# Morse index for the ground state in the energy supercritical
Gross–Pitaevskii equation

Dmitry E. Pelinovsky  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>and
Szymon Sobieszek  Department of Mathematics and Statistics, McMaster
University, Hamilton, Ontario, Canada, L8S 4K1<EMAIL_ADDRESS>
###### Abstract.

The ground state of the energy super-critical Gross–Pitaevskii equation with a
harmonic potential converges in the energy space to the singular solution in
the limit of large amplitudes. The ground state can be represented by a
solution curve which has either oscillatory or monotone behavior, depending on
the dimension of the system and the power of the focusing nonlinearity. We
address here the monotone case for the cubic nonlinearity in the spatial
dimensions $d\geq 13$. By using the shooting method for the radial Schrödinger
operators, we prove that the Morse index of the ground state is finite and is
independent of the (large) amplitude. It is equal to the Morse index of the
limiting singular solution, which can be computed from numerical
approximations. The numerical results suggest that the Morse index of the
ground state is one and that it is stable in the time evolution of the cubic
Gross–Pitaevskii equation in dimensions $d\geq 13$.

##  1\. Introduction

We consider the stationary Gross-Pitaevskii equation with a harmonic
potential,

$(-\Delta u+|x|^{2})u-|u|^{2p}u=\lambda u\,,$ (1.1)

where $x\in\mathbb{R}^{d}$, $\lambda\in\mathbb{R}$, and $u\in\mathbb{R}$.
Existence of its ground state (a positive and radially decreasing solution)
has been addressed before in the energy subcritical [7, 13], critical [20],
and supercritical [21, 22] regimes, where the critical exponent is
$p=\frac{2}{d-2}$ if $d\geq 3$. We are concerned here with the energy
supercritical case of the focusing Gross–Pitaevskii equation. Scattering in
the defocusing version of energy supercritical equations was studied in [15,
16, 17].

The stationary Gross–Pitaevskii equation (1.1) is the Euler–Lagrange equation
for the action functional $\Lambda_{\lambda}(u)=E(u)-\lambda M(u)$, where
$E(u)$ and $M(u)$ are the energy and mass given by

$M(w)=\int_{\mathbb{R}^{d}}|u|^{2}dx$ (1.2)

and

$E(w)=\int_{\mathbb{R}^{d}}\left(|\nabla
u|^{2}+|x|^{2}|u|^{2}-\frac{1}{p+1}|u|^{2p+2}\right)dx.$ (1.3)

The energy and mass are formally the conserved quantities in the evolution of
the time-dependent Gross–Pitaevskii equation. They are defined in the energy
space $\mathcal{E}\cap L^{2p+2}_{r}$, where

$\mathcal{E}:=\left\\{u\in L^{2}_{r}(\mathbb{R}^{+}):\quad u^{\prime}\in
L^{2}_{r}(\mathbb{R}^{+}),\quad ru\in L^{2}_{r}(\mathbb{R}^{+})\right\\}$
(1.4)

and $L^{q}_{r}$ denotes the space of radially symmetric
$L^{q}(\mathbb{R}^{d})$ functions.

The ground state of the stationary equation (1.1) can be obtained
variationally in the energy subcritical case $p(d-2)<2$, but the variational
methods are not applicable in the energy critical and supercritical cases
$p(d-2)\geq 2$ if $d\geq 3$. In all cases, the ground state forms a family
which appears as the curve on the $(\lambda,b)$ plane, where
$b:=u(0)\equiv\|u\|_{L^{\infty}}$ is referred to as the amplitude. The large-
amplitude limit is the limit as $b\to\infty$. In the energy supercritical
regime, it was proven in [22] that there exists $\lambda_{\infty}\in(0,d)$
such that $\lambda\to\lambda_{\infty}$ as $b\to\infty$ along the solution
curve. It was discovered in our recent work [1] that there exists $d_{*}(p)$
such that the solution curve is oscillatory for $2+\frac{2}{p}<d<d_{*}(p)$ and
monotone for $d>d_{*}(p)$.

For the sake of simplicity, we have been working with the cubic nonlinearity,
$p=1$, for which $d_{*}(p=1)=8+2\sqrt{6}$. We will continue working with the
cubic nonlinearity here. Figure 1.1 shows the dependence of $\lambda(b)$,
which is oscillatory for $5\leq d\leq 12$ and monotone for $d\geq 13$.

Figure 1.1. Graph of $\lambda$ versus $b$ for the ground state of the
boundary-value problem (1.5) for $d=5$ (left) and $d=13$ (right).

A similar duality between the oscillatory and monotone behaviors was
discovered for the classical Liouville–Bratu–Gerlfand problem in [12] and
explored in [2, 3, 4, 18] for the stationary focusing nonlinear Schrödinger
equation in a ball and without a harmonic potential. The similarity is
explained by the same linearization of the stationary equation near the origin
after the Emden–Fowler transformation [6]. Another example of the oscillatory
and monotone behaviors was considered in [5] for the Schrödinger–Newton–Hooke
model.

Let us define the ground state of the stationary equation (1.1) in radial
variable $r=|x|$ as a solution of the following boundary-value problem:

$\left\\{\begin{array}[]{ll}\mathfrak{u}^{\prime\prime}(r)+\frac{d-1}{r}\mathfrak{u}^{\prime}(r)-r^{2}\mathfrak{u}(r)+\lambda\mathfrak{u}(r)+\mathfrak{u}(r)^{3}=0,&r>0,\\\
\mathfrak{u}(r)>0,\qquad\qquad\mathfrak{u}^{\prime}(r)<0,&\\\
\lim\limits_{r\to
0}\mathfrak{u}(r)<\infty,\quad\lim\limits_{r\to\infty}\mathfrak{u}(r)=0.&\end{array}\right.$
(1.5)

Any solution of the boundary-value problem (1.5) belongs to $\mathcal{E}\cap
L^{\infty}$, since $\mathfrak{u}\in C^{2}(0,\infty)$ and $\mathfrak{u}(r)\to
0$ decays fast as $r\to\infty$.

As is well understood since the pioneering work in [12], the ground state of
the boundary-value problem (1.5) can be found from the family of solutions to
the following initial-value problem:

$\left\\{\begin{array}[]{ll}f_{b}^{\prime\prime}(r)+\frac{d-1}{r}f_{b}^{\prime}(r)-r^{2}f_{b}(r)+\lambda
f_{b}(r)+f_{b}(r)^{3}=0,&r>0,\\\ f_{b}(0)=b,\quad
f_{b}^{\prime}(0)=0,&\end{array}\right.$ (1.6)

where $b>0$ is an arbitrary parameter. By Theorem 1.1 in our previous work
[1], for any $b>0$ and $d\geq 4$, there exists some $\lambda\in(d-4,d)$ such
that the unique solution $f_{b}\in C^{2}(0,\infty)$ to the initial-value
problem (1.6) is monotonically decaying to zero as $r\to\infty$, making it a
ground state $\mathfrak{u}\equiv\mathfrak{u}_{b}\in\mathcal{E}\cap L^{\infty}$
of the boundary-value problem (1.5). That value of $\lambda$ is denoted as
$\lambda(b)$. The mapping $b\mapsto\lambda(b)$ defines a solution curve on the
$(\lambda,b)$ plane. Uniqueness of $\lambda(b)$ is an open problem for $d\geq
4$, whereas Figure 1.1 suggests that $\lambda(b)$ is unique for every $b>0$.

The purpose of this work is to study the Morse index of the ground state
$\mathfrak{u}_{b}$. It is defined as the number of negative eigenvalues of the
linearized operator $\mathcal{L}_{b}$ given by

$\mathcal{L}_{b}:=-\frac{d^{2}}{dr^{2}}-\frac{d-1}{r}\frac{d}{dr}+r^{2}-\lambda(b)-3\mathfrak{u}_{b}^{2}(r).$
(1.7)

Since $\mathcal{E}$ is the form domain of $\mathcal{L}_{b}$, we can write
$\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$, where $\mathcal{E}^{*}$
is the dual of $\mathcal{E}$ with respect to the scalar product in
$L^{2}_{r}$.

Assuming $C^{1}$ property of $\mathfrak{u}_{b}$ in $b$ and differentiating the
initial-value problem (1.6) with $\lambda=\lambda(b)$ in $b$, we can see that
$\mathcal{L}_{b}\partial_{b}\mathfrak{u}_{b}=\lambda^{\prime}(b)\mathfrak{u}_{b}$,
where $\partial_{b}\mathfrak{u}_{b}\in\mathcal{E}$. Hence, any value of $b$
for which $\lambda^{\prime}(b)=0$ corresponds to zero eigenvalue being in the
spectrum of $\mathcal{L}_{b}$ in $L^{2}_{r}$.

Although the converse is not known, this property implies that the oscillatory
case is very different from the monotone case, where the former has infinitely
many crossing of zero eigenvalue of $\mathcal{L}_{b}$ in the parameter
continuation in $b$ as $b\to\infty$ whereas the latter does not have any
eigenvalue crossing as $b\to\infty$, see also Figure 1.1. This suggests that
the Morse index should be well defined in the monotone case, independently of
$b$ for large values of $b$. This is in fact the main result which we
formulate as the following theorem.

######  Theorem 1.1.

For every $d\geq 13$, there exists $b_{0}>0$ such that the Morse index of
$\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ is finite and is
independent of $b$ for every $b\in(b_{0},\infty)$.

######  Remark 1.1.

Regarding the Morse index for the ground state in the energy supercrticial
case, we are only aware of the works [8, 14], where the Morse index was
estimated in the monotone case for the limiting singular solutions of the
Dirichlet problem in a ball. We believe that the conclusion of Theorem 1.1 and
the technique behind its proof remain valid for other problems in the monotone
case, e.g. for the nonlinear Schrödinger equation in a ball.

######  Remark 1.2.

By the Lyapunov–Schmidt reduction technique (see, e.g., [21]), the solution
curve satisfies $\lambda(b)\to d$ and $\mathfrak{u}_{b}\to 0$ as $b\to 0$,
where the Morse index of $\mathcal{L}_{b}$ in $L^{2}_{r}$ is equal to one. If
the Morse index is equal to one for $b>b_{0}$ in Theorem 1.1, then it is quite
possible that the Morse index remains one for every $b\in(0,\infty)$. Since
the ground state under these conditions is orbitally stable in the time
evolution of the Gross–Pitaevskii equation in $\mathcal{E}\cap L^{4}_{r}$ if
the mapping of $\lambda\mapsto\|\mathfrak{u}_{b}\|^{2}_{L^{2}_{r}}$ is
monotonically decreasing (see, e.g., Theorem 4.8 in [19]), it is rather
interesting to observe that the transition from the oscillatory case for
$5\leq d\leq 12$ to the monotone case $d\geq 13$ may enforce stability of the
ground state.

We shall now explain the strategy to prove Theorem 1.1. We use the limiting
singular solution $\mathfrak{u}_{\infty}\in\mathcal{E}\cap L^{4}_{r}$ which
exists for a particular value of $\lambda=\lambda_{\infty}$ if $d\geq 5$ [22].
It was also established in [22] that
$\mathfrak{u}_{b}\to\mathfrak{u}_{\infty}$ in $\mathcal{E}$ and
$\lambda(b)\to\lambda_{\infty}$ as $b\to\infty$. Uniqueness of
$\lambda_{\infty}$ is also an open problem, see the discussion in [1].

Figure 1.2 shows the ground state $\mathfrak{u}_{b}(r)$ for two values of $b$
and the limiting singular solution $\mathfrak{u}_{\infty}(r)$ for $d=13$. The
discrepancy between the two solutions moves to smaller values of $r$ if the
value of $b$ is increased. When $b=10$, the difference between
$\mathfrak{u}_{b}$ and $\mathfrak{u}_{\infty}$ becomes invisible on the scale
used in Figure 1.2.

Figure 1.2. Graph of the ground state $\mathfrak{u}_{b}$ for $b=1$ and $b=6$
in comparison with the limiting singular solution $\mathfrak{u}_{\infty}$ for
$d=13$.

Since the limiting singular solution $\mathfrak{u}_{\infty}$ satisfies the
following divergent behavior

$\mathfrak{u}_{\infty}(r)=\frac{\sqrt{d-3}}{r}\left[1+\mathcal{O}\left(r^{2}\right)\right],\quad\text{as
}r\to 0,$ (1.8)

it is natural to introduce $F_{\infty}(r):=r\mathfrak{u}_{\infty}(r)$ and
define it from the family of solutions to the following initial-value problem

$\left\\{\begin{array}[]{ll}F^{\prime\prime}(r)+\frac{d-3}{r}F^{\prime}(r)-\frac{d-3}{r^{2}}F(r)-r^{2}F(r)+\lambda
F(r)+\frac{1}{r^{2}}F(r)^{3}=0,&r>0,\\\ F(0)=\sqrt{d-3},\quad
F^{\prime}(0)=0.&\end{array}\right.$ (1.9)

By Theorem 1.2 in [1] (based on the earlier work in [22]), for any $d\geq 5$,
there exists some $\lambda_{\infty}\in(d-4,d)$ such that the unique solution
$F_{\infty}\in C^{2}(0,\infty)$ to the initial-value problem (1.9) is
monotonically decreasing to zero as $r\to\infty$, making it the limiting
singular solution $\mathfrak{u}_{\infty}\in\mathcal{E}\cap L^{4}_{r}$ by the
transformation $\mathfrak{u}_{\infty}(r)=r^{-1}F_{\infty}(r)$.

By Theorem 1.3 in [1], convergence of $\lambda(b)\to\lambda_{\infty}$ as
$b\to\infty$ is oscillatory for $5\leq d\leq 12$ and monotone for $d\geq 13$,
see Figure 1.1. The latter case is the only case we are interested in here.

In order to characterize the Morse index of $\mathcal{L}_{b}$, we use the
Emden–Fowler transformation [6] for the nonlinear equation in (1.9) and study
two families of solutions. One family is obtained from
$F_{b}(r):=r\mathfrak{u}_{b}(r)$ and is parametrized by its parameter $b$ from
the behavior as $r\to 0$. The other family is parametrized by another
parameter $c$ from the decaying behavior as $r\to\infty$. The second family is
considered in a local neighborhood of the limiting singular solution
$F_{\infty}(r)=r\mathfrak{u}_{\infty}(r)$. Both families have $C^{1}$ property
with respect to their parameters and their derivatives with respect to these
parameters are solutions of the homogeneous equation $\mathcal{L}_{b}v=0$
after the inverse Emden–Fowler transformation, e.g.,
$v(r)=r^{-1}\partial_{b}F_{b}(r)$. The proof of Theorem 1.1 is achieved from
the Sturm’s Oscillation Theorem (see, e.g., Theorem 3.5 in [23]) by showing
that the two derivatives have finitely many oscillations and there exists
$b_{0}>0$ such that the two derivatives are linearly independent for every
$b\in(b_{0},\infty)$.

As a by-product of our approach, we establish the equivalence of the Morse
index of $\mathcal{L}_{b}$ in $L^{2}_{r}$ with the Morse index of the limiting
operator $\mathcal{L}_{\infty}:=\lim\limits_{b\to\infty}\mathcal{L}_{b}$ which
is computed at the limiting singular solution for $d\geq 5$:

$\mathcal{L}_{\infty}:=-\frac{d^{2}}{dr^{2}}-\frac{d-1}{r}\frac{d}{dr}+r^{2}-\lambda_{\infty}-3\mathfrak{u}_{\infty}^{2}(r).$
(1.10)

Compared to $\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$, where the
potential $-3\mathfrak{u}_{b}^{2}(r)$ is bounded from below, the potential
$-3\mathfrak{u}_{\infty}^{2}(r)$ is unbounded from below, hence
$\mathcal{L}_{\infty}:\mathcal{E}_{\infty}\mapsto\mathcal{E}_{\infty}^{*}$,
where $\mathcal{E}_{\infty}=\\{u\in\mathcal{E}:\;r^{-1}u\in L^{2}_{r}\\}$ and
$\mathcal{E}_{\infty}^{*}$ is the dual of $\mathcal{E}_{\infty}$ with respect
to the scalar product in $L^{2}_{r}$.

