsilon}(0)$. From the definition of $P_{n+1}(\cdot;t)$ and
the orthogonality of $Q_{n}(\cdot;t)$ with respect to $\sigma$, it is easy to
see that

(21) $\displaystyle\int
P_{n+1}(e^{i\theta};t)\overline{g(e^{i\theta})}\,\mathrm{d}\sigma(\theta;t)=0,$

for any polynomial $g$ of degree at most $n$ vanishing at $z=0$. An easy
computation (see [6, (5)]) also shows that

(22)
$\displaystyle\int\dfrac{P_{n+1}(e^{i\theta};t)}{e^{i\theta}-\zeta(t)}\overline{h(e^{i\theta})}\mathrm{d}\sigma(\theta;t)=\overline{h(\zeta(t))}\displaystyle\int\dfrac{P_{n+1}(e^{i\theta};t)}{e^{i\theta}-\zeta(t)}\mathrm{d}\sigma(\theta;t),$

for any polynomial $h$ of degree at most $n$. We claim that777An alternative
proof can be given using Christoffel-Darboux’s formula as in the proof of [16,
Lemma 4.4].

(23)
$\displaystyle\int\dfrac{P_{n+1}(e^{i\theta};t)}{e^{i\theta}-\zeta(t)}\mathrm{d}\sigma(\theta;t)\neq
0.$

Indeed, replacing $h=h(\cdot;t)=Q_{n}(\cdot;t)$ into (22), we have

$\displaystyle\int\dfrac{P_{n+1}(e^{i\theta};t)}{e^{i\theta}-\zeta(t)}\mathrm{d}\sigma(\theta;t)$
$\displaystyle\quad=\frac{1}{\overline{Q_{n}(\zeta(t);t)}}\,\int\frac{P_{n+1}(e^{i\theta};t)}{e^{i\theta}-\zeta(t)}\overline{Q_{n}(e^{i\theta};t)}\mathrm{d}\sigma(\theta;t)\not=0,$

because $Q_{n}(\cdot;t)$ is orthogonal with respect to $\sigma$ and, by
Fejér’s theorem (see [25, Theorem 1.7.19]), the zeros of this polynomial are
in $\mathbb{D}$. Since the leading coefficient of $P_{n+1}(\cdot;t)$ does not
depend on $t$, replacing

$h(z;t)=\displaystyle\frac{\partial P_{n+1}}{\partial t}(z;t)$

into (22) and using (23), we obtain

(24) $\frac{\partial P_{n+1}}{\partial
t}(\zeta(t);t)=\frac{\displaystyle\int\frac{\overline{P_{n+1}(e^{i\theta};t)}}{\overline{e^{i\theta}-\zeta(t)}}\frac{\partial
P_{n+1}}{\partial
t}(e^{i\theta};t)\mathrm{d}\sigma(\theta;t)}{\displaystyle\int\frac{\overline{P_{n+1}(e^{i\theta};t)}}{\overline{e^{i\theta}-\zeta(t)}}\mathrm{d}\sigma(\theta;t)}.$

Moreover, one can check that

$\frac{z}{(z-\zeta(t))^{2}}\left(P_{n+1}(z;t)-(z-\zeta(t))\frac{\partial
P_{n+1}}{\partial z}(\zeta(t);t)\right)$

is a nonzero polynomial of degree $n$ vanishing at $z=0$. Hence, by (21),

(25) $\frac{\partial P_{n+1}}{\partial
z}(\zeta(t);t)=\frac{h_{\zeta}(t)}{\displaystyle\int\frac{\overline{P_{n+1}(e^{i\theta};t)}}{\overline{e^{i\theta}-\zeta(t)}}\mathrm{d}\sigma(\theta;t)},$

where

$h_{\zeta}(t)=\displaystyle\int\left|\frac{P_{n+1}(e^{i\theta};t)}{e^{i\theta}-\zeta(t)}\right|^{2}\mathrm{d}\sigma(\theta;t)>0.$

Substituting (24) and (25) into (20), and multiplying the result by
$-i\overline{\zeta(t)}$, we obtain

(26) $\displaystyle
h_{\zeta}(t)\frac{\mathrm{d}\varphi}{\mathrm{dt}}(t)=\displaystyle\int\frac{i\overline{\zeta(t)}}{\overline{e^{i\theta}-\zeta(t)}}\overline{P_{n+1}(e^{i\theta};t)}\frac{\partial
P_{n+1}}{\partial t}(e^{i\theta};t)\mathrm{d}\sigma(\theta;t).$

Assuming that $P_{n+1}(\overline{\zeta(t)};t)=0$, the equation (26) yields

(27)
$\displaystyle(h_{\zeta}(t)+h_{\overline{\zeta}}(t))\frac{\mathrm{d}\varphi}{\mathrm{dt}}(t)$
$\displaystyle\quad=2\,\Im(\zeta(t))\int\frac{e^{i\theta}P_{n+1}(e^{i\theta};t)}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}\overline{\frac{\partial
P_{n+1}}{\partial t}(e^{i\theta};t)}\mathrm{d}\sigma(\theta;t).$

We first prove the theorem for the case where

(28) $\displaystyle(c_{j})_{j\in\mathbb{N}}\in\ell_{1}.$

In this case, combining the Radon-Nikodym and Lebesgue decomposition theorems,
we can write the measure in the form

$\mathrm{d}\sigma(\theta;t)=\omega(\theta;t)\frac{\mathrm{d}\theta}{2\pi}+\mathrm{d}\sigma_{s}(t),$

where $\sigma_{s}$ is singular relative to the Lebesgue measure and $\omega\in
L^{1}(\mathbb{S}^{1},\sigma)$. Of course,
$(c_{j})_{j\in\mathbb{Z}}\in\ell_{1}$ gives that $\sigma$ is absolutely
continuous with continuous density $\omega$, which equals

