††thanks:

# Spectral analysis for singularity formation of the two dimensional Keller-
Segel system

Charles Collot  Courant Institute of Mathematical Sciences, New York
University, 251 Mercer Street, New York, NY 10003, United States of America.
<EMAIL_ADDRESS>Tej-Eddine Ghoul  Department of Mathematics, New York
University in Abu Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi, United
Arab Emirates<EMAIL_ADDRESS>Nader Masmoudi  Department of Mathematics, New
York University in Abu Dhabi, Saadiyat Island, P.O. Box 129188, Abu Dhabi,
United Arab Emirates<EMAIL_ADDRESS>Van Tien Nguyen  Department of
Mathematics, New York University in Abu Dhabi, Saadiyat Island, P.O. Box
129188, Abu Dhabi, United Arab Emirates<EMAIL_ADDRESS>
###### Abstract

We analyse an operator arising in the description of singular solutions to the
two-dimensional Keller-Segel problem. It corresponds to the linearised
operator in parabolic self-similar variables, close to a concentrated
stationary state. This is a two-scale problem, with a vanishing thin
transition zone near the origin. Via rigorous matched asymptotic expansions,
we describe the eigenvalues and eigenfunctions precisely. We also show a
stability result with respect to suitable perturbations, as well as a
coercivity estimate for the non-radial part. These results are used as key
arguments in a new rigorous proof of the existence and refined description of
singular solutions for the Keller-Segel problem by the authors [Collot et
al.(2019a)Collot, Ghoul, Masmoudi, and Nguyen]. The present paper extends the
result by Dejak, Lushnikov, Yu, Ovchinnikov and Sigal [Dejak et
al.(2012)Dejak, Lushnikov, Ovchinnikov, and Sigal]. Two major difficulties
arise in the analysis: this is a singular limit problem, and a degeneracy
causes corrections not being polynomial but logarithmic with respect to the
main parameter.

###### keywords:

Keller-Segel system, Blowup solution, Blowup profile, Stability, Construction,
spectral analysis

## 1\. Introduction

We describe in this paper a detailed spectral analysis for the linear operator

$\mathscr{L}^{z}f=\Delta
f-\nabla\cdot(f\nabla\Phi_{U_{\nu}}+U_{\nu}\nabla\Phi_{f})-\beta\nabla\cdot(zf)$
(1.1)

in the radial and non-radial settings, where

$\Phi_{f}=-\frac{1}{2\pi}\log|z|\ast f,\quad
U_{\nu}(z)=\frac{8\nu^{2}}{(\nu^{2}+|z|^{2})^{2}},\quad\nabla\Phi_{U_{\nu}}(z)=-\frac{4z}{\nu^{2}+|z|^{2}},$

$\beta>0$ is a fixed constant and $0<\nu\ll 1$ is the main parameter of the
problem.

### 1.1. Origin of the spectral problem

The linear operator $\mathscr{L}^{z}$ appears in the study of singularities of
the following two dimensional parabolic-elliptic Keller-Segel system:

$\left\\{\begin{array}[]{ll}\partial_{t}u=\Delta
u-\nabla\cdot\big{(}u\nabla\Phi_{u}\big{)},\\\
\Phi_{u}=-\frac{1}{2\pi}\log|x|\ast
u,\end{array}\right.\quad(x,t)\in\mathbb{R}^{2}\times[0,T),$ (1.2)

see [Keller and Segel(1970)], [Keller and Segel(1971a)], [Keller and
Segel(1971b)], [Patlak(1953)], and [Horstmann(2003)] for a survey of the
problem. It is well known (see for example, [Jäger and Luckhaus(1992)],
[Childress and Percus(1981)], [Childress(1984)], [Diaz et al.(1998)Diaz,
Nagai, and Rakotoson], [Blanchet et al.(2006)Blanchet, Dolbeault, and
Perthame], [Blanchet et al.(2012)Blanchet, Carlen, and Carrillo] and
references therein) that the problem (1.2) exhibits finite time blowup
solutions if the initial datum satisfies $u_{0}\geq 0$, some localisation
assumptions and

$M=\int_{\mathbb{R}^{2}}u_{0}(x)dx>8\pi.$

The threshold $8\pi$ is related to the family of stationary solutions
$(U_{\eta})_{\eta>0}$ of (1.2), where

$U_{\eta}(x)=\frac{1}{\eta^{2}}U\big{(}\frac{x}{\eta}\big{)}\quad\textup{with}\quad
U(x)=\frac{8}{(1+|x|^{2})^{2}}\quad\textup{and}\quad\int_{\mathbb{R}^{2}}U_{\eta}(x)dx=8\pi.$
(1.3)

The parameter $\eta$ is linked to the scaling symmetry of the problem: if $u$
is a solution to (1.2), then for any $\eta>0$, $u_{\eta}$ defined by