The following theorem gives the precise result on the Morse index of the two
linear operators.

######  Theorem 1.2.

For every $d\geq 13$, there exists $b_{0}>0$ such that the Morse index of
$\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ coincides with the Morse
index of
$\mathcal{L}_{\infty}:\mathcal{E}_{\infty}\mapsto\mathcal{E}^{*}_{\infty}$.

######  Remark 1.3.

If the norm convergence of the resolvent for $\mathcal{L}_{b}$ to the
resolvent for $\mathcal{L}_{\infty}$ can be established as $b\to\infty$, this
would imply the result of Theorem 1.2. We do not study the norm convergence of
resolvents here as our methods are based on analysis of differential
equations.

######  Remark 1.4.

The result of Theorem 1.2 suggests a simple way to obtain the Morse index for
the ground state $\mathfrak{u}_{b}$ in the monotone case for large $b$ from
the Morse index for the limiting singular solution $\mathfrak{u}_{\infty}$.
The latter one can be approximated numerically with good accuracy.

Figure 1.3 shows uniquely normalized solutions $v(r)$ of $\mathcal{L}_{b}v=0$
with $b=1$ and $\mathcal{L}_{\infty}v=0$ such that $v(r)\to 0$ as
$r\to\infty$. Both solutions diverge as $r\to 0$ with different divergence
rates. Since there exists only one zero for each solution on $(0,\infty)$,
Sturm’s Oscillation Theorem (Theorem 3.5 in [23]) asserts that the Morse index
of both $\mathcal{L}_{b}$ and $\mathcal{L}_{\infty}$ in $L^{2}_{r}$ is equal
to one.

Figure 1.3. Graph of the uniquely normalized solutions $v(r)$ of
$\mathcal{L}_{b}v=0$ with $b=1$ and $\mathcal{L}_{\infty}v=0$ satisfying
$v(r)\to 0$ as $r\to\infty$ for $d=13$. Figure 1.4. Mass
$\|\mathfrak{u}_{b}\|^{2}_{L^{2}_{r}}$ of the ground state $\mathfrak{u}_{b}$
for $d=13$ as a function of $\lambda$ together with the mass
$\|\mathfrak{u}_{\infty}\|^{2}_{L^{2}_{r}}$ of the limiting singular solution
$\mathfrak{u}_{\infty}$.

By the Vakhitov–Kolokolov stability criterion (Theorem 4.8 in [19]), if the
Morse index is equal to one, the ground state $\mathfrak{u}_{b}$ is orbitally
stable in the time evolution of the Gross–Pitaevskii equation in
$\mathcal{E}\cap L^{4}_{r}$ for $b\in(b_{0},\infty)$ if the mapping of
$\lambda\mapsto\|\mathfrak{u}_{b}\|^{2}_{L^{2}_{r}}$ is monotonically
decreasing. Figure 1.4 shows the dependence of the mass
$M(\mathfrak{u}_{b})=\|\mathfrak{u}_{b}\|^{2}_{L^{2}_{r}}$ versus $\lambda$
for $\lambda=\lambda(b)$. The red dot depicts the finite value of the limiting
mass $M(\mathfrak{u}_{\infty})=\|\mathfrak{u}_{\infty}\|^{2}_{L^{2}_{r}}$.
Since the mapping is monotonically decreasing, the Vakhitov–Kolokolov
stability criterion asserts that the ground state $\mathfrak{u}_{b}$ is
orbitally stable for $d=13$ if the time evolution of the Gross–Pitaevskii
equation is locally well-posed in $\mathcal{E}\cap L^{4}_{r}$.

Note that the same conclusion holds for other values of $d$ in the monotone
case $d\geq 13$. We have also checked other values of $b$ and found no points
of bifurcations along the solution family $\lambda(b)$ where $\mathcal{L}_{b}$
admits zero eigenvalue in $L^{2}_{r}$. This suggests that the monotone
dependence of $\lambda(b)$ with no critical points, where
$\lambda^{\prime}(b)$ vanishes, implies no bifurcation points. This useful
property has not been proven in the literature.

The article is organized as follows. Section 2 gives the review of preliminary
results. Sections 3 and 4 contain estimates of the derivatives of the two
families of solutions described above. The proof of Theorems 1.1 and 1.2 is
described in Section 5.

##  2\. Preliminary results

We begin by introducing the Emden-Fowler transformation

$r=e^{t},\quad F(r)=\Psi(t),\quad F^{\prime}(r)=e^{-t}\Psi^{\prime}(t),$ (2.1)

which transforms the differential equation

$F^{\prime\prime}(r)+\frac{d-3}{r}F^{\prime}(r)-\frac{d-3}{r^{2}}F(r)-r^{2}F(r)+\lambda
F(r)+\frac{1}{r^{2}}F(r)^{3}=0,\quad r>0$ (2.2)

to the equivalent form

$\Psi^{\prime\prime}(t)+(d-4)\Psi^{\prime}(t)+(3-d)\Psi(t)+\Psi(t)^{3}=-\lambda
e^{2t}\Psi(t)+e^{4t}\Psi(t),\quad t\in\mathbb{R}.$ (2.3)

For fixed $d\geq 5$ and $\lambda\in(0,d)$, two one-parameter families of
solutions to the second-order differential equation (2.3) have been
constructed in [1] according to their asymptotic behaviors as $t\to-\infty$
and $t\to+\infty$, respectively.

The first family of solutions to the differential equation (2.3), denoted as
$\left\\{\Psi_{b}\right\\}_{b\in\mathbb{R}}$, corresponds to solutions of the
initial-value problem (1.6) after applying the transformation
$\Psi_{b}(t)=e^{t}f_{b}(e^{t})$. By Lemmas 3.2 and 3.4 in [1], $\Psi_{b}\in
C^{2}(\mathbb{R})$ satisfies the asymptotic behavior

$\Psi_{b}(t)=be^{t}-(\lambda
b+b^{3})(2d)^{-1}e^{3t}+\mathcal{O}(e^{5t}),\quad\text{as }t\to-\infty,$ (2.4)

where the expansion can be differentiated in $t$. These solutions depend on
$\lambda$ as well, and for $\lambda=\lambda(b)$ and $b>0$, $\Psi_{b}(t)$ gives
a solution to the boundary-value problem (1.1), after the transformation
$\mathfrak{u}_{b}(r)=r^{-1}\Psi_{b}(\log r)$. For other values of $\lambda$,
$\Psi_{b}(t)$ generally diverges as $t\to+\infty$.

The second family of solutions to the differential equation (2.3), denoted as
$\left\\{\Psi_{c}\right\\}_{c\in\mathbb{R}}$, decays to zero as $t\to+\infty$.
By Lemmas 3.3 and 3.4 in [1], $\Psi_{c}\in C^{2}(\mathbb{R})$ satisfies the
asymptotic behavior

$\Psi_{c}(t)\sim
ce^{\frac{\lambda-d+2}{2}t}e^{-\frac{1}{2}e^{2t}},\quad\text{as }t\to+\infty,$
(2.5)

where the sign $\sim$ denotes the asymptotic correspondence which can be
differentiated in $t$. Each $\Psi_{c}(t)$ generally diverges as $t\to-\infty$,
except when $\lambda=\lambda(b)$ and $c=c(b)$ for some value of $c(b)$ for
which it concides with $\Psi_{b}(t)=e^{t}\mathfrak{u}_{b}(e^{t})$:

$\lambda=\lambda(b):\quad\Psi_{b}(t)=\Psi_{c(b)}(t),\quad\mbox{for
all}\;\;t\in\mathbb{R}.$ (2.6)

Each family of solutions is differentiable with respect to parameters
$\lambda$ and either $b$ or $c$ due to smoothness of the differential equation
(2.3). Their derivatives decay to zero as $t\to-\infty$ and $t\to+\infty$
respectively, but generally diverge at the other infinities.

Let us define linearizations of the second-order equation (2.3) at the two
families of solutions:

$\displaystyle\mathcal{M}_{b}$ $\displaystyle:=$
$\displaystyle\frac{d^{2}}{dt^{2}}+(d-4)\frac{d}{dt}+(3-d)+3\Psi_{b}^{2}+\lambda
e^{2t}-e^{4t},$ (2.7) $\displaystyle\mathcal{M}_{c}$ $\displaystyle:=$
$\displaystyle\frac{d^{2}}{dt^{2}}+(d-4)\frac{d}{dt}+(3-d)+3\Psi_{c}^{2}+\lambda
e^{2t}-e^{4t}.$ (2.8)

Then, differentiating the second-order equation (2.3) with respect to $b$ and
$c$ at fixed $\lambda$ yields

$\displaystyle\mathcal{M}_{b}\partial_{b}\Psi_{b}=0,\quad\mathcal{M}_{c}\partial_{c}\Psi_{c}=0,$
(2.9)

where $\partial_{b}\Psi_{b}(t)\to 0$ as $t\to-\infty$ and
$\partial_{c}\Psi_{c}(t)\to 0$ as $t\to+\infty$.

The first family $\left\\{\Psi_{b}\right\\}_{b\in\mathbb{R}}$ is defined in a
neighborhood of a heteroclinic orbit $\Theta$ connecting the saddle point
$(0,0)$ and the stable point $(\sqrt{d-3},0)$ of the truncated autonomous
version of equation (2.3) given by

$\Theta^{\prime\prime}(t)+(d-4)\Theta^{\prime}(t)+(3-d)\Theta(t)+\Theta(t)^{3}=0.$
(2.10)

By Lemma 6.1 in [1], there exists a heteroclinic orbit between $(0,0)$ and
$(\sqrt{d-3},0)$ which is defined uniquely (module to the translation in $t$)
by the asymptotic behavior

$\Theta(t)=e^{t}-(2d)^{-1}e^{3t}+\mathcal{O}(e^{5t}),\quad\text{as
}t\to-\infty.$ (2.11)

The following proposition presents the main result of Lemmas 6.2, 6.5, and 6.8
in [1].

######  Proposition 2.1.

Fix $d\geq 5$ and $\lambda\in\mathbb{R}$. For every $T>0$ and $a\in(0,1)$,
there exist $(T,a)$-independent constants $b_{0}>0$, $C_{0}>0$ and
$(T,a)$-dependent constants $b_{T,a}>0$, $C_{T,a}>0$ such that the unique
solution $\Psi_{b}$ to the differential equation (2.3) with the asymptotic
behavior (2.4) satisfies for every $b\in(b_{0},\infty)$

$\sup_{t\in(-\infty,0]}|\Psi_{b}(t-\log b)-\Theta(t)|\leq C_{0}b^{-2}e^{3t}$
(2.12)

and for every $b\in(b_{T,a},\infty)$

$\sup_{t\in[0,T+a\log b]}|\Psi_{b}(t-\log b)-\Theta(t)|\leq
C_{T,a}b^{-2(1-a)}.$ (2.13)

The heteroclinic orbit of the truncated equation (2.10) connects the saddle
point $(0,0)$ associated with the characteristic exponents $\kappa_{1}=1$ and
$\kappa_{2}=3-d$ and the stable point $(\sqrt{d-3},0)$ associated with the
characteristic exponentis $\kappa_{+}$ and $\kappa_{-}$ given by

$\kappa_{\pm}=-\frac{1}{2}(d-4)\pm\frac{1}{2}\sqrt{d^{2}-16d+40}.$ (2.14)

For $d\geq 13$, the characteristic exponents are real and satisfy
$\kappa_{-}<\kappa_{+}<0$. We make the following assumption on how the
heteroclinic orbit converges to the stable point $(\sqrt{d-3},0)$.

######  Assumption 2.1.

Assume that there exists $A_{0}\neq 0$ such that

$\Theta(t)=\sqrt{d-3}+A_{0}e^{\kappa_{+}t}+\mathcal{O}(e^{\kappa_{-}t},e^{2\kappa_{+}t})\quad\mbox{\rm
as}\quad t\to+\infty.$ (2.15)

######  Remark 2.1.

Assumption 2.1 implies that $\Theta(t)$ converges to $\sqrt{d-3}$ as
$t\to+\infty$ according to the slowest decay rate given by $\kappa_{+}$. It is
not a priori clear why the constant $A_{0}$ could not be zero in exceptional
cases, for which $\Theta(t)$ converges to $\sqrt{d-3}$ as $t\to+\infty$
according to the fastest decay rate given by $\kappa_{-}$.

The second family $\left\\{\Psi_{c}\right\\}_{c\in\mathbb{R}}$ is defined in a
neighborhood of the special solution
$\Psi_{\infty}(t):=e^{t}\mathfrak{u}_{\infty}(e^{t})$ obtained from the
limiting singular solution $\mathfrak{u}_{\infty}\in\mathcal{E}\cap
L^{4}_{r}$. This special solution corresponds to the values of
$\lambda=\lambda_{\infty}$ and $c=c_{\infty}$ so that

$\lambda=\lambda_{\infty}:\quad\Psi_{\infty}(t)=\Psi_{c_{\infty}}(t),\quad\mbox{\rm
for all \;}t\in\mathbb{R}.$ (2.16)

The solution $\Psi_{\infty}$ satisfies the asymptotic behaviors

$\Psi_{\infty}(t)=\sqrt{d-3}\left[1-\frac{\lambda_{\infty}}{4d-10}e^{2t}+\mathcal{O}(e^{4t})\right],\quad\text{as
}t\to-\infty$ (2.17)

and

$\Psi_{\infty}(t)\sim
c_{\infty}e^{\frac{\lambda_{\infty}-d+2}{2}t}e^{-\frac{1}{2}e^{2t}},\quad\text{as
}t\to+\infty,$ (2.18)

The following proposition presents a modification of Lemmas 6.6 and 6.9 in
[1]. Since the statement was not proven in [1], we give the precise proof of
this result in Appendix A.

######  Proposition 2.2.

Fix $d\geq 13$ and $a\in(0,1)$. There exist constants $b_{0}>0$, $C_{0}>0$,
and $\epsilon_{0}>0$, such that for every $\epsilon\in(0,\epsilon_{0})$,
$b\in(b_{0},\infty)$, and $(\lambda,c)\in\mathbb{R}^{2}$ satisfying

$|\lambda-\lambda_{\infty}|+|c-c_{\infty}|\leq\epsilon b^{\kappa_{-}(1-a)},$
(2.19)

it is true for every $t\in[(a-1)\log b,0]$ that

$|\Psi_{c}(t)-\Psi_{\infty}(t)|\leq C_{0}\epsilon
b^{\kappa_{-}(1-a)}e^{\kappa_{-}t}.$ (2.20)

######  Remark 2.2.

Note that the divergent behavior of $e^{\kappa_{-}t}$ for large negative $t$
in (2.20) is cancelled by the decay of $b^{\kappa_{-}(1-a)}$ on any fixed
interval $[(a-1)\log b,0]$. Thus, bound (2.20) implies

$\sup_{t\in[(a-1)\log b,0]}|\Psi_{c}(t)-\Psi_{\infty}(t)|\leq C_{0}\epsilon,$
(2.21)

for every $(\lambda,c)\in\mathbb{R}^{2}$ satisfying (2.19).