$\sum_{j=-\infty}^{\infty}c_{j}e^{ij\theta}$

in this case and obeys $\min_{\theta}\,\omega(\theta;t)>0.$ Since the set
$\\{(\theta;t):\theta\in[\theta_{0},\theta_{0}+2\pi],\,t\in
I_{\epsilon}(0)\\}$ is closed and bounded,

(29) $\displaystyle\frac{\partial\omega}{\partial
t}(\theta;t)=(2-\delta_{k,0})\,\Re\left(\frac{\mathrm{d}c_{k}}{\mathrm{dt}}(t)\,e^{ik\theta}\right),$

and the range of integration is finite, we can take the partial derivative of
(21) with respect to $t$, which gives

(30) $\displaystyle\int\frac{\partial P_{n+1}}{\partial
t}(e^{i\theta};t)\overline{g(e^{i\theta})}\,\omega(\theta;t)\frac{\mathrm{d}\theta}{2\pi}=-\int
P_{n+1}(e^{i\theta};t)\overline{g(e^{i\theta})}\,\frac{\partial\omega}{\partial
t}(\theta;t)\frac{\mathrm{d}\theta}{2\pi}.$

Since

$\displaystyle\frac{zP_{n+1}(z;t)}{(z-\zeta(t))(z-\overline{\zeta(t)})}$

is a nonzero polynomial of degree $n$ vanishing at $z=0$, the equation (30)
yields

(31)
$\displaystyle\int\frac{e^{i\theta}P_{n+1}(e^{i\theta};t)}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}\overline{\frac{\partial
P_{n+1}}{\partial
t}(e^{i\theta};t)}\omega(\theta;t)\frac{\mathrm{d}\theta}{2\pi}$
$\displaystyle=-\int\frac{e^{i\theta}\left|P_{n+1}(e^{i\theta};t)\right|^{2}}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}\frac{\partial\omega}{\partial
t}(\theta;t)\frac{\mathrm{d}\theta}{2\pi}.$

Combining (31) and (27), we can assert that

(32)
$(h_{\zeta}(t)+h_{\overline{\zeta}}(t))\frac{\mathrm{d}\varphi}{\mathrm{dt}}(t)=-2\,\Im(\zeta(t))\displaystyle\int\frac{e^{i\theta}\left|P_{n+1}(e^{i\theta};t)\right|^{2}}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}\frac{\partial\omega}{\partial
t}(\theta;t)\frac{\mathrm{d}\theta}{2\pi}.$

Write

(33)
$\frac{e^{i\theta}\left|P_{n+1}(e^{i\theta};t)\right|^{2}}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}=\displaystyle\sum_{j=-n}^{n}d_{j}(t)e^{ij\theta}.$

We claim that $d_{-k}(t)=\overline{d_{k}(t)}$. Indeed, multiplying both sides
of (33) by $e^{-ik\theta}$ and then integrating on $\mathbb{S}^{1}$ yields

(34) $\displaystyle
d_{k}(t)=\displaystyle\int\frac{e^{-i(k-1)\theta}}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}\left|P_{n+1}(e^{i\theta};t)\right|^{2}\dfrac{\mathrm{d}\theta}{2\pi}.$

Since

$\frac{e^{-i(k-1)\theta}}{(e^{i\theta}-\zeta(t))(e^{i\theta}-\overline{\zeta(t)})}=\frac{e^{-ik\theta}}{\cos\theta-\cos\varphi(t)},$

our claim follows from (34). Substituting (29) and (33) into (32), we obtain

$\displaystyle(h_{\zeta}(t)+h_{\overline{\zeta}}(t))\frac{\mathrm{d}\varphi}{\mathrm{dt}}(t)$
$\displaystyle\quad=-2(2-\delta_{k,0})\,\Im(\zeta(t))\displaystyle\int\displaystyle\sum_{j=-n}^{n}d_{j}(t)e^{ij\theta}\Re\left(\frac{\mathrm{d}c_{k}}{\mathrm{dt}}(t)\,e^{ik\theta}\right)\frac{\mathrm{d}\theta}{2\pi}$
(35)
$\displaystyle\quad=-2(2-\delta_{k,0})\,\Im(\zeta(t))\,\Re\left(\overline{d_{k}(t)}\,\frac{\mathrm{d}c_{k}}{\mathrm{dt}}(t)\right).$

Let us now note that the assumption (28) can be suppressed. Indeed, by
Bernstein-Szegő’s approximation (see [25, Theorem 1.7.8]),
$(Q_{j}(\cdot;t))_{j=0}^{n}$ are OPUC for any fixed $m\in\mathbb{N}$ such that

$\mathrm{d}\sigma_{m}(\theta;t)=\frac{1}{|q_{m}(e^{i\theta};t)|^{2}}\,\frac{\mathrm{d}\theta}{2\pi},\quad
m\geq n.$

Consequently, $Q_{j}(\cdot;t)$ and $P_{j+1}(\cdot;t)$ are OPUC and POPUC,
respectively, for both $\mathrm{d}\sigma$ and $\mathrm{d}\sigma_{m}$.
Combining Geronimus’ theorem (see [25, Theorem 1.7.5]) and Baxter’s theorem
(see [25, Theorem 5.2.1]) gives

$(c_{j}(\mathrm{d}\sigma_{m}))_{j\in\mathbb{N}}\in\ell_{1}.$

Taking into account the notation fixed in (6) and (7), the second part of the
theorem for conjugate zeros follows from (35). Similar arguments apply under
the assumption $P_{n+1}(e^{-i\varphi(t)};t)=0$ replaced by
$P_{n+1}(e^{i\theta_{0}};t)=0$. The details are left to the reader, as an
exercise.