$u_{\eta}(x,t)=\frac{1}{\eta^{2}}u\left(\frac{x}{\eta},\frac{t}{\eta^{2}}\right)$
(1.4)

is a solution as well. As the mass $M$ which is a conserved quantity for (1.2)
is invariant under the above transformation, the problem is called critical. A
key problem in understanding singular solutions is to analyse their asymptotic
self-similarity. If a solution to (1.2) is both singular at $t=0$ and
invariant under the transformation (1.4), it would be of the form
$(-t)^{-1}w(x/\sqrt{-t})$; non-degenerate self-similarity would then refer to
blowup solutions satisfying $u\sim(T-t)^{-1}w(x/\sqrt{T-t})$. However, one of
the remarkable facts about finite time blowup solutions of (1.2) is that they
present a degenerate self-similarity. Precisely, they are of type II blowup
(see Theorem 8.19 in [Suzuki and Senba(2011)] and Theorem 10 in [Naito and
Suzuki(2008)] for such a statement) in the following sense. A solution $u(t)$
of (1.2) exhibits type I blowup at $t=T$ if there exists a constant $C>0$ such
that

$\limsup_{t\to T}(T-t)\|u(t)\|_{L^{\infty}(\mathbb{R}^{2})}\leq C,$ (1.5)

otherwise, the blowup is of type II. Equivalently, in the parabolic self-
similar variables

$u(x,t)=\frac{1}{\mu^{2}}w(z,\tau),\quad\Phi_{u}(x,t)=\Phi_{w}(y,s),\quad
z=\frac{x}{\mu},\quad\frac{d\tau}{dt}=\frac{1}{\mu^{2}},\quad\mu(t)=\sqrt{T-t},$
(1.6)

where $w(z,\tau)$ solves the equation

$\partial_{\tau}w=\nabla\cdot\big{(}\nabla
w-w\nabla\Phi_{w}\big{)}-\beta\nabla\cdot(zw)\quad\textup{with}\quad\beta=-\frac{\mu_{\tau}}{\mu}=\frac{1}{2},$
(1.7)

$u$ is a type II finite time blowup solution of (1.2) if and only if $w$ is a
global but unbounded solution of (1.7). The mechanism of singularity formation
then involves crucially the above family of solutions $U_{\eta}$, see for
example, [HV1, Velázquez(2002), Velázquez(2004b), Velázquez(2004a), Deja14,
Dyachenko13, Raphaël and Schweyer(2014)] and references therein. The key idea
is that in equation (1.2) the time variation $\partial_{t}u$ is asymptotically
of lower order, the solution approaches the family of stationary states $u\sim
U_{\eta}$ and a scaling instability drives the parameter $\eta$ to $0$ as
$t\rightarrow T$. This motivates the study of a solution in the variables
(1.6) having the form

$w(z,\tau)=U_{\nu}(z)+\varepsilon(z,\tau),$

where $\nu=\eta/\sqrt{T-t}$ is time dependent with $\nu(\tau)\rightarrow 0$ as
$\tau\rightarrow\infty$, and $\varepsilon$ is a lower order perturbation
solving:

$\partial_{\tau}\varepsilon=\mathscr{L}^{z}\varepsilon-\nabla\cdot(\varepsilon\nabla\Phi_{\varepsilon})+\left(\frac{\nu_{\tau}}{\nu}-\beta\right)\nabla\cdot(zU_{\nu}).$
(1.8)

Above, $\mathscr{L}^{z}$ is precisely the operator introduced in (1.1) that we
aim at studying in the present paper. The importance of its study is motivated
by the following.The first rigorous construction of a blow-up solution by
Herrero and Velázquez for (1.2), [HV1], does not provide its stability, which
is formally obtained in [Velázquez(2002)]. The work [Velázquez(2002)] shows
linear stability in the inner zone $|z|\sim\nu$ if the scaling term is
neglected, and gives an expansion for
$\partial_{\tau}\varepsilon=\mathscr{L}^{z}\varepsilon$ in the parabolic zone
$|z|\sim 1$ via formal series and matched asymptotics. A rigorous radial
stability result is given by Raphaël and Schweyer in [Raphaël and
Schweyer(2014)] in which the solution is studied in blow up variables
$y=\frac{z}{\nu}\sim 1$ where a refined description is obtained. The
description involves parameters, and their evolution (the modulation laws) is
computed based on so called tail-dynamics, relying on suitable cancellations
in the parabolic zone $|z|\sim 1$. The analysis of the tail-dynamics is
however heavy, as it does not involve a refined understanding of the solution
in self-similar variables. Our precise spectral study for the operator (1.8),
however, gives a framework to control the solution accurately, _on both scales
simultaneously_ , and the temporal evolution of the parameters is easily
related to the projection of the dynamics on its eigenmodes. The present paper
is a key result in this new approach to the construction of singular solutions
to (1.2) that is implemented in [Collot et al.(2019a)Collot, Ghoul, Masmoudi,
and Nguyen], and allows to obtain a refined description (see Remark 1.3).