######  Remark 2.3.

Since $\Psi_{c}$ is smooth in $\lambda$ and $c$ and has the same decay (2.5)
as $t\to+\infty$ in comparison with (2.18) for $\Psi_{\infty}$, it is true for
every $(\lambda,c)$ in a local neighborhood of $(\lambda_{\infty},c_{\infty})$
that

$\sup_{t\in[0,\infty)}|\Psi_{c}(t)-\Psi_{\infty}(t)|\leq
C_{0}(|\lambda-\lambda_{\infty}|+|c-c_{\infty}|).$ (2.22)

Linearization of the second-order equation (2.3) at $\Psi_{\infty}$ is given
by

$\mathcal{M}_{\infty}:=\frac{d^{2}}{dt^{2}}+(d-4)\frac{d}{dt}+(3-d)+3\Psi_{\infty}^{2}+\lambda_{\infty}e^{2t}-e^{4t}.$
(2.23)

Differentiating the second-order equation (2.3) with respect to $c$ at fixed
$\lambda$ and then substituting $c=c_{\infty}$ and $\lambda=\lambda_{\infty}$
gives

$\displaystyle\mathcal{M}_{\infty}\partial_{c}\Psi_{\infty}=0,$ (2.24)

where $\partial_{c}\Psi_{\infty}$ is a short notation for
$\partial_{c}\Psi_{c}|_{(\lambda,c)=(\lambda_{\infty},c_{\infty})}$. The
function $\partial_{c}\Psi_{\infty}(t)$ decays fast as $t\to+\infty$ according
to (2.5), but generally diverges as $t\to-\infty$. Since
$\Psi_{\infty}(t)\to\sqrt{d-3}$ as $t\to-\infty$, the divergence of
$\partial_{c}\Psi_{\infty}(t)$ as $t\to-\infty$ is defined by the same two
characteristic exponents $\kappa_{+}$ and $\kappa_{-}$ given by (2.14). We
make the following assumption on the divergence of this solution.

######  Assumption 2.2.

Assume that there exists $L_{\infty}\neq 0$ such that

$\partial_{c}\Psi_{\infty}(t)=L_{\infty}e^{\kappa_{-}t}+\mathcal{O}(e^{\kappa_{+}t},e^{(\kappa_{-}+2)t})\quad\mbox{\rm
as}\quad t\to-\infty.$ (2.25)

######  Remark 2.4.

Assumption 2.2 implies that $\partial_{c}\Psi_{\infty}(t)$ diverges as
$t\to-\infty$ with the fastest growth rate given by $\kappa_{-}$. Again, it is
not a priori clear why the constant $L_{\infty}$ could not be zero in
exceptional cases, for which $\partial_{c}\Psi_{\infty}(t)$ diverges as
$t\to-\infty$ with the slowest growth rate given by $\kappa_{+}$.

Figure 2.1. Graph of the solution $\Psi_{b}$ for $b=1$ and $b=6$ in comparison
with the solution $\Psi_{\infty}$ for $d=13$. Figure 2.2. Graph of
$\partial_{c}\Psi_{c(b)}$ with $b=1$ and $\partial_{c}\Psi_{\infty}$ for
$d=13$.

Figure 2.1 shows $\Psi_{b}(t)$ for two values of $b$ and $\Psi_{\infty}(t)$
for $d=13$. After the inverse Emden-Fowler transformation (2.1) and the
transformation $\mathfrak{u}(r)=r^{-1}F(r)$, these functions give
$\mathfrak{u}_{b}(r)$ and $\mathfrak{u}_{\infty}(r)$, as shown on Figure 1.2.

Figure 2.2 shows $\partial_{c}\Psi_{c(b)}(t)$ with $b=1$ and
$\partial_{c}\Psi_{\infty}(t)$ for $d=13$. These functions are solutions of
the homogeneous equaitons $\mathcal{M}_{b}\partial_{c}\Psi_{c(b)}=0$ and
$\mathcal{M}_{\infty}\partial_{c}\Psi_{\infty}=0$. After the transformation
$v(r)=r^{-1}\partial_{c}\Psi_{c(b)}(\log r)$, these functions give solutions
of $\mathcal{L}_{b}v=0$ and $\mathcal{L}_{\infty}v=0$ that decay to zero as
$r\to\infty$, as shown in Figure 1.3. Since $\partial_{c}\Psi_{c(b)}(t)$ and
$\partial_{c}\Psi_{\infty}(t)$ have only one zero on $\mathbb{R}$, the
corresponding functions $v(r)$ have only one zero on $(0,\infty)$, so that the
Morse index of $\mathcal{L}_{b}$ and $\mathcal{L}_{\infty}$ is equal to one.

##  3\. Derivative of the $b$-family of solutions

Here we describe the asymptotic behavior of $\partial_{b}\Psi_{b}$. The
following lemma shows that after translation by $-\log b$,
$\partial_{b}\Psi_{b}$ converges to $b^{-1}\Theta^{\prime}$ on the negative
$t$-axis. Moreover, the estimate can be extended from $(-\infty,0]$ to
$[0,T+a\log b]$ for fixed $T\in\mathbb{R}$ and $a\in(0,1)$ and for
sufficiently large $b$ at the expense of slower convergence rate.

######  Lemma 3.1.

Let $d\geq 13$ and $\lambda\in\mathbb{R}$. For every $T>0$ and $a\in(0,1)$,
there exist $(T,a)$-independent constants $b_{0}>0$, $C_{0}>0$ and
$(T,a)$-dependent constants $b_{T,a}>0$, $C_{T,a}>0$ such that
$\partial_{b}\Psi_{b}$ satisfies for every $b\in(b_{0},\infty)$

$\sup_{t\in(-\infty,0]}|\partial_{b}\Psi_{b}(t-\log
b)-b^{-1}\Theta^{\prime}(t)|+\sup_{t\in(-\infty,0]}|\partial_{b}\Psi_{b}^{\prime}(t-\log
b)-b^{-1}\Theta^{\prime\prime}(t)|\leq C_{0}b^{-3}$ (3.1)

and for every $b\in(b_{T,a},\infty)$

$\sup_{t\in\left[0,T+a\log b\right]}|\partial_{b}\Psi_{b}(t-\log
b)-b^{-1}\Theta^{\prime}(t)|+\sup_{t\in\left[0,T+a\log
b\right]}|\partial_{b}\Psi_{b}^{\prime}(t-\log
b)-b^{-1}\Theta^{\prime\prime}(t)|\leq C_{T,a}b^{-2(1-a)-1}.$ (3.2)

###### Proof.

We begin by introducing $\gamma(t):=\partial_{b}\Psi_{b}(t-\log
b)-b^{-1}\Theta^{\prime}(t)$, where $\Psi_{b}$ is the unique solution to (2.3)
with the asymptotic behavior (2.4) and $\Theta$ is the unique solution to
(2.10) with the asymptotic behavior (2.11). Since $\partial_{b}\Psi_{b}$
satisfies $\mathcal{M}_{b}\partial_{b}\Psi_{b}=0$ and $\Theta^{\prime}$
satisfies $\mathcal{M}_{0}\Theta^{\prime}=0$, where $\mathcal{M}_{b}$ is given
by (2.7) and

$\mathcal{M}_{0}:=\frac{d^{2}}{dt^{2}}+(d-4)\frac{d}{dt}+(3-d)+3\Theta^{2},$
(3.3)

the difference term $\gamma(t)$ satisfies the following equation:

$\mathcal{M}_{0}\gamma=f_{b}(b^{-1}\Theta^{\prime}+\gamma),$ (3.4)

where

$\displaystyle f_{b}(t):=3(\Theta(t)^{2}-\Psi_{b}(t-\log b)^{2})-\lambda
b^{-2}e^{2t}+b^{-4}e^{4t}.$ (3.5)

Note that $\mathcal{M}_{0}-f_{b}$ gives $\mathcal{M}_{b}$ after translation
$t\mapsto t+\log b$.

Proof of the bound (3.1). Two linearly independent solutions of
$\mathcal{M}_{0}\gamma=0$ are given by $\Theta^{\prime}(t)$ and another
function $\Xi(t)$, which can be found from the Wronskian relation

$W(\Theta^{\prime},\Xi)(t)=\Theta^{\prime}(t)\Xi^{\prime}(t)-\Theta^{\prime\prime}(t)\Xi(t)=W_{0}e^{(4-d)t},$
(3.6)

for some constant $W_{0}\neq 0$. We take $W_{0}=1$ in order to normalize
$\Xi(t)$ uniquely. Since $\Theta^{\prime}(t)\to 0$ as $t\to-\infty$ according
to the asymptotic expansion

$\Theta^{\prime}(t)=e^{t}+\mathcal{O}(e^{3t}),\quad\text{as }t\to-\infty,$
(3.7)

we have $\Xi(t)\to\infty$ as $t\to-\infty$ according to the asymptotic
expansion

$\Xi(t)=(2-d)^{-1}e^{(3-d)t}+\mathcal{O}(e^{(5-d)t}),\quad\text{as
}t\to-\infty.$ (3.8)

In order to estimate the supremum-norm of $\gamma(t)$ for $t\in(-\infty,0]$,
we first rewrite the differential equation (3.4) as an integral equation

$\gamma(t)=\int_{-\infty}^{t}e^{(d-4)t^{\prime}}[\Theta^{\prime}(t^{\prime})\Xi(t)-\Theta^{\prime}(t)\Xi(t^{\prime})]f_{b}(t^{\prime})[b^{-1}\Theta^{\prime}(t^{\prime})+\gamma(t^{\prime})]dt^{\prime},$
(3.9)

where the free solution $c_{1}\Theta^{\prime}(t)+c_{2}\Xi(t)$ is set to zero
from the requirement that $\gamma(t)=\mathcal{O}(e^{3t})$ as $t\to-\infty$.
The integral kernel in (3.9) becomes bounded if we introduce the
transformation $\tilde{\gamma}(t)=e^{-t}\gamma(t)$. The integral equation
corresponding to $\tilde{\gamma}$ is

$\tilde{\gamma}(t)=\int_{-\infty}^{t}K_{1}(t,t^{\prime})f_{b}(t^{\prime})[b^{-1}e^{-t^{\prime}}\Theta^{\prime}(t^{\prime})+\tilde{\gamma}(t^{\prime})]dt^{\prime},$
(3.10)

where

$K_{1}(t,t^{\prime}):=[e^{-t^{\prime}}\Theta^{\prime}(t^{\prime})][e^{(d-3)t}\Xi(t)]e^{(d-2)(t^{\prime}-t)}-[e^{-t}\Theta^{\prime}(t)][e^{(d-3)t^{\prime}}\Xi(t^{\prime})].$
(3.11)

Since $\Theta^{\prime}(t)=\mathcal{O}(e^{t})$ and
$\Xi(t)=\mathcal{O}(e^{(3-d)t})$ as $t\to-\infty$, the integral kernel
$K_{1}(t,t^{\prime})$ is a bounded function for all
$t,t^{\prime}\in(-\infty,0]$.

By using bound (2.12) of Proposition 2.1, we obtain from (3.5) that

$|f_{b}(t)|\leq C_{0}b^{-2}e^{2t},\quad t\in(-\infty,0],$ (3.12)

where the constant $C_{0}$ is independent of $b$ for sufficiently large $b$
and may change from one line to another line. Boundedness of $K_{1}$ in the
integral equation (3.10) on $(-\infty,0]\times(-\infty,0]$ and the estimate
(3.12) allow us to estimate the supremum norm of $\tilde{\gamma}(t)$ on
$(-\infty,0]$ as follows

$\|\tilde{\gamma}\|_{\infty}\leq
C_{0}b^{-2}(b^{-1}\|e^{-t}\Theta^{\prime}\|_{\infty}+\|\tilde{\gamma}\|_{\infty}).$
(3.13)

Due to smallness of $b^{-2}$ and boundness of $b$-independent
$\|e^{-t}\Theta^{\prime}\|_{\infty}$, this estimate implies that

$\sup_{t\in(-\infty,0]}|\tilde{\gamma}(t)|\leq C_{0}b^{-3}.$ (3.14)

Since $|\gamma(t)|\leq|\tilde{\gamma}(t)|$ for all $t\in(-\infty,0]$, we
obtain the first part of bound (3.1). Since $\tilde{\gamma}\in
C^{1}(-\infty,0)$, we obtain the second part of bound (3.1) by differentiting
equation (3.10) and using (3.14).

Proof of the bound (3.2). In order to estimate $|\gamma(t)|$ for sufficiently
large positive $t$, we need to define solutions to $\mathcal{M}_{0}\gamma=0$
from their behavior as $t\to+\infty$. Since $(\sqrt{d-3},0)$ is a stable node
of the nonlinear equation (2.10), we can pick two linearly independent
solutions $\gamma_{1}(t)$ and $\gamma_{2}(t)$ from their decaying behavior

$\gamma_{1}(t)=\mathcal{O}(e^{\kappa_{-}t}),\quad\gamma_{2}(t)=\mathcal{O}(e^{\kappa_{+}t})\quad\mbox{\rm
as}\;\;t\to+\infty,$ (3.15)

where $\kappa_{-}<\kappa_{+}<0$ are given by (2.14). The Liouville’s formula
yields the Wronskian relation

$W(\gamma_{1},\gamma_{2})(t)=\gamma_{1}(t)\gamma_{2}^{\prime}(t)-\gamma_{1}^{\prime}(t)\gamma_{2}(t)=W_{0}e^{(4-d)t},$
(3.16)

for some constant $W_{0}\neq 0$, and by normalizing $\gamma_{1}(t)$ and
$\gamma_{2}(t)$ we can assume that $W_{0}=1$. In order to derive supremum-norm
estimates for $\gamma(t)$, we once again rewrite differential equation (3.4)
as an integral equation

$\gamma(t)=\gamma(0)[\gamma_{1}(t)\gamma_{2}^{\prime}(0)-\gamma_{1}^{\prime}(0)\gamma_{2}(t)]+\gamma^{\prime}(0)[\gamma_{1}(0)\gamma_{2}(t)-\gamma_{1}(t)\gamma_{2}(0)]\\\
+\int_{0}^{t}e^{(d-4)t^{\prime}}[\gamma_{1}(t^{\prime})\gamma_{2}(t)-\gamma_{1}(t)\gamma_{2}(t^{\prime})]f_{b}(t^{\prime})[b^{-1}\Theta^{\prime}(t^{\prime})+\gamma(t^{\prime})]dt^{\prime},$
(3.17)

this time for $t\in[0,T+a\log b]$. From bound (3.1) we obtain existence of a
constant $C_{0}>0$ and $b_{0}>0$, such that

$|\gamma(0)|+|\gamma^{\prime}(0)|\leq C_{0}b^{-3},\quad\text{for all }b\geq
b_{0}.$ (3.18)

Due to the decay of $\gamma_{1}(t)$ and $\gamma_{2}(t)$ as $t\to+\infty$, the
kernel of the integral equation (3.17) behaves like
$e^{\kappa_{+}(t-t^{\prime})}$ and $e^{\kappa_{-}(t-t^{\prime})}$, and is thus
bounded as $t\to+\infty$ since $t^{\prime}\leq t$. By using bound (2.13) of
Proposition 2.1, we obtain from (3.5) that

$\sup_{t\in[0,T+a\log b]}|f_{b}(t)|\leq C_{T,a}b^{-2(1-a)},$ (3.19)

where the $b$-independent constant $C_{T,a}$ may change from one line to
another line. Using estimates (3.15), (3.18), and (3.19), we obtain from the
integral equation (3.17) the following bound on the supremum-norm of
$\gamma(t)$ on $[0,T+a\log b]$:

$\|\gamma\|_{\infty}\leq
C_{T,a}\left(b^{-3}+b^{-2(1-a)-1}\|\Theta^{\prime}\|_{L^{1}}+(T+a\log
b)b^{-2(1-a)}\|\gamma\|_{\infty}\right).$ (3.20)

Due to smallness of $(T+a\log b)b^{-2(1-a)}$ and boundedness of
$b$-independent $\|\Theta^{\prime}\|_{L^{1}}$, this estimate implies that

$\sup_{t\in[0,T+a\log b]}|\gamma(t)|\leq C_{T,a}b^{-2(1-a)-1}.$ (3.21)

By differentiting equation (3.17) and using (3.21), we obtain a similar bound
on $\gamma^{\prime}(t)$, which together with (3.21) gives us bound (3.2). ∎

Bound (3.2) of Lemma 3.1 and the expansion (2.15) imply the following
important representation of $\partial_{b}\Psi_{b}(t)$ at $t=T+(a-1)\log b$.