∎

## Acknowledgements

This work is partially supported by the Centre for Mathematics of the
University of Coimbra (funded by the Portuguese Government through FCT/MCTES,
DOI 10.54499/UIDB/00324/2020). The first author is supported by FCT through
DOI 10.54499/2022.00143.CEECIND/CP1714/CT0002. The second author is supported
by FCT grant 2021.05089.BD.

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#  Unraveling the global behavior of equation of state
by explicit finite nuclei constraints

Anagh Venneti<EMAIL_ADDRESS>Sakshi Gautam  Sarmistha
Banik  B. K. Agrawal

###### Abstract

We obtain posterior distribution of equations of state (EOSs) across a broad
range of density by imposing explicitly the constraints from precisely
measured fundamental properties of finite nuclei, in combination with the
experimental data from heavy-ion collisions and the astrophysical observations
of radius, tidal deformability and minimum-maximum mass of neutron stars. The
acquired EOSs exhibit a distinct global behavior compared to those usually
obtained by imposing the finite nuclei constraints implicitly through
empirical values of selected key parameters describing symmetric nuclear
matter and symmetry energy in the vicinity of the saturation density. The
explicit treatment of finite nuclei constraints yields softer EOSs at low
densities which eventually become stiffer to meet the maximum mass criteria.
The Kullback-Leibler divergence has been used to perform a quantitative
comparison of the distributions of neutron star properties resulting from the
EOSs obtained from implicit and explicit finite nuclei constraints.

###### keywords:

Finite Nuclei Properties , Neutron Star Observables , Equation of State ,
Symmetry Energy

††journal: Physics Letters B

[first]organization=Department of Physics, BITS-Pilani Hyderabad
campus,city=Hyderabad, postcode=500078, country=India
[second]organization=Department of Physics, Panjab University,city=Chandigarh,
postcode=160014, country=India [third]organization=Saha Institute of Nuclear
Physics,addressline=1/AF Bidhannagar, city=Kolkata, postcode=700064,
state=West Bengal, country=India [fourth]organization=Homi Bhabha National
Institute,addressline=Anushakti Nagar, city=Mumbai, postcode=400094,
state=Maharashtra, country=India

Introduction
The wealth of data [1] stemming from nuclear physics experiments corresponding
to fundamental properties of finite nuclei and heavy-ion collisions (HICs)
together with astrophysical observations has spurred comprehensive
investigations aimed at constraining the equation of state (EOS) for neutron
star matter over a wide range of density. The nuclear physics experiments,
determining the binding energies [2], charge radii [3], neutron skin thickness
[4], giant resonances [5], and other properties of the finite nuclei, probe
the EOS in the vicinity of the nuclear saturation density ($\rho_{0}\sim 0.16$
fm-3). The HIC experiments [6, 7] also give deep insights into the EOS of
symmetric nuclear matter and symmetry energy at densities up to 2$\rho_{0}$.
On the other hand, neutron stars (NS) serve as a unique gateway into the
exploration of neutron-rich nuclear matter EOS at densities not yet possible
in terrestrial experiments. The EOS of dense nuclear matter significantly
shapes the properties of NS, such as mass, radius, and tidal deformability.
These properties have been observed in pulsar mass measurements [8],
simultaneous mass and radius measurements from NICER [9, 10, 11, 12], and
detection of gravitational waves from binary neutron star (BNS) merger events
[13]. The third-generation gravitational-wave observatories, Cosmic Explorer
and Einstein Telescope, are expected to provide more accurate constraints on
the neutron star equation of state [14].

Several efforts have been made to constrain the EOS for neutron star matter
combining the results from finite nuclei, heavy-ion collisions as well as
astrophysical observations [15, 16, 17]. The constraints from finite nuclei
properties have been incorporated implicitly through empirical values of
selected nuclear matter parameters (NMPs). These parameters are instrumental
in governing the EOS for symmetric nuclear matter and the symmetry energy,
around the saturation density, which are the main ingredients for our
understanding of the EOS for neutron star matter. Of late, the posterior
distribution of EOSs was obtained using a comprehensive set of data from
nuclear physics experiments and recent astrophysical observations. The inputs
from nuclear physics implicitly included the constraints, on the symmetry
energy and symmetry energy pressure, empirically derived using the nuclear
masses, neutron skin thickness and iso-vector giant dipole resonances. The
experimental data on the HICs were considered in terms of the symmetric
nuclear matter pressure and the symmetry energy in the range 0.2-2$\rho_{0}$
[18].

Often, the constraints from finite nuclei are incorporated implicitly through
the empirical values of binding energy per nucleon, incompressibility
coefficient for the symmetric nuclear matter, and the symmetry energy, at the
saturation density, determined by accurately calibrating the mean-field models
to the experimental data on binding energies, charge radii, and iso-scalar
giant monopole resonance (ISGMR) in several nuclei. Their values are found to
vary over narrow ranges across various models. Consequently, the values of
binding energy per nucleon and saturation density are usually kept fixed
around $-16$ MeV and $0.16$ fm-3, respectively. One usually overlooks the
small variations required in these quantities specific to a given model, which
may subsequently influence the values of symmetry energy parameters. The
uncertainty in the symmetry energy parameters may propagate to NS properties
such as radius and tidal deformability [15, 19, 20, 21, 22]. It would be ideal
to allow all the NMPs to vary and be explicitly subjected to the constraints
on the experimental and astrophysical observations, narrowing the parameter
space and preserving their inter-correlations. [23, 24, 25].