######  Corollary 3.1.

Under Assumption 2.1, there exist some constant $a_{0}\in(0,0.5)$ such that
for every $a\in(0,a_{0})$ and $T>0$, there exist some $(T,a)$-dependent
constants $b_{T,a}>0$ and $C_{T,a}>0$ such that for every
$b\in(b_{T,a},\infty)$ we have

$|\partial_{b}\Psi_{b}(T+(a-1)\log
b)-A_{0}\kappa_{+}b^{a\kappa_{+}-1}e^{\kappa_{+}T}|\leq
C_{T,a}\max\left\\{b^{-2(1-a)-1},b^{a\kappa_{-}-1},b^{2a\kappa_{+}-1}\right\\}.$
(3.22)

###### Proof.

Bound (3.22) follows from the expansion (2.15) at large positive $t=T+a\log
b$, the bound (3.2) at $t=T+(a-1)\log b$, and the triangle inequality if

$b^{a\kappa_{+}-1}\gg b^{-2(1-a)-1}.$

This constraint is satisfied if $a\in(0,a_{0})$, where

$a_{0}:=\frac{2}{2+|\kappa_{+}|}=\frac{4}{d-\sqrt{d^{2}-16d+40}}=\frac{d+\sqrt{d^{2}-16d+40}}{2(2d-5)}.$
(3.23)

Note that $a_{0}\in(0,0.5)$ for every $d\geq 13$. ∎

######  Remark 3.1.

Lemma 3.1 suggests that both bounds (2.12) and (2.13) of Proposition 2.1 can
be differentiated in $b$ after translation: $t\mapsto t+\log b$. This $C^{1}$
property would also follow from applications of Banach fixed-point theorem in
[1] due to contraction of integral operators and smoothness of the vector
fields. Since the $C^{1}$ property was not written explicitly in [1], we
provided the precise proof of Lemma 3.1.

##  4\. Derivative of the $c$-family of solutions

Here we describe the asymptotic behavior of $\partial_{c}\Psi_{c}$. Since
$\Psi_{c}$ is smooth in $\lambda$ and $c$ and has the same decay as
$t\to+\infty$ as the limiting solution $\Psi_{\infty}$ according to (2.5) and
(2.18), $\partial_{c}\Psi_{c}$ converges to $\partial_{c}\Psi_{\infty}$ on
$[0,\infty)$. To be precise, there exists a constant $C_{0}>0$ such that for
every $(\lambda,c)$ in a local neighborhood of
$(\lambda_{\infty},c_{\infty})$, we have

$\sup_{t\in[0,\infty)}|\partial_{c}\Psi_{c}(t)-\partial_{c}\Psi_{\infty}(t)|+\sup_{t\in[0,\infty)}|\partial_{c}\Psi_{c}^{\prime}(t)-\partial_{c}\Psi_{\infty}^{\prime}(t)|\leq
C_{0}(|\lambda-\lambda_{\infty}|+|c-c_{\infty}|).$ (4.1)

The following lemma extends the estimate on the difference
$|\partial_{c}\Psi_{c}(t)-\partial_{c}\Psi_{\infty}(t)|$ from $[0,\infty)$ to
$[(a-1)\log b,0]$ for fixed $a\in(0,1)$ and for sufficiently large $b$
provided that $(\lambda,c)$ are sufficiently close to
$(\lambda_{\infty},c_{\infty})$.

######  Lemma 4.1.

Fix $d\geq 13$. For fixed $a\in(0,1)$, there exist $b_{0}>0$, $C_{0}>0$, and
$\epsilon_{0}>0$, such that for every $\epsilon\in(0,\epsilon_{0})$ and for
every $(\lambda,c)\in\mathbb{R}^{2}$ satisfying

$|\lambda-\lambda_{\infty}|+|c-c_{\infty}|\leq\epsilon b^{\kappa_{-}(1-a)},$
(4.2)

it is true for every $b\geq b_{0}$ and every $t\in[(a-1)\log b,0]$ that

$|\partial_{c}\Psi_{c}(t)-\partial_{c}\Psi_{\infty}(t)|+|\partial_{c}\Psi_{c}^{\prime}(t)-\partial_{c}\Psi_{\infty}^{\prime}(t)|\leq
C_{0}\epsilon e^{\kappa_{-}t}.$ (4.3)

###### Proof.

Let $r(t):=\partial_{c}\Psi_{c}(t)-\partial_{c}\Psi_{\infty}(t)$. Since
$\partial_{c}\Psi_{c}$ satisfies $\mathcal{M}_{c}\partial_{c}\Psi_{c}=0$ and
$\partial_{c}\Psi_{\infty}$ satisfies
$\mathcal{M}_{\infty}\partial_{c}\Psi_{\infty}=0$, the difference term $r$
satisfies the following equation:

$\mathcal{M}_{\infty}r=f_{c}+g_{c}r,$ (4.4)

where

$\displaystyle f_{c}(t)$
$\displaystyle:=3(\Psi_{\infty}(t)^{2}-\Psi_{c}(t)^{2})\partial_{c}\Psi_{\infty}(t)-(\lambda-\lambda_{\infty})e^{2t}\partial_{c}\Psi_{\infty}(t),$
(4.5) $\displaystyle g_{c}(t)$
$\displaystyle:=3(\Psi_{\infty}(t)^{2}-\Psi_{c}(t)^{2})+(\lambda_{\infty}-\lambda)e^{2t}.$
(4.6)

Note that $\mathcal{M}_{c}=\mathcal{M}_{\infty}-g_{c}$.

As in Appendix A, we pick two linearly independent solutions $r_{1}$, $r_{2}$
to $\mathcal{M}_{\infty}r=0$ such that

$r_{1}(t)=\mathcal{O}(e^{\kappa_{-}t}),\quad
r_{2}(t)=\mathcal{O}(e^{\kappa_{+}t}),\quad\text{as }t\to-\infty,$ (4.7)

where $\kappa_{-}<\kappa_{+}<0$ are given by (2.14). Using the method of
variation of parameters, we rewrite the differential equation (4.4) as an
integral equation for every $t\in[(a-1)\log b,0]$:

$r(t)=r(0)[r_{1}(t)r_{2}^{\prime}(0)-r_{1}^{\prime}(0)r_{2}(t)]+r^{\prime}(0)[r_{1}(0)r_{2}(t)-r_{1}(t)r_{2}(0)]\\\
+\int_{t}^{0}e^{(d-4)t^{\prime}}[r_{1}(t)r_{2}(t^{\prime})-r_{1}(t^{\prime})r_{2}(t)][f_{c}(t^{\prime})+g_{c}(t^{\prime})r(t^{\prime})]dt^{\prime},$
(4.8)

where we have used the normalization of the Wronskian
$W(r_{1},r_{2})(t)=e^{-(d-4)t}$ between the two solutions $r_{1}$ and $r_{2}$
as in (A.3).

In order to elliminate the divergent behavior of the kernel in (4.8) as
$t\to-\infty$, we introduce the transformation
$\tilde{r}(t)=e^{-\kappa_{-}t}r(t)$, which results in the following integral
equation for $\tilde{r}$:

$\tilde{r}(t)=r(0)e^{-\kappa_{-}t}[r_{1}(t)r_{2}^{\prime}(0)-r_{1}^{\prime}(0)r_{2}(t)]+r^{\prime}(0)e^{-\kappa_{-}t}[r_{1}(0)r_{2}(t)-r_{1}(t)r_{2}(0)]\\\
+\int_{t}^{0}K_{2}(t,t^{\prime})[e^{-\kappa_{-}t^{\prime}}f_{c}(t^{\prime})+g_{c}(t^{\prime})\tilde{r}(t^{\prime})]dt^{\prime},$
(4.9)

where the kernel $K_{2}(t,t^{\prime})$ is the same as in (A.6):

$K_{2}(t,t^{\prime}):=e^{\kappa_{-}(t^{\prime}-t)-(\kappa_{+}+\kappa_{-})t^{\prime}}\left[r_{1}(t)r_{2}(t^{\prime})-r_{1}(t^{\prime})r_{2}(t)\right].$
(4.10)

The kernel is bounded for every $-\infty<t\leq t^{\prime}\leq 0$ as in (A.7).
It follows from (4.1) that

$|r(0)|+|r^{\prime}(0)|\leq
C_{0}(|\lambda-\lambda_{\infty}|+|c-c_{\infty}|)\leq C_{0}\epsilon
b^{\kappa_{-}(1-a)},$ (4.11)

where the $(\epsilon,b)$-independent constant $C_{0}$ can change from one line
to another line. It follows from the expansion (2.25) in Assumption 2.2 that
$\partial_{c}\Psi_{\infty}(t)=\mathcal{O}(e^{\kappa_{-}t})$ as $t\to-\infty$.
Therefore, we get by using bounds (2.20) and (2.21):

$\int_{t}^{0}e^{-\kappa_{-}t^{\prime}}|f_{c}(t^{\prime})|dt^{\prime}\leq
C_{0}\left(\epsilon b^{\kappa_{-}(1-a)}\int_{(a-1)\log
b}^{0}e^{\kappa_{-}t^{\prime}}dt^{\prime}+|\lambda-\lambda_{\infty}|\right)\leq
C_{0}\epsilon.$ (4.12)

On the other hand, for every $\tilde{r}\in L^{\infty}((a-1)\log b,0)$, we get
by using bounds (2.20) and (2.21):

$\int_{(a-1)\log b}^{0}|g_{c}(t^{\prime})\tilde{r}(t^{\prime})|dt^{\prime}\leq
C_{0}\epsilon||\tilde{r}||_{\infty}.$ (4.13)

Putting estimates (4.7), (4.11), (4.12), and (4.13) together in the integral
equation (4.9) yields

$\sup_{t\in[(a-1)\log b,0]}|\tilde{r}(t)|\leq C_{0}\epsilon,$ (4.14)

which is the first part of bound (4.3) after going back to the original
variable $r(t)$. The second part of bound (4.3) is obtained by differentiating
(4.9) in $t$ and using bound (4.14). ∎

Bound (4.3) of Lemma 4.1 and the expansion (2.25) imply the following
important representation of $\partial_{c}\Psi_{c}(t)$ at $t=T+(a-1)\log b$.

######  Corollary 4.1.

Under Assumption 2.2, for every $a\in(0,1)$ and $T>0$, there exist some
$(T,a)$-dependent constants $b_{T,a}>0$ and $C_{T,a}>0$ such that for every
$b\in(b_{T,a},\infty)$ we have

$\displaystyle|\partial_{c}\Psi_{c}(T+(a-1)\log
b)-L_{\infty}e^{\kappa_{-}T}b^{-\kappa_{-}(1-a)}|\leq C_{T,a}\max\\{\epsilon
b^{-\kappa_{-}(1-a)},b^{-\kappa_{+}(1-a)},b^{-(2+\kappa_{-})(1-a)}\\}.$ (4.15)

###### Proof.

Bound (4.15) follows from the bound (4.3) at $t=T+(a-1)\log b$ for fixed
$T>0$, $a\in(0,1)$, and sufficiently large $b>0$ after
$\partial_{c}\Psi_{\infty}(t)$ for large negative $t$ is expressed from the
expansion (2.25). ∎

######  Remark 4.1.

Lemma 4.1 can be obtained from the $C^{1}$ property of $\Psi_{c}$ in
$(\lambda,c)$ after some transformations. It follows from the proof of
Proposition 2.2 in Appendix A that

$|\Psi_{c}(t)-\Psi_{\infty}(t)|=\mathcal{O}(\epsilon)\quad t\in[(a-1)\log
b,0],$

and the asymptotic expansion can be differentiated in $\epsilon$. Parameter
$\epsilon$ determines the size of the distance $|\lambda-\lambda_{\infty}|$
and $|c-c_{\infty}|$ so that we can write $c-c_{\infty}=\mathcal{O}(\epsilon
b^{-\kappa_{-}(1-a)})$ and differentiate it in $\epsilon$. By taking
derivative in $c$ and using the chain rule, this yields

$|\partial_{c}\Psi_{c}(t)-\partial_{c}\Psi_{\infty}(t)|=\mathcal{O}(\epsilon
b^{-\kappa_{-}(1-a)})\quad t\in[(a-1)\log b,0],$

which is equivalent to the bound (4.3). Since taking derivatives in $c$ and
using the chain rule are not obvious from the proof of Proposition 2.2, we
gave the precise proof of Lemma 4.1.

##  5\. Proofs of the main results

We recall that $\mathcal{M}_{b}=\mathcal{M}_{c(b)}$ for $\lambda=\lambda(b)$
since $\Psi_{b}(t)=\Psi_{c(b)}(t)$ for every $t\in\mathbb{R}$. Hence, both
$\partial_{b}\Psi$ and $\partial_{c}\Psi_{c(b)}$ are solutions of the same
homogeneous equation $\mathcal{M}_{b}\gamma=0$ for $\lambda=\lambda(b)$. The
following lemma shows that these two solutions are linearly independent for
sufficiently large values of $b$.

######  Lemma 5.1.

Fix $d\geq 13$. Under Assumptions 2.1 and 2.2, for every $a\in(0,a_{0})$ with
$a_{0}$ given by (3.23), there exists $b_{0}>0$ such that for every
$b\in(b_{0},\infty)$, there exists no $C\in\mathbb{R}$ such that

$C\partial_{c}\Psi_{c(b)}(t)=\partial_{b}\Psi_{b}(t),\quad\text{for all
}t\in\mathbb{R}.$ (5.1)

###### Proof.

In order to get a contradiction, suppose that relation (5.1) holds for some
constant $C\in\mathbb{R}$. The results of Corollaries 3.22 and 4.1 apply for
$t=T+(a-1)\log b$ for fixed $a\in(0,a_{0})$, $T>0$, and sufficiently large
$b$. Substituting bounds (3.22) and (4.15) into (5.1) yields

$CL_{\infty}e^{\kappa_{-}T}b^{-\kappa_{-}(1-a)}\left[1+\mathcal{O}(\epsilon,b^{-(\kappa_{+}-\kappa_{-})(1-a)},b^{-2(1-a)})\right]\\\
=A_{0}\kappa_{+}b^{a\kappa_{+}-1}e^{\kappa_{+}T}\left[1+\mathcal{O}(b^{-2(1-a)-a\kappa_{+}},b^{-a(\kappa_{+}-\kappa_{-})},b^{a\kappa_{+}})\right],$
(5.2)

where we recall that $2(1-a)+a\kappa_{+}>0$ if $a\in(0,a_{0})$, where $a_{0}$
is given by (3.23) in Corollary 3.22. Since $A_{0}\neq 0$ and $L_{\infty}\neq
0$ by Assumptions 2.1 and 2.2, we obtain from (5.2) that

$C\left[1+\mathcal{O}(\epsilon,b^{-(\kappa_{+}-\kappa_{-})(1-a)},b^{-2(1-a)})\right]\\\
=L_{\infty}^{-1}A_{0}\kappa_{+}b^{a(\kappa_{+}-\kappa_{-})+\kappa_{-}-1}e^{(\kappa_{+}-\kappa_{-})T}\left[1+\mathcal{O}(b^{-2(1-a)-a\kappa_{+}},b^{-a(\kappa_{+}-\kappa_{-})},b^{a\kappa_{+}})\right].$
(5.3)

Since the remainder terms on both sides of (5.3) are smaller than the leading-
order terms and $\kappa_{+}\neq\kappa_{-}$, this gives a $T$-dependent
coefficient $C$, which is a contradiction with the relation (5.1) for all
$t\in\mathbb{R}$ and hence for all $T>0$. ∎

From Lemma 5.1, we can now prove Theorem 1.1 which states that the Morse index
of $\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ is finite and is
independent of $b$ for every $b\in(b_{0},\infty)$.

  * Proof of Theorem 1.1 For every $b\in(0,\infty)$, the potential $-3\mathfrak{u}^{2}_{b}(r)$ in $\mathcal{L}_{b}$ is bounded from below on $[0,\infty)$. The Schrödinger operator $-\Delta_{r}+r^{2}:\mathcal{E}\mapsto\mathcal{E}^{*}$ is strictly positive with a purely discrete spectrum. Since $\mathcal{L}_{b}=-\Delta_{r}+r^{2}-\lambda(b)-3\mathfrak{u}_{b}^{2}(r)$ is bounded from below, the number of negative eigenvalues (the Morse index) of $\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ is finite by Theorem 10.7 in [11].

It remains to show that the Morse index is independent of $b$ for every
$b\in(b_{0},\infty)$. Let us recall the Emden–Fowler transformation (2.1),
which relates solutions of $\mathcal{L}_{b}v=0$ with solutions of
$\mathcal{M}_{b}\gamma=0$ by $v(r)=r^{-1}\gamma(\log r)$. The spectrum of
$\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ includes the zero
eigenvalue if and only if there exists $v\in\mathcal{E}$ satisfying
$\mathcal{L}_{b}v=0$. This is impossible due to Lemma 5.1 according to the
following arguments.

As $t\to-\infty$, there exist two linearly independent solutions to
$\mathcal{M}_{b}\gamma=0$ and the decaying solution is

$\partial_{b}\Psi_{b}(t)=e^{t}+\mathcal{O}(e^{3t}),\quad\mbox{\rm as
\;\;}t\to-\infty.$

The other solution is growing as $e^{(3-d)t}$ which corresponds to $v(r)\sim
r^{2-d}$ so that

$r^{d-1}|v(r)|^{2}\sim r^{3-d}$

is not integrable near $r=0$ for $d\geq 4$. Hence, the corresponding $v(r)$ is
not in $L^{2}_{r}$ and if there exists nonzero $v\in\mathcal{E}$ satisfying
$\mathcal{L}_{b}v=0$, then there exists a constant $C_{-}\neq 0$ such that

$v(r)=C_{-}r^{-1}\partial_{b}\Psi_{b}(\log r).$

As $t\to+\infty$, there exist two linearly independent solutions to
$\mathcal{M}_{b}\gamma=0$ and the decaying solution is

$\partial_{c}\Psi_{c(b)}(t)\sim
e^{\frac{\lambda(b)-d+2}{2}}e^{-\frac{1}{2}e^{2t}},\quad\mbox{\rm as
\;\;}t\to\infty.$

The other solution is growing as $e^{\frac{1}{2}e^{2t}}$, which corresponds to
$v(r)\sim e^{\frac{1}{2}r^{2}}$, clearly not in $L^{2}_{r}$. If there exists
nonzero $v\in\mathcal{E}$ satisfying $\mathcal{L}_{b}v=0$, then there exists a
constant $C_{+}\neq 0$ such that

$v(r)=C_{+}r^{-1}\partial_{c}\Psi_{c(b)}(\log r).$

Since $C_{-},C_{+}\neq 0$, if there exists nonzero $v\in\mathcal{E}$, then
$\partial_{b}\Psi_{b}$ and $\partial_{c}\Psi_{c(b)}$ are linearly dependent,
which results in a contradiction with Lemma 5.1 for every
$b\in(b_{0},\infty)$. Hence $0\notin\sigma(\mathcal{L}_{b})$ for every
$b\in(b_{0},\infty)$ so that the Morse index of
$\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ is independent of $b$ for
every $b\in(b_{0},\infty)$.

By Lemma 4.1, $\partial_{c}\Psi_{c}(t)$ converges to
$\partial_{c}\Psi_{\infty}$ on $[(a-1)\log b,\infty)$ as
$(\lambda,c)\to(\lambda_{\infty},c_{\infty})$. Each zero of either
$\partial_{c}\Psi_{c}$ or $\partial_{c}\Psi_{\infty}$ is simple since they are
solutions of the second-order linear homogeneous equations
$\mathcal{M}_{c}\partial_{c}\Psi_{c}=0$ and
$\mathcal{M}_{\infty}\partial_{c}\Psi_{\infty}=0$. Consequently, the number of
nodal domains of $\partial_{c}\Psi_{c}$ in $[(a-1)\log b,\infty)$ coincides
with that of $\partial_{c}\Psi_{\infty}$ in $[(a-1)\log b,\infty)$.

The following lemma shows that $\partial_{c}\Psi_{c(b)}$ does not have
additional nodal domains in the interval $(-\infty,(a-1)\log b)$ for
sufficiently large $b$.

######  Lemma 5.2.

Fix $d\geq 13$. Under Assumption 2.2, for every $a\in(0,1)$, there exists
$b_{0}>0$ such that for every $b\in(b_{0},\infty)$, there exists $C_{0}>0$
such that

$|\partial_{c}\Psi_{c(b)}(t)|\geq C_{0},\quad t\in(-\infty,(a-1)\log b).$
(5.4)

###### Proof.

Recall that $\mathcal{M}_{b}=\mathcal{M}_{c(b)}$ for $\lambda=\lambda(b)$ and
$\mathcal{M}_{b}\partial_{b}\Psi_{b}=0$ with

$\partial_{b}\Psi_{b}(t)=e^{t}+\mathcal{O}(e^{3t}),\quad\mbox{\rm
as}\;\;t\to-\infty.$ (5.5)

Similar to the proof of Lemma 3.1, we denote the second linearly independent
solution of $\mathcal{M}_{b}\gamma=0$ by $\Phi$ and normalize it such that

$\Phi(t)=(2-d)^{-1}e^{(3-d)t}+\mathcal{O}(e^{(5-d)t}),\quad\text{as
}t\to-\infty.$ (5.6)

We are interested in the behavior of $\partial_{c}\Psi_{c(b)}$ for
$t\in(-\infty,t_{0})$, where $t_{0}:=(a-1)\log b$. It follows from (2.25) and
(4.3) that for sufficiently large $b$, we have

$\displaystyle\partial_{c}\Psi_{c(b)}(t_{0})$
$\displaystyle=L_{\infty}e^{\kappa_{-}t_{0}}\left[1+\mathcal{O}\left(\epsilon,e^{(\kappa_{+}-\kappa_{-})t_{0}},e^{2t_{0}}\right)\right],$
(5.7) $\displaystyle\partial_{c}\Psi^{\prime}_{c(b)}(t_{0})$
$\displaystyle=L_{\infty}\kappa_{-}e^{\kappa_{-}t_{0}}\left[1+\mathcal{O}\left(\epsilon,e^{(\kappa_{+}-\kappa_{-})t_{0}},e^{2t_{0}}\right)\right].$
(5.8)

Since $\partial_{c}\Psi_{c(b)}$ is a linear combination of
$\partial_{b}\Psi_{b}$ and $\Phi$ by the linear superposition principle, we
can express $\partial_{c}\Psi_{c(b)}$ as

$\partial_{c}\Psi_{c(b)}(t)=e^{(d-4)t_{0}}\big{[}(\partial_{c}\Psi_{c(b)}(t_{0})\Phi^{\prime}(t_{0})-\partial_{c}\Psi^{\prime}_{c(b)}(t_{0})\Phi(t_{0}))\partial_{b}\Psi_{b}(t)\\\
+\left(\partial_{c}\Psi^{\prime}_{c(b)}(t_{0})\partial_{b}\Psi_{b}(t_{0})-\partial_{c}\Psi_{c(b)}(t_{0})\partial_{b}\Psi^{\prime}_{b}(t_{0})\right)\Phi(t)\big{]},$
(5.9)

where we have used the normalization $W(\partial_{b}\Psi_{b},\Phi)=e^{(4-d)t}$
of the Wronskian between the two solutions $\partial_{b}\Psi_{b}$ and $\Phi$.
Since $t_{0}\to-\infty$ as $b\to\infty$, we can use asymptotics (5.5) and
(5.6) as well as the boundary condtions (5.7) and (5.8) to obtain for every
$t\in(-\infty,t_{0})$:

$\partial_{c}\Psi_{c(b)}(t)=(d-2)^{-1}L_{\infty}e^{\kappa_{-}t_{0}}\left[(\kappa_{-}+d-3)e^{t-t_{0}}+(1-\kappa_{-})e^{(3-d)(t-t_{0})}\right]\\\
\times\left[1+\mathcal{O}\left(\epsilon,e^{(\kappa_{+}-\kappa_{-})t_{0}},e^{2t_{0}}\right)\right]\left[1+\mathcal{O}(e^{2t})\right],$
(5.10)

where $1-\kappa_{-}>0$ and

$\kappa_{-}+d-3=\frac{1}{2}\left(d-2-\sqrt{d^{2}-16d+40}\right)>0$

for every $d\geq 13$. Thus, the sign of $\partial_{c}\Psi_{c(b)}(t)$ for every
$t\in(-\infty,t_{0})$ coincides with the sign of $L_{\infty}$ so that the
bound (5.4) holds. ∎

From Lemmas 4.1 and 5.4, we can now prove Theorem 1.2 which states that the
Morse index of $\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ coincides
with the Morse index of
$\mathcal{L}_{\infty}:\mathcal{E}_{\infty}\mapsto\mathcal{E}_{\infty}^{*}$ for
every $b\in(b_{0},\infty)$.

  * Proof of Theorem 1.2 By Sturm’s Oscillation Theorem (see, e.g., Theorem 3.5 in [23]), the Morse index of $\mathcal{L}_{b}:\mathcal{E}\mapsto\mathcal{E}^{*}$ coincides with the number of zeros of the function $v(r)$ on $(0,\infty)$ satisfying $\mathcal{L}_{b}v=0$ and $v(r)\to 0$ as $r\to\infty$. Due to the Emden–Fowler transformation (2.1), the number of zeros of $v(r)$ on $(0,\infty)$ coincides with the number of zeros of $\partial_{c}\Psi_{c(b)}(t)$ on $\mathbb{R}$ since $\mathcal{M}_{b}\partial_{c}\Psi_{c(b)}=0$ and $\partial_{c}\Psi_{c(b)}(t)\to 0$ as $t\to\infty$.

By Lemma 5.4, all zeros of $\partial_{c}\Psi_{c(b)}$ are located in the
interval $[(a-1)\log b,\infty)$ for fixed $a\in(0,1)$ and sufficiently large
$b>0$. By Lemma 4.1 and simplicity of the zeros of $\partial_{c}\Psi_{c(b)}$
and $\partial_{c}\Psi_{\infty}$, the number of zeros of
$\partial_{c}\Psi_{c(b)}$ and $\partial_{c}\Psi_{\infty}$ in $[(a-1)\log
b,\infty)$ coincides since $\lambda(b)\to\lambda_{\infty}$ and $c(b)\to
c_{\infty}$ as $b\to\infty$. All zeros of $\partial_{c}\Psi_{\infty}$ are
located in the interval $[(a-1)\log b,\infty)$ by Assumption 2.2 with the
expansion (2.25) and give the Morse index of
$\mathcal{L}_{\infty}:\mathcal{E}_{\infty}\mapsto\mathcal{E}_{\infty}^{*}$.
Hence, the Morse indices of the two operators are equal for every
$b\in(b_{0},\infty)$.

##  Appendix A Proof of Proposition 2.2

Let $\Sigma(t):=\Psi_{c}(t)-\Psi_{\infty}(t)$. It follows from (2.3) that
$\Sigma$ satisfies the following equation:

$\mathcal{M}_{\infty}\Sigma=\mathcal{F}(\Sigma)(t),$ (A.1)

where $\mathcal{M}_{\infty}$ is defined by (2.23) and

$\mathcal{F}(\Sigma)(t):=-(\lambda-\lambda_{\infty})e^{2t}(\Psi_{\infty}(t)+\Sigma(t))-3\Psi_{\infty}(t)\Sigma(t)^{2}-\Sigma(t)^{3}.$

Since $\Psi_{\infty}(t)\to\sqrt{d-3}$ as $t\to-\infty$, as it follows from
(2.17), we can pick two linearly independent solutions $r_{1}$, $r_{2}$ to
$\mathcal{M}_{\infty}r=0$ such that

$r_{1}(t)=\mathcal{O}(e^{\kappa_{-}t}),\quad
r_{2}(t)=\mathcal{O}(e^{\kappa_{+}t}),\quad\text{ as }t\to-\infty,$ (A.2)

where $\kappa_{-}<\kappa_{+}<0$ are given by (2.14). Using the Liouville’s
formula, we normalize the Wronskian according to the relation:

$W(r_{1},r_{2})(t)=r_{1}(t)r_{2}^{\prime}(t)-r_{1}^{\prime}(t)r_{2}(t)=e^{-(d-4)t}.$
(A.3)

By the variation of parameters method, we rewrite the differential equation
(A.1) as an integral equation for every $t\in[(a-1)\log b,0]$:

$\Sigma(t)=\Sigma(0)\left[r_{1}(t)r_{2}^{\prime}(0)-r_{1}^{\prime}(0)r_{2}(t)\right]+\Sigma^{\prime}(0)\left[r_{1}(0)r_{2}(t)-r_{1}(t)r_{2}(0)\right]\\\
+\int_{t}^{0}e^{(d-4)t^{\prime}}\left[r_{1}(t)r_{2}(t^{\prime})-r_{1}(t^{\prime})r_{2}(t)\right]\mathcal{F}(\Sigma)(t^{\prime})dt^{\prime}.$
(A.4)

In order to use Banach fixed-point iterations, we introduce
$\tilde{\Sigma}(t):=e^{-\kappa_{-}t}\Sigma(t)$, which satisfies
$\tilde{\Sigma}(t)=\mathcal{A}(\tilde{\Sigma})(t)$, where

$\displaystyle\begin{split}\mathcal{A}(\tilde{\Sigma})(t)&=\Sigma(0)e^{-\kappa_{-}t}\left[r_{1}(t)r_{2}^{\prime}(0)-r_{1}^{\prime}(0)r_{2}(t)\right]+\Sigma^{\prime}(0)e^{-\kappa_{-}t}\left[r_{1}(0)r_{2}(t)-r_{1}(t)r_{2}(0)\right]\\\
&-(\lambda-\lambda_{\infty})\int_{t}^{0}K_{2}(t,t^{\prime})e^{2t^{\prime}}\left[e^{-\kappa_{-}t^{\prime}}\Psi_{\infty}(t^{\prime})+\tilde{\Sigma}(t^{\prime})\right]dt^{\prime}\\\
&-\int_{t}^{0}K_{2}(t,t^{\prime})\left[3e^{\kappa_{-}t^{\prime}}\Psi_{\infty}(t^{\prime})\tilde{\Sigma}^{2}(t^{\prime})+e^{2\kappa_{-}t^{\prime}}\tilde{\Sigma}^{3}(t^{\prime})\right]dt^{\prime},\end{split}$
(A.5)

where the kernel $K_{2}(t,t^{\prime})$ is defined as

$K_{2}(t,t^{\prime}):=e^{\kappa_{-}(t^{\prime}-t)-(\kappa_{+}+\kappa_{-})t^{\prime}}\left[r_{1}(t)r_{2}(t^{\prime})-r_{1}(t^{\prime})r_{2}(t)\right].$
(A.6)

It follows from (A.2) that $K_{2}(t,t^{\prime})\sim
1+e^{(\kappa_{+}-\kappa_{-})(t-t^{\prime})}$ as $t-t^{\prime}\to-\infty$,
which means that there exists some constant $K_{0}>0$, such that

$\sup_{-\infty<t\leq t^{\prime}\leq 0}|K_{2}(t,t^{\prime})|\leq K_{0}.$ (A.7)

It follows from (2.22) that there exists some constant $C_{0}>0$ such that

$|\Sigma(0)|+|\Sigma^{\prime}(0)|\leq
C_{0}\left(|\lambda-\lambda_{\infty}|+|c-c_{\infty}|\right).$

The integral operator $\mathcal{A}(\tilde{\Sigma})$ in (A.5) is estimated for
every $\tilde{\Sigma}\in L^{\infty}((a-1)\log b,0)$ as

$\|\mathcal{A}(\tilde{\Sigma})\|_{\infty}\leq
C_{0}\Big{[}|\lambda-\lambda_{\infty}|+|c-c_{\infty}|+|\lambda-\lambda_{\infty}|\|\tilde{\Sigma}\|_{\infty}\\\
+b^{-\kappa_{-}(1-a)}\|\tilde{\Sigma}\|^{2}_{\infty}+b^{-2\kappa_{-}(1-a)}\|\tilde{\Sigma}\|^{3}_{\infty}\Big{]}.$
(A.8)

Similar estimate applies to
$\|\mathcal{A}(\tilde{\Sigma}_{1})-\mathcal{A}(\tilde{\Sigma}_{2})\|_{\infty}$.
The estimates show that the integral operator $\mathcal{A}(\tilde{\Sigma})$ is
closed and is a contraction in the ball $B_{\delta}\subset
L^{\infty}((a-1)\log b,0)$ of the small radius $\delta:=2C_{0}\epsilon
b^{\kappa_{-}(1-a)}$, provided that $(\lambda,c)$ satisfy the bound (2.19) and
$\epsilon>0$ is sufficiently small. By the Banach fixed-point theorem, there
exists a unique fixed point of $\mathcal{A}(\tilde{\Sigma})$ satisfying

$\|\tilde{\Sigma}\|_{\infty}\leq 2C_{0}\epsilon b^{\kappa_{-}(1-a)},$ (A.9)

which proves the bound (2.20) after going back to the original variable
$\Sigma$ and redefining $C_{0}$.