The precisely measured values of finite nuclei properties, such as binding
energies, charge radii, and giant resonances, when considered explicitly,
impose strong constraints on the radius and tidal deformability of NS with
canonical mass (1.4 $M_{\odot}$). The radius $R_{1.4}=10.6-12.6$ km and
dimensionless tidal deformability $\Lambda_{1.4}=125-410$ were estimated
solely from the constraints imposed by the finite nuclei properties and
minimum-maximum mass of NS [26]. Similar analysis with further inclusion of
low-density EOS for pure neutron matter from ab-initio calculations yielded
$R_{1.4}=12.3-12.5$ km and $\Lambda_{1.4}=380-460$ [27]. Recently, the
connection between finite nuclei properties and NS with canonical mass has
been investigated by employing a group of more than 400 mean field models,
derived from the Skyrme-type effective nucleon-nucleon interactions and the
effective Lagrangian describing nucleon interactions through $\sigma$,
$\omega$, and $\rho$ mesons [28]. The analysis reveals that the models adept
at capturing the binding energies, charge radii and ISGMR energies of
symmetric and asymmetric finite nuclei display a strong correlation between
tidal deformability of canonical mass neutron star and the slope of symmetry
energy at the saturation density.

In our present work, we obtain a posterior distribution of EOSs for neutron
star matter, consistent with the measured values of basic properties of finite
nuclei, such as binding energies and charge radii, together with the
comprehensive set of experimental data from HICs and astrophysical
observations of radius, tidal deformability and minimum-maximum mass of
neutron stars. We compare these EOSs with their counterpart obtained by
implicitly imposing the finite nuclei constraints. The statistical distance
between the distributions of EOSs resulting from these two distinct scenarios
is quantified using the Kullback-Leibler divergence. Additionally, the
distributions of radius and tidal deformability across a wide range of neutron
star masses are also calculated.

Methodology
The nuclear matter is widely described by the formalism of relativistic mean
field (RMF) theory, based on quantum hadrodynamics. In the RMF theory, the
neutrons and protons, constituents of the nuclear matter, interact through the
exchange of $\sigma$, $\omega$, and $\rho$ mesons. The $\sigma$ and $\omega$
mesons respectively mediate the medium-range attractive and short-range
repulsive nature of nuclear interactions, respectively, while the $\rho$ meson
governs the iso-vector properties. Additionally, the theory incorporates cross
interaction between $\omega$ and $\rho$ mesons as well as the non-linear self-
couplings of the $\sigma$ and $\omega$ mesons. The Lagrangian of this nuclear
matter is given as [29, 30],

$\mathcal{L}_{NL}=\mathcal{L}_{nm}+\mathcal{L}_{\sigma}+\mathcal{L}_{\omega}+\mathcal{L}_{\rho}+\mathcal{L}_{int},$
(1)

where,

$\displaystyle\mathcal{L}_{nm}$ $\displaystyle=$
$\displaystyle\\!\bar{\psi}\left(i\gamma^{\mu}\partial_{\mu}-m\right)\psi+\\!\\!g_{\sigma}\sigma\bar{\psi}\psi-\\!\\!g_{\omega}\bar{\psi}\gamma^{\mu}\omega_{\mu}\psi$
$\displaystyle-\frac{g_{\rho}}{2}\bar{\psi}\gamma^{\mu}\vec{\rho}_{\mu}\vec{\tau}\psi,$
$\displaystyle\mathcal{L}_{\sigma}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\partial^{\mu}\sigma\partial_{\mu}\sigma-m^{2}_{\sigma}\sigma^{2}\right)-\frac{A}{3}\sigma^{3}-\frac{B}{4}\sigma^{4},$
$\displaystyle\mathcal{L}_{\omega}$ $\displaystyle=$
$\displaystyle\\!-\frac{1}{4}\Omega^{\mu\nu}\Omega_{\mu\nu}+\frac{1}{2}m^{2}_{\omega}\omega^{\mu}\omega_{\mu}+\frac{C}{4}\left(g_{\omega}^{2}\omega_{\mu}\omega^{\mu}\right)^{2},$
$\displaystyle\mathcal{L}_{\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\vec{B}^{\mu\nu}\vec{B}_{\mu\nu}+\frac{1}{2}m^{2}_{\rho}\vec{\rho}_{\mu}\vec{\rho}^{\mu},$
$\displaystyle\mathcal{L}_{int}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\Lambda_{v}g^{2}_{\omega}g^{2}_{\rho}\omega_{\mu}\omega^{\mu}\vec{\rho}_{\mu}\vec{\rho}^{\mu}.$

In the above equations,
$\Omega_{\mu\nu}=\partial_{\nu}\omega_{\mu}-\partial_{\mu}\omega_{\nu}$ and
$\vec{B}_{\mu\nu}=\partial_{\nu}\vec{\rho}_{\mu}-\partial_{\mu}\vec{\rho}_{\nu}-g_{\rho}\left(\vec{\rho}_{\mu}\times\vec{\rho}_{\nu}\right)$.
The masses of the nucleon, $\sigma$, $\omega$ and $\rho$ mesons are denoted by
$m$, $m_{\sigma}$, $m_{\omega}$, and $m_{\rho}$ respectively. The coupling
strengths $g_{\sigma}$, $g_{\omega}$, $g_{\rho}$, $A$, $B$, $C$ and
$\Lambda_{v}$ are usually calibrated to yield the measured values of the bulk
properties of finite nuclei and neutron stars accurately. The finite nuclei
properties are calculated by numerically solving the equations of motion for
the mesons and nucleons within the basis expansion method [31, 32, 33].