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#  Sieve methods and the twin prime conjecture

Mbakiso Fix Mothebe
Department of Mathematics, University of Botswana, Pvt Bag 00704,
Gaborone, Botswana
e-mail add<EMAIL_ADDRESS>
###### Abstract

For $n\geqslant 3,$ let $p_{n}$ denote the $n^{\rm th}$ prime number. Let
$[\;]$ denote the floor or greatest integer function. For a positive integer
$m,$ let $\pi_{2}(m)$ denote the number of twin primes not exceeding $m.$ The
twin prime conjecture states that there are infinitely many prime numbers $p$
such that $p+2$ is also prime. In this paper we state a conjecture to the
effect that given any integer $a>0$ there exists an integer $N_{2}(a)$ such
that

$\left[\frac{ap^{2}_{n+1}}{2(n+1)}\right]\leqslant\pi_{2}\left(p^{2}_{n+1}\right)$

for all $n\geqslant N_{2}(a)$ and prove the conjecture in the case $a=1.$
This, in turn, establishes the twin prime conjecture.

##  1 Introduction and main results

An integer $p\geqslant 2$ is called a prime if its only positive divisors are
$1$ and $p.$ The prime numbers form a sequence:

$2,\;3,\;5,\;7,\;11,\;13,\;17,\;19,\;23,\;29,\;31,\;37,\;41,\;43,\;47\;\ldots.$
(1.1)

Euclid (300 B.C.) considered prime numbers and proved that there are
infinitely many. Prime numbers are odd except $2$ and the only consecutive
prime numbers are $2$ and $3.$ Any two odd prime numbers in the sequences
(1.1) differ by at least $2.$ Pairs of prime numbers that differ by $2$ as,
for example, in the sequence below

$(3,5),\;(5,7),\;(11,13),\;(17,19),\;(29,31),\;(41,43),\;\ldots$ (1.2)

are said to be twin primes.

######  Conjecture 1.1.

There exist infinitely many twin primes.

The conjecture is still open. The first known published reference to this
question was made by Alphonse de Polignac in $1849,$ who conjectured that for
every even number $k,$ there are infinitely many pairs of prime numbers $p$
and $p^{\prime}$ such that $p^{\prime}-p=k$ (see [7]). The case $k=2$ is the
twin prime conjecture. The conjecture has not yet been proven or disproven for
a given value of $k.$ In $2013,$ an important breakthrough was made by Yitang
Zhang who proved the conjecture for some value of $k<70\;000\;000$ (see [11]).
Later that same year, James Maynard announced a related breakthrough which
proved the conjecture for some $k<600$ (see [5]). In 2014 D.H.J. Polymath
proved the conjecture for some $k\leqslant 246.$ (see [8])

In this paper, we prove that Conjecture 1.1 is true.

Let $[\;]$ denote the floor or greatest integer function and let $\pi_{2}(m)$
denote the number of twin primes not exceeding the positive integer $m.$ Our
conjecture is the following;

AMS Subject Classification: 11N05; 11N36.

Keywords: Primes, Twin primes, Sieve methods.

######  Conjecture 1.2.

For $n\geqslant 3,$ let $p_{n}$ denote the $n^{\rm th}$ prime. Then for each
integer $a>0$ there exists an integer $N_{2}(a)$ such that

$\left[\frac{ap^{2}_{n+1}}{2(n+1)}\right]\leqslant\pi_{2}\left(p^{2}_{n+1}\right)$

for all $n\geqslant N_{2}(a).$

The following is our main result:

######  Theorem 1.3.

Conjecture 1.2 is true in the case $a=1.$

Thus our main result is that there exists an integer $N_{2}(1)$ such that

$\left[\frac{p^{2}_{n+1}}{2(n+1)}\right]\leqslant\pi_{2}\left(p^{2}_{n+1}\right)$

for all $n\geqslant N_{2}(1).$ We shall show later that $N_{2}(1)=20.$

The sequence $\left[\frac{p^{2}_{n+1}}{2(n+1)}\right]$ is unbounded. As a
consequence, Conjecture 1.1 holds if there are infinitely many integers $n$
for which
$\left[\frac{p^{2}_{n+1}}{2(n+1)}\right]\leqslant\pi_{2}\left(p^{2}_{n+1}\right).$

In [6] we claim to prove the following result which also implies the twin
prime conjecture.

######  Theorem 1.4.

For $n\geqslant 2,$ let $p_{n}$ denote the $n^{\rm th}$ prime. Then

$\left[\frac{p^{2}_{n+3}}{3(n+2)}\right]\leqslant\pi_{2}\left(p^{2}_{n+3}\right)$

for all $n\geqslant 2.$

Our attempt to prove the above result has, however, not been deemed
sufficiently clear. In this paper we use a similar approach as in [6] to
derive our main result, namely Theorem 1.3. However our method of proof is by
mathematical induction. The result of Theorem 1.3 supersedes that of Theorem
1.4 and therefore establishes its validity.

Our work is organized as follows: In Section 2, we recall the definition of
the well known sieve of Eratosthenes and state some preliminary results. We
show, as one of our main results, that

$\left[\frac{p^{2}_{n+1}}{n+1}\right]\leqslant\pi\left(p^{2}_{n+1}\right)$
(1.3)

for all integers $n\geqslant 1.$ Here, $\pi(m)$ denotes the number of primes
not exceeding the integer $m.$ Finally, a proof for Theorem 1.3 is presented
in Section 3.

The usage of the word sieve in some cases of this work refers to subtracting
magnitudes, as opposed to the usual reference to sifting out of integer
positions. One of the basic principles applied in our methods of proof is that
if two uneven sieves are individually applied to two finite sets of the same
order, then after some finite time the order of the residue of the more porous
one will be less that that of the other sieve. This principle is applied
inductively and yields some of our results in this paper.

##  2 Preliminary Results

The concepts required are elementary and can be obtained from introductory
texts on number theory, discrete mathematics and set theory ([2],[3], [10]).

Eratosthenes ($276-194$ B.C.) was a Greek mathematician whose work in number
theory remains significant. Consider the following lemma:

######  Lemma 2.1.

Let $a>1$ be an integer. If $a$ is not divisible by a prime number
$p\leqslant\sqrt{a},$ then a is a prime.

Eratosthenes used the above lemma as a basis of a technique called “Sieve of
Eratosthenes” for finding all the prime numbers less than a given integer $x$.
The algorithm calls for writing down the integers from $2$ to $x$ in their
natural order. The composite numbers in the sequence are then sifted out by
crossing off from $2,$ every second number (all multiples of two) in the list,
from the next remaining number, $3,$ every third number, from the next
remaining number, $5,$ every fifth number, and so on for all the remaining
prime numbers less than or equal to $\sqrt{x}.$ The integers that are left on
the list are primes. We shall refer to the set of integers left as the residue
of the sieve. Thus the order of the residue set is equal to $\pi(x).$ In our
application of the sieve of Eratosthenes the prime numbers $2,3,5,\ldots,p$
are also sifted out from the sequence so that if $p$ is the $n^{\rm th}$
prime, then the residue has order $\pi(x)-n.$

We shall require the following results of J.B. Rosser and L. Schoenfeld (see
[9] page 69).

######  Theorem 2.2.

(J.B.Rosser, L. Schoenfeld) Let $n\geqslant 1$ be an integer. Then:

  * (i)

$\frac{n}{(\log n-\frac{1}{2})}<\pi(n)$ for $n\geqslant 67,$

  * (ii)

$\pi(n)<\frac{n}{(\log n-\frac{3}{2})}$ for $n\geqslant e^{\frac{3}{2}}.$

######  Corollary 2.3.

Let $n\geqslant 1$ be an integer. Then:

  * $\frac{n}{\log n}<\pi(n)\;\;$ for $n\geqslant 17.$

######  Theorem 2.4.

(J.B.Rosser, L. Schoenfeld) Let $n\geqslant 1$ be an integer. Then:

  * (i)

$n(\log n+\log\log n-{\frac{3}{2}})<p_{n}$ for $n\geqslant 2,$

  * (ii)

$p_{n}<n(\log n+\log\log n-{\frac{1}{2}})$ for $n\geqslant 20.$

######  Corollary 2.5.

Let $n\geqslant 1$ be an integer. Then:

  * (i)

$n(\log n)<p_{n}$ for $n\geqslant 1,$

  * (ii)

$p_{n}<n(\log n+\log\log n)$ for $n\geqslant 6.$

Let $[\;]$ denote the floor or greatest integer function. The following is one
of our main results.

######  Theorem 2.6.

Let $p_{n}$ denote the $n^{\rm th}$ prime. Then

$\left[\frac{p^{2}_{n+1}}{n+1}\right]\leqslant\pi\left(p^{2}_{n+1}\right)$

for all integers $n\geqslant 1.$

The following result gives an equivalent criterion for the validity of the
inequality below. If $n\geqslant 2,$ then

$\frac{p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right).$ (2.4)

######  Lemma 2.7.

Let $n\geqslant 2$ be a fixed integer, let $p_{n}$ denote the $n^{\rm th}$
prime number and for each $s,\;1\leqslant s\leqslant n,$ let $m^{s}_{r}$
denote the $r^{\rm th}$ multiple in the ordered sequence of all products,
$p_{s}\prod_{i\geqslant 1}{p_{s_{i}}},$ where $p_{s_{i}}$ are primes not less
than $p_{s}.$ For each $s,$ let $\overline{\\{{m^{s}_{r}}\\}}$ denote the
terms of the sequence $\\{{m^{s}_{r}}\\}_{r\geqslant 1}$ which are less than
$p^{2}_{n+1}.$ Then $\frac{p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$ if
and only if
$|\bigcup_{s=1}^{n}\overline{\\{{m^{s}_{r}}\\}}|<\frac{n(p^{2}_{n+1})}{n+1}+2.$

###### Proof.

Note that $\frac{1}{n+1}=1-\frac{n}{n+1}=1-\sum_{s=1}^{n}\frac{1}{s(s+1)}.$
Therefore $\frac{p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$ if and only if
$p^{2}_{n+1}-\sum_{s=1}^{n}\frac{p^{2}_{n+1}}{s(s+1)}<p^{2}_{n+1}-2-|\bigcup_{s=1}^{n}\overline{\\{{m^{s}_{r}}\\}}|,$
that is, if and only if
$|\bigcup_{s=1}^{n}\overline{\\{{m^{s}_{r}}\\}}|<\frac{n(p^{2}_{n+1})}{n+1}+2$
since
$\sum_{s=1}^{n}\frac{p^{2}_{n+1}}{s(s+1)}=\frac{n(p^{2}_{n+1})}{n+1}.$ ∎

As a consequence of Lemma 2.7 we have the following result which forms the
basis of our arguments in this paper.

For each integer $s\geqslant 1,$ let $p_{s}$ denote the $s^{\rm th}$ prime
number. Let $n\geqslant 1$ be an integer. For each integer $x\geqslant
p^{2}_{n+1},$ let $S(x,n)$ denote the sum

$S(x,n)=x+\sum_{j=1}^{n}(-1)^{j}\left\\{\sum_{1\leqslant
s_{1}<\cdots<s_{j}\leqslant
n}\left[\frac{x}{\prod_{i=1}^{j}p_{s_{i}}}\right]\right\\}.$ (2.5)

The sum $S(x,n)$ is based on the inclusion-exclusion principle and can be
considered as a sieve on the sequence of integers;

$1,2,3,4,5,\ldots,x$

which sifts out all integers $y$ for which g.c.d.$(y,p_{s})\neq 1$ for some
$s,\;1\leqslant s\leqslant n.$ Thus for each value of $x,$ $S(x,n)$ is the
order of the set, ${\cal S}(x,n),$ of positive integers not exceeding $x$
which are relatively prime to the primes $p_{s},\;1\leqslant s\leqslant n.$ We
have:

######  Theorem 2.8.

For each $n\geqslant 2,$ $\frac{x}{n+1}\leqslant S(x,n)$ for all integers
$x=x_{n+1}\geqslant p^{2}_{n+1}.$

###### Proof.

Our proof is by induction on $n.$ The theorem certainly holds for $n=2.$ If
$n=3,$ then $S(x,3)$ may be closely approximated by

$x-\left(\left[\frac{x}{2}\right]+\left[\frac{x}{6}\right]+\left[\frac{x}{15}\right]\right)$
(2.6)

while

$\frac{x}{4}=x-\left(\frac{x}{2}+\frac{x}{6}+\frac{x}{12}\right)$ (2.7)

and certainly $\frac{x}{4}<S(x,3)$ for all $x\geqslant 49.$ So the result of
the theorem is true when $n=2,3.$ Assume that $\frac{x}{n+1}<S(x,n)$ for some
integer $n\geqslant 3$ whenever $x=x_{n+1}\geqslant p^{2}_{n+1}.$ To show that
we must have $\frac{x}{n+2}<S(x,n+1),$ whenever $x=x_{n+2}\geqslant
p^{2}_{n+2}$ it is sufficient to establish that the result of subtracting
$K(x,n)=\frac{x}{(n+1)(n+2)}$ from $\frac{x}{n+1}$ is less than that of
subtracting

$T(x,n)=\left[\frac{x}{p_{n+1}}\right]+\sum_{j=1}^{n}(-1)^{j}\left\\{\sum_{1\leqslant
s_{1}<\cdots<s_{j}\leqslant
n}\left[\frac{x}{{p_{n+1}}\prod_{i=1}^{j}p_{s_{i}}}\right]\right\\}$

from $S(x,n).$ Note that $T(x,n)$ sifts out $p_{n+1}$ and the composite
numbers $\\{{m^{n+1}_{r}}\\}_{r\geqslant 1}$ in the residue of $S(x,n)$ which
are less than $x_{n+2}.$

To prove that $K(x,n)\geqslant T(x,n)$ if $x\geqslant p^{2}_{n+2},$ let
$a_{n}(x)$ be the real number $\frac{(n+1)S(x,n)}{x}.$ We may assume that
$a_{n}(x)>1$ since this is the case when $n=3.$ Then $\frac{a_{n}(x)x}{n+1}$
is equal to $S(x,n).$ ${\cal S}(x,n+1)$ is obtained from ${\cal S}(x,n)$ as
follows: By definition $m^{n}_{r}=p_{n}q^{n}_{r}$ where $q^{n}_{r}=1$ or is a
product of primes greater than or equal to $p_{n},$ while
$m^{n+1}_{r}=p_{n+1}q^{n+1}_{r}$ where $q^{n+1}_{r}=1$ or is a product of
primes greater than or equal to $p_{n+1}.$ Thus, for $x\geqslant p^{2}_{n+2},$
${\cal S}(x,n+1)$ is obtained from ${\cal S}(x,n)$ by sifting out all products
$qp_{n+1}$ less than $x=x_{n+2},$ where $q$ is an element of ${\cal S}(x,n).$
On the other hand $\frac{a_{n}(x)x}{n+2}$ is obtained from
$\frac{a_{n}(x)x}{n+1}$ by sifting out all integer multiples of $(n+1)(n+2)$
less than $a_{n}(x)x,$ that is,
$\frac{a_{n}(x)x}{n+2}=\frac{a_{n}(x)x}{n+1}-\frac{a_{n}(x)x}{(n+1)(n+2)}$ or,
equivalently, $\frac{x}{\frac{1}{a_{n}(x)}(n+2)}$ is obtained from
$\frac{x}{\frac{1}{a_{n}(x)}(n+1)}$ by sifting out all integer multiples of
$\frac{1}{a_{n}(x)}(n+1)(n+2)$ less than $x=x_{n+2}.$ Let
$q^{n}_{r},\;r\geqslant 1$ denote the elements of ${\cal S}(x,n),$ so that
$q^{n}_{1}=1,q^{n}_{2}=p_{n},q^{n}_{3}=p_{n+1},\ldots.$ Since $n+2<p_{n+1}$
and $\frac{a_{n}(x)x}{n+1}$ is equal to $S(x,n),$ the number of integer
multiples of $\frac{1}{a_{n}(x)}(n+1)(n+2)$ less that $x=x_{n+2}$ is greater
than the number of products $p_{n+1}q^{n}_{r}$ less than $x=x_{n+2},$ that is,
$r(\frac{1}{a_{n}(x)}(n+1)(n+2))<p_{n+1}q^{n}_{r}$ for all integers
$r\geqslant 2.$ That this statement is valid can easily be checked for small
values of $n$ up to $x=\prod_{s=1}^{n}p_{s}.$ This validates the statement for
all values of $x$ greater than $\prod_{s=1}^{n}p_{s}$ since the Sieve (2.5) is
“cyclic” over each interval of this length. As $n$ increases $p_{n+1}$ is much
larger than $n+2$ while $\frac{1}{a_{n}(x)}(n+1)$ increases by small amounts
so the products $p_{n+1}q^{n}_{r}$ are much larger than
$r(\frac{1}{a_{n}(x)}(n+1)(n+2))$ for each $r\geqslant 2.$ Thus for all
$x\geqslant p^{2}_{n+2},$

$\frac{x}{n+2}=\frac{x}{n+1}-\frac{x}{(n+1)(n+2)}<\frac{a_{n}(x)x}{n+2}=\frac{a_{n}(x)x}{n+1}-\frac{a_{n}(x)x}{(n+1)(n+2)}<S(x,n)-T(x,n)=S(x,n+1).$
(2.8)

Note that we then have $a_{n+1}(x)>1$ and this completes the induction. ∎

As a consequence of Theorem 2.8 we have for $x=p^{2}_{n+1},$
$\left[\frac{x}{n+1}\right]<S(x,n)+\pi\left(p_{n}\right)=\pi\left(p^{2}_{n+1}\right)$
for all $n\geqslant 2$ which establishes Theorem 2.6. We note that $a_{n}(x)$
is a ratio that measures the increase of $S(n,x)$ relative to $\frac{x}{n+1}.$
From our discussion it is clear that $a_{n}(x)$ increases with $n,$ so as $n$
increases $S(n,x)$ grows faster than $\frac{x}{n+1}.$ In the example below, we
compute $a_{n}(x)$ in some simple cases when $n=70.$

######  Example 2.9.

Let $n=70.$ The table below shows the values of $a_{70}(x)$ for some integers
$x,$
$p_{71}^{2}=124609\leqslant x\leqslant\prod_{s=1}^{70}p_{s}.$ For the last
value of $x$ in the table, we use the formula:

$a_{n}(\prod_{s=1}^{70}p_{s})=\frac{(71)\prod_{s=1}^{70}(p_{s}-1)}{\prod_{s=1}^{70}p_{s}}.$

$x$ | $a_{n}(x)$
---|---
$124\;609$ | $6.6278$
$654\;480$ | $6.4641$
$1\;885\;128$ | $6.6046$
$3\;279\;720$ | $6.6707$
$5\;870\;928$ | $6.7240$
$9\;480\;240$ | $6.7566$
$13\;890\;528$ | $6.7745$
$29\;626\;248$ | $6.7911$
$62\;710\;560$ | $6.7869$
$223\;092\;870$ | $6.7650$
$6\;469\;693\;230$ | $6.7482$
$\prod_{s=1}^{70}p_{s}$ | $6.7514$

The above computations together with the result of Theorem 2.8, suffice to
show that $a_{70}(x)>6$ for all $x\geqslant 124609.$

The following result shows that for the values $x=p^{2}_{n+1},$ the sequence
of real numbers $a_{n}(x),\;n\geqslant 4$ defined in the proof of Theorem 2.8
is unbounded if we replace $S(p^{2}_{n+1},n)$ by
$\pi\left(p^{2}_{n+1}\right).$

######  Theorem 2.10.

For $n\geqslant 3$ let $p_{n}$ denote the $n^{\rm th}$ prime. Then for each
integer $b>0$ there exists an integer $N(b)$ such that

$\frac{bp^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$

for all $n\geqslant N(b).$

###### Proof.

Let $b>0$ be an integer. By Corollary 2.3, $\pi(x)>\frac{x}{{\rm log}x}$ for
$x>17.$ It therefore suffices to show that there exists an integer $N(b)$ such
that

$\frac{p_{n+1}^{2}}{(2{\rm log}(p_{n+1}))}>\frac{bp_{n+1}^{2}}{(n+1)}$

for all $n\geqslant N(b)$ or, equivalently, such that ${\rm
log}p_{n+1}<\frac{(n+1)}{2b}.$ By Corollary 2.5 (ii),

$p_{n+1}<(n+1)({\log}(n+1)+{\log}{\log}(n+1))$

for $n>6.$ Thus, it is enough to find $N(b)$ such that

${\rm log}((n+1)({\rm log}(n+1)+{\rm log}{\rm log}(n+1)))<\frac{(n+1)}{2b}.$

This is always possible to achieve since if we treat ${\rm log}((n+1)({\rm
log}(n+1)+{\rm log}{\rm log}(n+1)))$ as a function of $n,$ we get its
derivative to be less than $\frac{3}{(n+1)}$ which is smaller than
$\frac{1}{2b}$ for $n$ large enough. Thus for values of $n$ large enough ${\rm
log}((n+1)({\rm log}(n+1)+{\rm log}{\rm log}(n+1)))$ is less than
$\frac{(n+1)}{2b}.$ ∎

For example, $\frac{2p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$ for all
$n\geqslant 12,$ $\frac{3p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$ for
all $n\geqslant 23,$ $\frac{4p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$
for all $n\geqslant 35$ and so on.

The result of Theorem 2.10, in particular, implies that
$\left[\frac{p^{2}_{n+1}}{n+1}\right]$ is a weak lower bound for
$\pi\left(p^{2}_{n+1}\right).$

A famous conjecture of Hardy and Littlewood states that

$\pi_{2}\left(n\right)\approx 2C_{2}\int_{2}^{n}\frac{dt}{\log^{2}t}=L_{2}(n)$
(2.9)

where $C_{2}=0.661618155....$ $L_{2}(n)$ may be estimated by

$2C_{2}\frac{n}{\log^{2}n}.$ (2.10)

Later we shall show that the Hardy-Littlewood conjecture implies Conjecture
1.2.

We note that the $n^{\rm th}$ Catalan number,
${\frac{1}{n+1}}{\binom{2n}{n}},$ may be expressed in the form

$\displaystyle{\frac{1}{n+1}}{\binom{2n}{n}}$ $\displaystyle=$
$\displaystyle\left(1-{\frac{n}{n+1}}\right){\binom{2n}{n}}$ $\displaystyle=$
$\displaystyle{\binom{2n}{n}}-{\binom{2n}{n+1}}$ $\displaystyle=$
$\displaystyle{\binom{2n}{n}}-\sum_{i=1}^{n}{2n-i\choose n}.$ (2.11)

We can therefore consider ${\frac{1}{n+1}}{\binom{2n}{n}}$ as representing the
residue of a sieve defined on the sequence of consecutive integers from $1$ to
${\binom{2n}{n}}$ in the following way. Sift out ${\binom{2n-1}{n}}$ numbers
from the sequence in an evenly distributed fashion. Since
${\binom{2n-1}{n}}={\frac{1}{2}}{\binom{2n}{n}},$ this is equivalent to
striking out all the even numbers. From the remaining integers, ${2n-2\choose
n}$ numbers in the sequence are sifted out in an evenly distributed fashion
and so on for each of the integers $i,\;3\leqslant i\leqslant n.$ In any case
the ${\frac{1}{n+1}}{\binom{2n}{n}}$ elements of the residue set are assumed
to be evenly distributed over the interval $1$ to ${2n\choose n}.$ Thus any
two consecutive elements in the residue of the sieve may be assumed to be
separated by an interval of $n+1.$ So $n+1$ is the average density of the
residue of the sieve of Equation (2).

Let ${C^{2n}_{n}}={\binom{2n}{n}}.$ Writing

$\displaystyle\frac{p^{2}_{n+1}}{n+1}$ $\displaystyle=$
$\displaystyle\frac{p^{2}_{n+1}}{(n+1)}\left(\frac{1}{C^{2n}_{n}}\binom{2n}{n}\right)$
$\displaystyle=$
$\displaystyle\frac{p^{2}_{n+1}}{C^{2n}_{n}}\left({\binom{2n}{n}}-\sum_{i=1}^{n}{2n-i\choose
n}\right),$

we see that we can treat $\left[\frac{p^{2}_{n+1}}{n+1}\right]$ as a residue
of the sieve of Equation (2) when restricted to the sequence of integers
$1,\ldots,{p^{2}_{n+1}}.$

##  3 Proof of Theorem 1.3

For $n\geqslant 3,$ let $x\geqslant p^{2}_{n+1}-1$ be an integer. In this
section we shall compare the sieve

$\displaystyle x-\sum_{s=1}^{n}\frac{x}{s(s+1)}$ (3.12)

with the Sieve (2.5), viewed over two sets of pairs of integers, the latter of
whose residue shall consist of twin primes when $x={p^{2}_{n+1}}-1.$ We shall
show that as $n$ increases the Sieve (3.12) is dominant or remains more porous
than the Sieve (2.5) over these corresponding sets of pairs of integers. The
residue of the Sieve (3.12) will then be seen to be unbounded for all values
of $x$ as $n$ increases and this, in turn, will establish the twin prime
conjecture.

We first prove the following preliminary result. After sifting out the
multiples of $2$ and $3$ from the sequence $1,2,3,\ldots,(p^{2}_{n+1}-1)$ the
residue set is of the form
$\\{1,(p^{2}_{n+1}-2)\\}\cup\\{(6t-1,6t+1)\;|\;1\leqslant
t<\frac{(p^{2}_{n+1}-1)}{6}\\}.$ The sieve of Eratosthenes extends naturally
to the set $\\{(6t-1,6t+1)\;|\;1\leqslant t<\frac{p^{2}_{n+1}-1}{6}\\}$ and
for each integer $n\geqslant 3,$ we let ${\cal S}(p^{2}_{n+1})$ denote the set

${\cal S}(p^{2}_{n+1})=\\{t\in\mathbb{N}\;|\;1\leqslant
t<\frac{(p^{2}_{n+1}-1)}{6},\;\;6t-1\;\mbox{or}\;6t+1\;\mbox{is a
composite}\\}.$

Note that $6t-1$ or $6t+1$ is a composite if and only if either is equal to
$m_{r}^{s}$ for some integers $r,s.$ Twin primes are of the form $(6t-1,6t+1)$
for some integer $t\geqslant 1$ apart from the pair $(3,5).$ Thus the order of
the complement $|\left({\cal
S}(p^{2}_{n+1})\right)^{\prime}|=\pi_{2}\left(p^{2}_{n+1}\right)-1.$ We have
the following:

######  Lemma 3.1.

For $n\geqslant 3,$ let $p_{n}$ denote the $n^{\rm th}$ prime. Then

$\frac{p^{2}_{n+1}}{2(n+1)}<\pi_{2}\left(p^{2}_{n+1}\right)$

if and only if

$|{\cal S}(p^{2}_{n+1})|<\frac{(n-2)p^{2}_{n+1}}{6(n+1)}+\frac{1}{6}.$

###### Proof.

Let $n\geqslant 3$ be an integer. We recall from the result of Lemma 2.7 that
if for each $s,\;1\leqslant s\leqslant n,$ we let $m^{s}_{r}$ denote the
$r^{\rm th}$ multiple in the ordered sequence of all products,
$p_{s}\prod_{i\geqslant 1}{p_{s_{i}}},$ where $p_{s_{i}}$ are primes not less
than $p_{s}$ and for each $s,$ let $\overline{\\{{m^{s}_{r}}\\}}$ denote the
terms of the sequence $\\{{m^{s}_{r}}\\}_{r\geqslant 1}$ which are less than
$p^{2}_{n+1},$ then $\frac{p^{2}_{n+1}}{n+1}<\pi\left(p^{2}_{n+1}\right)$ if
and only if
$|\bigcup_{s=1}^{n}\overline{\\{{m^{s}_{r}}\\}}|<\sum_{s=1}^{n}\frac{p^{2}_{n+1}}{s(s+1)}+2.$
Therefore for all values of $n\geqslant 1,$

$|\overline{\\{{m^{1}_{r}}\\}}|+|\overline{\\{{m^{2}_{r}}\\}}|=\left[\frac{p^{2}_{n+1}}{2}\right]+\left[\frac{p^{2}_{n+1}}{6}\right]-2=\sum_{s=1}^{2}\left[\frac{p^{2}_{n+1}}{s(s+1)}\right]-2.$

$\overline{\\{{m^{1}_{r}}\\}}$ and $\overline{\\{{m^{2}_{r}}\\}}$ are obtained
by sifting out the multiples of $2$ and $3$ in the sequence
$1,\ldots,(p^{2}_{n+1}-1).$ Thus the residue set consists of
$\\{1,(p^{2}_{n+1}-2)\\}\cup\\{(6t-1,6t+1)\;|\;1\leqslant
t<\frac{(p^{2}_{n+1}-1)}{6}\\}.$ On the other hand
$\sum_{s=1}^{2}\left[\frac{p^{2}_{n+1}}{s(s+1)}\right]$ sifts out integers in
the sequence $1,\ldots,(p^{2}_{n+1}-1)$ to leave any two consecutive elements
in the residue of the sieve separated by an interval of $3.$ Thus we can
assume that the residue consists of
$\\{1,(p^{2}_{n+1}-3)\\}\cup\\{(6t-2,6t+1)\;|\;1\leqslant
t<\frac{(p^{2}_{n+1}-1)}{6}\\}.$ Now extending the sieve of Eratosthenes to
the set