The properties of a static, non-rotating neutron star (NS), such as mass and
radius, can be determined by a pair of coupled differential equations called
the Tolman-Oppenheimer-Volkoff (TOV) equations [34, 35, 36]. The dimensionless
tidal deformability (in the natural units $G=c=1$) , defined as
$\Lambda=\frac{2}{3}k_{2}\left(\frac{R}{M}\right)^{5}$, $M$ and $R$ being the
mass and radius, is a measure of the tendency of the NS to deform under the
tidal forces. The tidal love number ($k_{2}$) can be computed by solving a
differential equation along with the TOV equations [37]. These equations
require the EOS for beta-equilibrated charge-neutral matter. The EOS for crust
is constructed from the Baym-Pethick-Sutherland [38] EOS and a polytropic fit
[19, 39]. The EOS of the core, which starts at 0.5 $\rho_{0}$, can be readily
obtained for the given Lagrangian (Eq.1) [29].

Table 1: The median and 68% CI for binding energies (B) and charge radii ($R_{ch}$) for a few selected nuclei were obtained using implicit and explicit FNCs. The ‘*’ highlights the experimental data [2, 3] for the nuclei used for the explicit FNC. | Implicit-FNC | Explicit-FNC | Experimental
---|---|---|---
|  B [MeV] |  $R_{ch}$ [fm] |  B [MeV] |  $R_{ch}$ [fm] |  B [MeV] |  $R_{ch}$ [fm]
16O |  -135.85${}^{+8.91}_{-9.29}$ |  2.62${}^{+0.03}_{-0.03}$ |  -129.4${}^{+1.14}_{-1.16}$ |  2.7${}^{+0.01}_{-0.01}$ |  -127.62 |  2.70
40Ca∗ |  -353.87${}^{+19.29}_{-19.64}$ |  3.34${}^{+0.05}_{-0.03}$ |  -342.73${}^{+1.63}_{-1.69}$ |  3.44${}^{+0.01}_{-0.01}$ |  -342.05 |  3.48
48Ca |  -420.24${}^{+22.28}_{-22.51}$ |  3.38${}^{+0.06}_{-0.03}$ |  -414.39${}^{+1.87}_{-2.2}$ |  3.47${}^{+0.01}_{-0.02}$ |  -416.00 |  3.48
132Sn |  -1096.68${}^{+56.56}_{-56.22}$ |  4.6${}^{+0.1}_{-0.05}$ |  -1103.84${}^{+5.64}_{-5.49}$ |  4.72${}^{+0.02}_{-0.02}$ |  -1102.84 |  4.71
208Pb∗ |  -1620.19${}^{+88.37}_{-87.82}$ |  5.38${}^{+0.12}_{-0.06}$ |  -1632.94${}^{+7.73}_{-7.66}$ |  5.53${}^{+0.02}_{-0.02}$ |  -1636.43 |  5.50

We follow the Bayesian approach in our statistical analysis of the EOSs.
Bayesian analysis utilizes the Bayes’ theorem, defined as,

$P(\bm{\Theta}|D)=\frac{\mathcal{L}(D|\bm{\Theta})P(\bm{\Theta})}{\mathcal{Z}},$
(2)

where the $\mathcal{L}(D|\bm{\Theta})$, $P(\bm{\Theta})$, and
$P(\bm{\Theta}|D)$ are the likelihood function, the prior and the posterior
for the nuclear matter parameters
$\Theta=\\{\rho_{0},E_{0},K_{0},Q_{0},m^{*}/m,J_{0},L_{0}\\}$, given the data
$D$. The parameters in $\Theta$ are used to determine the seven coupling
parameters $g_{\sigma}$, $g_{\omega}$, $g_{\rho}$, $A$, $B$, $C$, and
$\Lambda_{v}$ (as elaborated in Section A of the supplemental material (SM)).
These coupling parameters, in turn, are used to determine finite nuclei
properties, the EOS, and subsequently, NS properties. The priors and
posteriors are listed in Table S1 of the SM. We use a Gaussian likelihood,
defined as,

$\mathcal{L}(D|\bm{\Theta})=\prod_{j}\frac{1}{\sqrt{2\pi\sigma^{2}_{j}}}e^{-\frac{1}{2}\left(\frac{d_{j}-m_{j}(\Theta)}{\sigma_{j}}\right)^{2}},$
(3)

where $m_{j}$, $d_{j}$, and $\sigma_{j}$ are the predicted value for a given
set of the model parameters, the observed value, and the uncertainty
respectively. The index $j$ runs over the experimental and astrophysical
observations. To incorporate the observational constraints of tidal
deformability [13], we use an asymmetric Gaussian [16, 18]. The Bayesian
analysis is conducted utilizing Python libraries of BILBY [40] and pymultinest
[41].