$S=\\{(6t-1,6t+1)\;|\;1\leqslant t<\frac{(p^{2}_{n+1}-1)}{6}\\}$

leaves a residue set of order $\pi_{2}\left(p^{2}_{n+1}\right)-1.$ On the
other hand we have
$\frac{1}{6}-\sum_{s=3}^{n}\frac{1}{2s(s+1)}=\frac{1}{6}-\frac{(n-2)}{6(n+1)}=\frac{1}{2(n+1)}.$
Thus we see that $\frac{p^{2}_{n+1}}{2(n+1)}<\pi_{2}\left(p^{2}_{n+1}\right)$
if and only if
$\frac{p^{2}_{n+1}}{6}-\sum_{s=3}^{n}\frac{p^{2}_{n+1}}{2s(s+1)}<\frac{(p^{2}_{n+1}-1)}{6}-|{\cal
S}(p^{2}_{n+1})|,$ that is, if and only if

$|{\cal S}(p^{2}_{n+1})|<\frac{(n-2)p^{2}_{n+1}}{6(n+1)}+\frac{1}{6}$

since
$\sum_{s=3}^{n}\frac{p^{2}_{n+1}}{2s(s+1)}=\frac{(n-2)p^{2}_{n+1}}{6(n+1)}$
and $\frac{p^{2}_{n+1}}{6}=\frac{(p^{2}_{n+1}-1)}{6}+\frac{1}{6}.$ ∎

For each integer $s\geqslant 1,$ let $p_{s}$ denote the $s^{\rm th}$ prime
number. Let $n\geqslant 3$ be an integer. For each integer
$x=x_{n}=6k_{n}\geqslant p^{2}_{n+1}-1,$ let ${\cal R}(x,n)$ denote the set

${\cal R}(x,n)=\\{t\in\mathbb{N}\;|\;1\leqslant
t<k,\;\;\mbox{both}\;6t-1\;\mbox{and}\;6t+1\;\mbox{are not divible
by}\;\;p_{s},\;\;3\leqslant s\leqslant n\\}.$ (3.13)

After sifting out the multiples of $2$ and $3,$ the sieve

$S(x,n)=x+\sum_{j=1}^{n}(-1)^{j}\left\\{\sum_{1\leqslant
s_{1}<\cdots<s_{j}\leqslant
n}\left[\frac{x}{\prod_{i=1}^{j}p_{s_{i}}}\right]\right\\}$

on the sequence of integers

$1,2,3,4,5,\ldots,x$ (3.14)

extends naturally to a sieve on the sequence of integers

$1,5,7,11,13,\ldots,6t-1,6t+1,\ldots,6k-1.$ (3.15)

On the sequence (3.15) we have

$S(x,n)=\frac{x}{3}-\left\\{\sum_{r=3}^{n}\left(\left[\frac{x}{p_{r}}\right]+\sum_{j=1}^{r-1}(-1)^{j}\left\\{\sum_{1\leqslant
s_{1}<\cdots<s_{j}\leqslant
r-1}\left[\frac{x}{\prod_{i=3}^{j}p_{r}p_{s_{i}}}\right]\right\\}\right)\right\\}.$
(3.16)

Multiplying Equation (3.16) by $\frac{1}{2}$ we obtain

$\frac{1}{2}S(x,n)=\frac{x}{6}-\frac{1}{2}\left\\{\sum_{r=3}^{n}\left(\left[\frac{x}{p_{r}}\right]+\sum_{j=1}^{r-1}(-1)^{j}\left\\{\sum_{1\leqslant
s_{1}<\cdots<s_{j}\leqslant
r-1}\left[\frac{x}{\prod_{i=3}^{j}p_{r}p_{s_{i}}}\right]\right\\}\right)\right\\}.$
(3.17)

so that $\frac{1}{2}S(x,n)$ is the order of the residue of the Sequence (3.15)
when the terms are counted in pairs. Arranging the terms of the Sequence
(3.15) as pairs in the set

$S=\\{1,6t-1\\}\cup\\{(6t-1,6t+1)\;|\;1\leqslant t<k\\}$

we can consider Equation (3.16) as a sieve on the set $S$ with a residue of
order $\frac{1}{2}S(x,n),$ when the remaining elements are counted in pairs.
Thus the residue set will consist of pairs with singleton components
$(6t-1,\;),$ $(\;,6t+1)$ as well as ordered pairs $(6t-1,6t+1),$ where
$(6t-1,\;),$ $(\;,6t+1)$ means $6t+1$ or $6t-1,$ respectively, is divisible by
some prime $p_{s},\;3\leqslant s\leqslant n.$ In other words we do not regard
( 3.16) as sifting pairs. If we then sift out all pairs with singleton
components, then the resulting residue set is ${\cal R}(x,n).$ Thus $|{\cal
R}(x,n)|\leqslant\frac{1}{2}S(x,n).$

In the same vein if $x=x_{n+1}=6k_{n+1}\geqslant p^{2}_{n+2}-1,$ then the
effect of

$T(x,n)=\left[\frac{x}{p_{n+1}}\right]+\left(\sum_{j=1}^{n}(-1)^{j}\left\\{\sum_{1\leqslant
s_{1}<\cdots<s_{j}\leqslant
n}\left[\frac{x}{{p_{n+1}}\prod_{i=1}^{j}p_{s_{i}}}\right]\right\\}\right)$

on the respective Sequence (3.15) extends to a sieve on ${\cal R}(x,n)$ that
sifts out pairs $(6t-1,6t+1)$ from ${\cal R}(x,n)$ for which $6t-1$ or $6t+1$
is divisible by $p_{n+1}$ with ${\cal R}(x,n+1)$ as the resulting residue set.
Thus the Sieve (3.16) extends naturally to a sieve on ${\cal R}(x,n).$ Clearly
if $Q(x,n)$ is the number of pairs sifted out from ${\cal R}(x,n)$ in this
way, then $Q(x,n)\leqslant T(x,n).$ The values of $t,$ $1\leqslant t<k,$ in
$S$ are $\frac{1}{6}$ the distance between the pairs $(6t-1,6t+1)$ when
considered as pairs in the the sequence (3.14) over the interval $1\leqslant
i\leqslant x.$

We have:

######  Theorem 3.2.

For all $n\geqslant 70,$ $\frac{x}{2(n+1)}\leqslant|{\cal R}(x,n)|$ for all
integers $x=6k\geqslant p^{2}_{n+1}-1.$

###### Proof.

For $n\geqslant 3,$ let $x=6k\geqslant p^{2}_{n+1}-1.$ The sieve
$\sum_{s=3}^{n}\frac{x}{s(s+1)}$ can be considered as a restriction of the
sieve $\sum_{s=1}^{n}\frac{x}{s(s+1)}$ on the sequence

$1,2,3,4,5,\ldots,x$ (3.18)

to the sequence of integers

$1,4,7,10,13,\ldots,6t-2,6t+1,\ldots,6k-2.$ (3.19)

Therefore

$\frac{x}{3}-\sum_{s=3}^{n}\frac{x}{s(s+1)}=\frac{x}{(n+1)}.$ (3.20)

Multiplying Equation (3.20) by $\frac{1}{2}$ we obtain

$\frac{x}{6}-\sum_{s=3}^{n}\frac{x}{2s(s+1)}=\frac{x}{2(n+1)}$ (3.21)

so that $\frac{x}{2(n+1)}$ is the order of the residue of the Sequence (3.19)
when the terms are counted in pairs. Arranging the terms of the Sequence
(3.19) as pairs in the set

$W=\\{1,6t-2\\}\cup\\{(6t-2,6t+1)\;|\;1\leqslant t\leqslant k\\}$

we can consider Equation (3.21) as a sieve on the set $W$ with a residue of
order $\frac{x}{2(n+1)},$ when the remaining elements are counted in pairs.
Our aim is to compare $\frac{x}{2(n+1)}$ with $|{\cal R}(x,n)|$ as $n$
increases.

Since ${\cal R}(x,n+1)$ can be considered as a residue set of the Sieve (3.16)
on $S$ or on the integers $t,$ $1\leqslant t<k,$ we can compare the cumulative
effect of the Sieve (3.21) on $W$ to that of (3.16) on $S.$ Going back to our
proof of Theorem 2.8 suppose that there is an integer $n$ such that for all
integers $x=x_{n+1}=6k_{n+1}\geqslant p^{2}_{n+2}-1,$ we have
$\frac{x}{(n+1)(n+2)}\geqslant 6\times T(x,n),$ that is,
$\frac{x}{(n+1)(n+2)}$ sifts out more than six times as many terms from
$\frac{x}{n+1}$ as does $T(x,n)$ from the residue of $S(x,n).$ Then
$\frac{x}{3(n+1)(n+2)}$ will be greater than or equal to $2\times T(x,n),$
that is, $\frac{x}{3(n+1)(n+2)}$ will sift our more terms from
$\frac{x}{3(n+1)}$ than $2\times T(x,n)$ from the $n^{\rm th}$ residue of the
sequence (3.15), viewed over the interval $\frac{x}{3}$ or $2k.$ The result is
that $\frac{x}{6(n+1)(n+2)}$ will sift our more terms from $\frac{x}{6(n+1)}$
than $T(x,n)$ from the $n^{\rm th}$ residue of the set $S,$ viewed over the
interval $\frac{x}{6}$ or $k.$

Suppose further that for this value of $n$ we have $\frac{x}{2(n+1)}<|{\cal
R}(x,n)|$ or, equivalently,

$a_{2}(n,x)=\frac{2(n+1)|{\cal R}(x,n)|}{x}>1$

for all integers $x=6k\geqslant p^{2}_{n+1}-1.$ Then subtracting $T(x,n)$
terms from the $n^{\rm th}$ residue of the set $S$ has the same effect as
sifting out all pairs $(6t-1,6t+1)$ for which $6t-1$ or $6t+1$ is divisible by
$p_{n+1}.$ Further all terms $(6t-1,\;),$ $(\;,6t+1)$ with components
divisible by $p_{n+1},$ is each sifted out paired with another component not
divisible by $p_{n+1}$ in either a double or single component pair. The effect
is that we are left with pairs $(6t-1,6t+1)$ less than $x_{n+1}$ for which no
component divisible by $p_{n+1}.$ The same pairs are not sifted out from
${\cal R}(x,n)$ when we subtract $Q(x,n).$ But the cumulative residue of the
Sieve (3.21) is $\frac{x}{2(n+2)}.$ Thus if $a_{2}(n,x)>1,$ then since
$Q(x,n)\leqslant T(x,n)$ we would have

$\displaystyle\begin{array}[]{ll}\frac{x}{2(n+2)}&<|{\cal
R}(x,n)|-Q(x,n)\\\\[8.61108pt] &=|{\cal R}(x,n+1)|.\end{array}$

Thus it suffices to find an integer $n$ for which $a_{2}(n,x)>1$ and
$\frac{x}{(n+1)(n+2)}\geqslant 6\times T(x,n),$ for all $x=6k\geqslant
p^{2}_{n+1}-1.$ By the result of Theorem 2.8 the the condition
$\frac{x}{(n+1)(n+2)}\geqslant 6\times T(x,n),$ would remain true for all
integers greater than $m\geqslant n$ and the result of the theorem then
follows by induction. Clearly if $a_{n}(x)\geqslant 6$ for all $x=6k\geqslant
p^{2}_{n+1}-1,$ then we must also have $\frac{x}{(n+1)(n+2)}\geqslant 6\times
T(x,n)$ over all values $x=6k\geqslant p^{2}_{n+2}-1.$ But this is the case
for $n=70$ as can be seen from Examples 2.9, as it suffices to only consider
values of $x,$ $p_{71}^{2}=124609\leqslant x\leqslant\prod_{s=1}^{70}p_{s}.$
In Example 3.3, below, we show that $\frac{x}{2(n+1)}<|{\cal R}(x,n)|$ or,
equivalently, that $a_{2}(n,x)>1$ for values of $x,$
$p_{71}^{2}-1=124608\leqslant x\leqslant\prod_{s=1}^{70}p_{s},$ and clearly
this suffices to show that the same is true for all values of $x\geqslant
124608.$ This completes the proof of the theorem. ∎

As a consequence of Theorem 3.2 we have for $x=p^{2}_{n+1}-1,$

$\left[\frac{x}{2(n+1)}\right]<|{\cal
R}(x,n)|+\pi_{2}\left(p_{n}\right)=\pi_{2}\left(p^{2}_{n+1}\right)$

for all $n\geqslant 70.$

The cases $20\leqslant n\leqslant 69$ of Theorem 1.3 may be checked
independently.

######  Example 3.3.

In this case $n=70.$ The table below shows the values of $a_{2}(n,x)$ for some
integers $x,$
$p_{71}^{2}-1=124608\leqslant x\leqslant\prod_{s=1}^{70}p_{s}.$ For the last
value of $x$ in the table, we use the formula:

$a_{2}(n,\prod_{s=1}^{70}p_{s})=\frac{(142)\prod_{s=3}^{70}(p_{s}-2)}{\prod_{s=1}^{70}p_{s}}.$

$x$ | $a_{2}(n,x)$
---|---
$124\;608$ | $1.6501$
$654\;480$ | $1.5613$
$1\;885\;128$ | $1.6317$
$3\;279\;720$ | $1.6613$
$5\;870\;928$ | $1.6855$
$9\;480\;240$ | $1.7010$
$13\;890\;528$ | $1.7079$
$29\;626\;248$ | $1.7158$
$62\;710\;560$ | $1.7136$
$223\;092\;870$ | $1.7030$
$6\;469\;693\;230$ | $1.6943$
$\prod_{s=1}^{70}p_{s}$ | 1.6960

The above computations together with the result of Theorem 2.8, suffice to
show that $a_{2}(70,x)>1$ for all $x\geqslant 124608.$

We now show that the Hardy-Littlewood conjecture (2.9) implies Conjecture 1.2.
Accordingly to the Hardy-Littlewood conjecture $\pi_{2}\left(n\right)$ is
approximately equal to $2C_{2}\frac{n}{\log^{2}n}.$ Thus to show that

$a\frac{p^{2}_{n+1}}{2(n+1)}<\pi_{2}\left(p^{2}_{n+1}\right)$

it would suffice to prove that

$2C_{2}\frac{p_{n+1}^{2}}{(4{\log}^{2}(p_{n+1}))}>a\frac{p^{2}_{n+1}}{2(n+1)}$

for all $n$ sufficiently large, which is equivalent to
${\log}^{2}p_{n+1}<C_{2}\frac{(n+1)}{a}.$ By Corollary 2.5 (ii)

$p_{n+1}<(n+1)({\log}(n+1)+{\log}{\log}(n+1))$

for $n>6.$ Thus, it would be enough to show that
${\log}^{2}((n+1)({\log}(n+1)+{\log}{\log}(n+1)))<C_{2}\frac{(n+1)}{a}.$
Regarding

${\log}^{2}((n+1)({\log}(n+1)+{\log}{\log}(n+1)))$

as a function of $n,$ we get its derivative to be less than

$2{\log}((n+1)({\log}(n+1)+{\log}{\log}(n+1)))\left(\frac{3}{(n+1)}\right).$

Noting that $n+1>{\log}(n+1)$ and $n+1>{\log}{\log}(n+1)$ this in turn implies
that the derivative is less than

$\left(\frac{6}{(n+1)}\right)(\log 2+2\log(n+1))$

which is smaller than $\frac{C_{2}}{a}$ for all $n$ sufficiently large. Thus
for values of $n$ large enough

${\log}^{2}((n+1)({\log}(n+1)+{\log}{\log}(n+1)))$

is less than $C_{2}\frac{(n+1)}{a}.$ This shows that in general
$\frac{p^{2}_{n+1}}{2(n+1)}$ is a weak lower bound for
$2C_{2}\frac{p_{n+1}^{2}}{(4{\log}^{2}(p_{n+1}))}.$

######  Acknowledgment.

We would like to thank the following people for their helpful comments and for
identifying errors and oversights in earlier versions of this work; Dang Vo
Phuc, Stephan Wagner, Berndt Gensel, Shalin Singh and Abebe Tufa. We are also
thankful to P. Kaelo for his assistance with several computer programmes that
provided the empirical results in this paper.

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