Results and Discussions
Following Tsang et al [18], we place the constraints on the symmetry energy
$J(\rho)$ (Eq.S4 of SM), pressure of symmetric nuclear matter $P_{SNM}(\rho)$
(Eq.S5 of SM), the symmetry pressure $P_{sym}(\rho)$ (Eq.S6 of SM), and the NS
observables such as radius, tidal deformability, and minimum-maximum mass. The
analyses of dipole polarizability of 208Pb [42], and isospin diffusion data
and single ratios of neutron-proton spectra [43, 44] provide us with empirical
constraints on $J(\rho)$ at densities $<0.5\rho_{0}$. At densities
$0.5-1.5\rho_{0}$, analyses of nuclear masses [45, 46, 47], PREX-II [4], and
pion spectra from heavy ion collisions (HICs) [48] have placed constraints on
$J(\rho)$ and $P_{sym}(\rho)$. The ISGMR [5] places the limits on the
incompressibility coefficient ($K_{0}$). $P_{SNM}(\rho)$ and $P_{sym}(\rho)$
are constrained by HICs at the densities $2\rho_{0}$ and $\sim 1.5\rho_{0}$.
The astrophysical observations on the minimum-maximum mass [8], radius [9, 10,
11, 12], and tidal deformability [13] of NS place constraints on the EOS for
densities beyond $2\rho_{0}$. We label the set of all the aforementioned
constraints as implicit finite nuclei constraints (FNC).

We then consider the scenario in which the EOSs are subjected to direct
experimental constraints from binding energies [2] and charge radii [3] for
40Ca and 208Pb, which can be computed from the Lagrangian (Eq.1) [31, 32]. We
incorporate these in place of the constraints from nuclear masses in the case
of implicit FNC. This set of constraints is labeled as explicit FNC. We
present a comprehensive overview of all the constraints, along with the
corresponding median and 68% confidence interval (CI) obtained from posterior
distributions for both scenarios, in Tables S3 and S4 of the SM.

Figure 1: The 68% CIs of symmetry energy, for the cases of implicit (red) and
explicit (violet) FNCs, in the density ranges between (a) 0.15 - 1$\rho_{0}$
and (b) 1 - 2$\rho_{0}$. The field symbols and the vertical lines indicate the
medians and 68% CIs of constraints from dipole polarizability (brown), nuclear
masses (black), and HICs (dark green).

Properties of finite nuclei
In Table 1, we present the median and 68% CI of binding energies and charge
rms radii, for some selected nuclei, obtained using the posterior
distributions of coupling parameters (listed in Table S2 of SM) resulting from
the implicit and explicit FNCs. The explicit FNC yields the binding energies
and charge radii closer to the corresponding measured values. It is evident
that the implicit FNC yields over-binding in the case of lighter nuclei and
under-binding for the heavier nuclei. The implicit FNC also yields smaller
charge rms radii. The deviation in medians of binding energies ranges from
about 6% for 16O nucleus to about 1% for 208Pb. On the contrary, employing
explicit FNC has led to a notable reduction in the deviation, about 1.5% for
16O, and less than 0.5% for all other nuclei under consideration.

Figure 2: Posterior distributions of nuclear matter parameters (NMPs) for the
two different scenarios of implicit and explicit FNC. The 1$\sigma$ CI is
displayed as vertical dashed lines in the marginalized posterior distributions
of the NMPs. All are in the units of MeV except for the saturation density
$\rho_{0}$ (in fm-3) and $m^{*}/m$ (dimensionless).

Symmetry Energy
Figs. 1a and 1b show 68% CI for the symmetry energy across the density ranges
of 0.15 - 1$\rho_{0}$ and 1 - 2$\rho_{0}$, respectively. The vertical dashed
lines represent the empirical constraints imposed on the symmetry energy at
different densities. The symmetry energy, $J(\rho)$, for the Lagrangian (Eq.
1), is as following, [29]

$J\left(\rho\right)=\frac{k^{2}_{f}}{6E^{*}_{f}}+\frac{1}{8}\left(\frac{g_{\rho}}{m^{*}_{\rho}}\right)^{2}\rho,$
(4)

where $E^{*}_{f}=\left(k^{2}_{f}+m^{*2}\right)^{1/2}$, and
$\left(m^{*}_{\rho}\right)^{2}=m^{2}_{\rho}+\Lambda_{v}g^{2}_{\omega}g^{2}_{\rho}\omega^{2}_{0}$
for Fermi momentum $k_{f}$, effective nucleon mass $m^{*}=m-g_{\sigma}\sigma$
and effective $\rho$-meson mass $m^{*}_{\rho}$. Evidently, the symmetry energy
distribution resulting from explicit FNC exhibits softer behavior at densities
$>0.5\rho_{0}$, in contrast to the behavior of those from the implicit FNC.
This distinction arises from incorporating the limits on binding energies and
charge radii explicitly in place of those on $E_{0}$, $\rho_{0}$, and
$J(\rho)$ as employed in the case of implicit FNC. The softness in symmetry
energy in the case of explicit FNC persists beyond 1.5$\rho_{0}$.
Nevertheless, the two scenarios are in similar agreement with respect to the
constraints from HICs and dipole polarizability at densities 0.2 -
0.55$\rho_{0}$. The medians and 68% CIs of the symmetry energy along with
those of $P_{SNM}(\rho)$, $K_{0}$ and $P_{sym}(\rho)$ obtained from the
posterior distribution of EOSs for both the scenarios are presented in Table
S3 of the SM.

Posterior Distribution of NMPs
The corner plots illustrating the marginalized posterior distributions for
both scenarios are presented in Figs.2. On the diagonals, the marginalized
posterior distributions for individual NMPs are depicted with vertical dashed
lines representing the 1$\sigma$ interval. The lower off-diagonal plots
display pairwise probability distributions with contours enclosing the
1$\sigma$, 2$\sigma$, and 3$\sigma$ CIs. The NMPs are better constrained for
the explicit FNC scenario, that exhibits a preference for lower values of the
parameters, with the exception of the incompressibility coefficient $K_{0}$,
skewness coefficient $Q_{0}$ and the curvature of symmetry energy $K_{sym,0}$,
when compared to the implicit FNC scenario. A NS of mass $\geq 2.08M_{\odot}$
[8] requires a stiff EOS at high densities. Consequently, the higher values of
$K_{0}$, $Q_{0}$, and $K_{sym,0}$ are needed to counterbalance the softness in
the symmetry energy observed for the case of explicit constraints (Fig.1).

The explicit FNC exhibit more inter-correlations, which are eventually
stronger, in comparison to those for the implicit FNC. It may be noted that
$J_{0}-K_{sym,0}$, $L_{0}-K_{sym,0}$, $K_{0}-E_{0}$, among other correlations,
show significant improvements in the explicit-FNC despite a narrowing down of
the variation in the parameters. $K_{sym,0}$ develops stronger correlations
with other parameters for the case of explicit FNC. For instance, the
correlation of $K_{sym,0}$ with $J_{0}$ becomes stronger (Pearson correlation
coefficient $r\sim-0.7$) in comparison to the negligible correlation observed
in the case of implicit FNC. Additionally, the nature of the correlation
between $K_{0}$ and $\rho_{0}$ transforms from a positive correlation ($r\sim
0.41$) in the implicit FNC to an anti-correlation ($r\sim-0.35$) in the
explicit FNC. The pairwise Pearson correlation coefficients ($r$) among the
NMPs, and the properties of the finite nuclei are shown in the Figs.S1 and S2
of the SM for both the scenarios.

Figure 3: (a) Plots for EOSs displaying a 95% CI for the scenarios with
implicit (red hatch) and explicit (violet hatch) finite nuclei constraints.(b)
The Kullback-Leibler divergence (DKL in bits) was obtained using the
distributions of EOSs for the two scenarios.

EOS and NS properties
Fig.3a shows the 95% CI for the posterior distributions of EOSs resulting from
explicit and implicit FNCs. In general, the explicit FNC displays a narrower
spread as compared to the implicit FNC. The two scenarios display distinct
behaviors at different densities. At low densities ($<\rho_{0}$), the EOSs
generated for the case of explicit constraints are softer than those for the
implicit constraints. The EOSs obtained from the former exhibit significant
stiffness at intermediate densities ($\sim$1.25-2.5$\rho_{0}$) in comparison
to those from the latter. At higher densities ($>$2.5$\rho_{0}$), the
distribution of the EOSs soften once more, overlapping with the distribution
of EOSs from implicit constraints.

In Fig.3b, we plot the values of Kullback-Leibler (KL) divergence ($D_{KL}$)
[49, 50], for a quantitative comparison of the distributions of the EOSs for
the two different scenarios considered. The $D_{KL}$ is calculated as,

$\displaystyle
D_{KL}\left(P||Q\right)=\int^{\infty}_{-\infty}p(x)log_{2}\left(\frac{p(x)}{q(x)}\right)dx,$

where P and Q represent the probability distributions for the pressure at a
given number density for explicit FNC and implicit FNC scenarios,
respectively. The statistical distance $D_{KL}$ quantifies the difference
between the distributions of the pressure at a given number density, for the
two different scenarios considered. The divergence, $D_{KL}$, approaches 0
when the distributions $p(x)$ and $q(x)$ are similar to each other. The value
of $D_{KL}$ is observed to vary across the density range. The values of KL
divergence is significant at the intermediate densities (around $\sim
2\rho_{0}$), indicating noticeable differences between the two distributions.
It approaches a minimum value at low ($\sim 0.8\rho_{0}$) and high densities
($2.75-4.25\rho_{0}$), indicating an evident overlap of these two
distributions at these densities (Fig.3a).

Figure 4: (a) The 95% CI neutron star mass-radius plots with implicit (red)
and explicit (violet) finite nuclei constraints. The radius inferred from
GW170817 (dark blue contours) [13], NICER observations (orange contours) for
PSR J0030+0451 ($\sim 1.44M_{\odot}$) [9, 11] and PSR J0740+6620 ($\sim
2.08M_{\odot}$)[10, 12] are also shown. (b) The Kullback-Leibler divergence
(DKL in bits) was obtained using the distributions of mass-radius for the two
scenarios. Figure 5: (a) The 95% CI neutron star mass-tidal deformability with
implicit (red hatch) and explicit (violet hatch) constraints. The canonical
tidal deformability from GW170817[13] is shown (horizontal black line). (b)
The Kullback-Leibler divergence (DKL in bits) was obtained using the
distributions of mass-tidal deformability for the two scenarios.

The 95 % CI of the posterior distributions of mass-radius of NS are plotted in
Fig.4a for both the implicit and explicit scenarios. The mass-radius
distributions corresponding to the EOSs from both explicit and implicit FNCs
exhibit distinct behavior across different mass ranges. Particularly, in the
case of explicit FNC, the median values of posterior radius distributions for
intermediate ($\sim 1.4M_{\odot}$ [9, 11]) and high-mass ($\sim 2.08M_{\odot}$
[10, 12]) NSs (as listed in Table S4 of the SM) tend to be larger by $\sim
0.4$ km. In Fig.4b, the KL divergence between the radius distributions of
explicit and implicit FNCs at different NS masses is shown. It varies across
the mass range, reaching a minimum value at $\sim 1.0M_{\odot}$. At both low
and high masses, the KL divergence reaches a value of $\sim 0.75$ bits.
However, despite these relatively low KL divergence values, the median values
of the radius distributions of the explicit and implicit scenarios show a
significant distinction for NSs of these masses.

In Fig.5a, the 95% CIs for dimensionless tidal deformability as a function of
NS mass are presented. The observational constraint (90% CI [13]) on the
canonical tidal deformability ($\Lambda_{1.4}$) aligns more closely with the
posteriors of implicit FNC than those of the explicit FNC. The values of KL
divergence, shown in Fig.5b, quantify the differences between the
distributions of tidal deformability for the two scenarios. The divergence
reaches a maximum value ($\sim 3$bits) at $\sim 1.5M_{\odot}$, indicating
significant differences between the distributions. This difference is less for
NSs with high and low masses, signifying an overlap of the distributions of
tidal deformability in these mass ranges. The posterior medians and 68% CIs of
NS radius and tidal deformability, for both cases, are listed in Table S4 of
the SM.

Figure 6: The correlation heat map between the nuclear matter parameters and
the neutron star properties for implicit finite nuclei constraints and
explicit finite nuclei constraints. The correlations with absolute values
$>0.3$ are displayed. The negative (positive) correlation is shown in the
boxes with shades of blue (red), with darker shades indicating a higher
negative (positive) correlation.

Finally, in Fig.6, the values of Pearson correlation coefficients ($r$)
between nuclear matter parameters and neutron star properties for both
scenarios are presented. The explicit FNC produces stronger correlations among
a larger number of pairs of NMPs and NS properties. The correlations involving
parameters $J_{0}$ and $L_{0}$, among others, show significant improvement.
For instance, $J_{0}$ \- $\Lambda_{1.4}$ correlation is strong ($r\sim 0.8$)
for explicit FNC as compared to the negligible correlation in the case of
implicit FNC. Additionally, $E_{0}$ and $\rho_{0}$ show weak correlations with
NS properties, which were absent in the case of implicit FNC. Interestingly,
the signs of the correlations in some cases are opposite for these two
scenarios. E.g., the $K_{sym,0}$ \- $\Lambda_{1.4}$ correlation. The stronger
correlations as depicted by the explicit FNC scenario would enable one to
place tighter bounds on the values of some of the NMPs from the precise
astrophysical observations of the NS properties.

We have also repeated our analysis for the RMF model without the inclusion of
vector self-coupling , i.e. $C=0$. The trends are qualitatively almost similar
to those obtained for the model with the $\omega$-meson self-coupling.
Regardless of the model under consideration, the explicit constraints yields
EOSs that are stiffer at intermediate densities (around $\sim 2\rho_{0}$) than
those obtained for implicit constraints. However, the stiffness of the EOSs
persists at higher densities for the model without the vector self coupling
(Fig S3 of the SM). Intermediate mass ($\sim 1.44M_{\odot}$) NSs, for the
model without vector self-coupling, exhibit smaller radius (by $\sim 0.5$ km)
as compared to the model incorporating the vector self-coupling. The radius of
higher NS mass ($\sim 2.08M_{\odot}$), on the other hand, is predicted to be
higher by the same amount. The results for the model without the vector self-
coupling is presented in section 3 of the SM. Nevertheless, irrespective of
the RMF model under consideration, explicit constraints on finite nuclei
properties, imposed at low densities, supplemented with astrophysical
constraints at high densities exhibit a global effect on the EOS and NS
properties.
Summary and outlook
We have confined the distributions of EOSs, derived from the relativistic
mean-field model, under two distinct scenarios regarding finite nuclei
constraints. One of the scenarios explicitly considers precisely measured
binding energies and charge radii for 40Ca and 208Pb nuclei, along with the
experimental data from heavy-ion collisions, other finite nuclei observables
and astrophysical observations of neutron stars, to establish posterior
distributions for the EOSs. We compared our results with the commonly employed
scenario in which finite nuclei constraints are imposed implicitly through
specific properties describing symmetric nuclear matter and the density
dependence of symmetry energy at sub-saturation densities. The two different
scenarios yield differing distributions of the NMPs as well as the
corresponding EOSs. The correlations among NMPs and NS properties are found to
be stronger in the case of explicit scenario. The distributions of EOSs for
the two different scenarios considered are employed to obtain the NS
properties. A quantitative comparison of the distributions of the EOSs and the
NS properties is performed through the Kullback-Leibler (KL) divergence.

It is observed that the implicit constraints significantly over bind the light
nuclei (up to $\sim 7\%$) and under bind the heavier nuclei (by $\sim 1\%$)
while also yielding smaller charge radii. The deviations, from the
experimentally measured values, in the case of explicit treatment of finite
nuclei constraints are significantly smaller. The EOSs resulting from it
exhibit considerable stiffness at intermediate densities ($1.25-2.5\rho_{0}$).
These features are quantitatively reflected in the values of KL divergence,
which is significant at these densities. The properties of NSs are also
significantly affected. The median values of the radius distributions for
explicit and implicit scenarios, at higher NS mass ($\sim 2.08M_{\odot}$) and
intermediate NS mass ($\sim 1.4M_{\odot}$), differ by $\sim 0.4$ km. This
difference is reflected at these masses by the values of KL divergence, which
also indicate similar differences at lower masses. The distinction between the
scenarios is further highlighted in the posterior distribution of tidal
deformability. The explicit FNC also predicts higher canonical tidal
deformability ($\Lambda_{1.4}$), as compared to the implicit FNC, displaying a
smaller overlap with the observational constraints on $\Lambda_{1.4}$ [13].
The maximum value of the KL divergence around $\sim 1.4M_{\odot}$ emphasizes
this difference between the two approaches.

The present investigation may be extended to include other finite nuclei
observables, such as iso-scalar and iso-vector giant resonances, explicitly
for a diverse set of nuclei to constraint the EOS.
Acknowledgements
AV would like to acknowledge the CSIR for the support through the CSIR-JRF
09/1026(16303)/2023-EMR-I.

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