It is remarkable that in the radial setting, the nonlocal operator
$\mathscr{L}^{z}$ reduces to a local one in terms of the partial mass

$m_{f}(\zeta)=\frac{1}{2\pi}\int_{B(0,\zeta)}f(z)zdz,\quad\zeta=|z|,$ (1.9)

where $B(0,\zeta)$ the ball centered at $0$ of radius $\zeta$. Indeed, if $f$
is spherically symmetric, then we have the relation

$\mathscr{L}^{z}f(\zeta)=\frac{1}{\zeta}\partial_{\zeta}\Big{(}\mathscr{A}^{\zeta}m_{f}(\zeta)\Big{)},$

where $\mathscr{A}^{\zeta}$ is the linear operator defined by

$\mathscr{A}^{\zeta}=\mathscr{A}^{\zeta}_{0}-\beta\zeta\partial_{\zeta}\quad\textup{with}\quad\mathscr{A}_{0}^{\zeta}=\partial_{\zeta}^{2}-\frac{1}{\zeta}\partial_{\zeta}+\frac{Q_{\nu}}{\zeta}\partial_{\zeta}+\frac{\partial_{\zeta}(Q_{\nu})}{\zeta}\quad\textup{and}\quad
Q_{\nu}(\zeta)=\frac{4\zeta^{2}}{\zeta^{2}+\nu^{2}}.$ (1.10)

Hence, in the radial setting $\mathscr{L}^{z}$ and $\mathscr{A}^{\zeta}$ share
the same spectrum and if $\varphi$ and $\phi$ are the radial eigenfunctions of
$\mathscr{L}^{z}$ and $\mathscr{A}^{\zeta}$ respectively, we have the relation

$\varphi(\zeta)=\frac{\partial_{\zeta}\phi(\zeta)}{\zeta},\quad\Phi_{\varphi}(\zeta)=-\frac{\phi(\zeta)}{\zeta}.$

Therefore, we are interested in the eigenproblem

$\mathscr{A}^{\zeta}\phi(\zeta)=\lambda\phi(\zeta),\quad\zeta\in[0,\infty),$
(1.11)

in the regime

$\beta\sim 1,\quad 0<\nu\ll 1.$ (1.12)

Note that the constant $\beta$ is not necessarily close to 1, it can be any
fixed positive constant.

### 1.2. Main results

Our first result concerns the spectrum of $\mathscr{L}^{z}$ in the radial
setting. Its analysis has been done by Dejak, Lushnikov, Yu, Ovchinnikov and
Sigal [Dejak et al.(2012)Dejak, Lushnikov, Ovchinnikov, and Sigal] via matched
asymptotic expansions. Our approach, similar in spirit to [Dejak et
al.(2012)Dejak, Lushnikov, Ovchinnikov, and Sigal], is inspired by the work of
Collot, Merle, and Raphaël [Collot et al.(2017)Collot, Merle, and Raphael] for
the study of type II supercritical singularities of the semilinear heat
equation $u_{t}=\Delta u+|u|^{p-1}u$ (see also [Collot et al.(2019b)Collot,
Raphaël, and Szeftel, Hadžić and Raphaël(2019), Merle Raphael Szeftel (2018)]
for related problems). The strategy is to construct suitable eigenfunctions
near the origin and away from the origin, and to match them rigorously to
produce a full eigenfunction. Differentiating the matching condition then
provides information on the dependence of the eigenfunctions on the
parameters. The current work extends this approach to a critical problem,
showing its robustness. Solving (1.11), though, is not just a mere adaptation
the techniques of [Collot et al.(2017)Collot, Merle, and Raphael] because of
the following points.

This critical case displays two new degeneracies. First, this is a singular
limit problem. Indeed, from the explicit formula (1.10) for $Q_{\nu}$, we note
that the operator $\mathscr{A}^{\zeta}$ converges to a limit operator
pointwise outside the origin, namely that for any smooth function $f$ and at
any fixed $\zeta>0$, we have

$\mathscr{A}^{\zeta}f(\zeta)\rightarrow\partial_{\zeta}^{2}f(\zeta)+\frac{3}{\zeta}\partial_{\zeta}f(\zeta)-\beta\zeta\partial_{\zeta}f(\zeta)\quad\mbox{
as }\nu\rightarrow 0.$

The limit operator
$\partial_{\zeta}^{2}+3/\zeta\partial_{\zeta}-\beta\zeta\partial_{\zeta}$ is
well understood, its spectrum is $\\{0,-2\beta,-4\beta,-6\beta,...\\}$ and its
eigenfunctions are Hermite polynomials. However, the limit $\nu\rightarrow 0$
for the problem (1.11) is a singular one. The problem involves two scales: one
is $\zeta\sim 1$ and the other is $\zeta\sim\nu$. What happens at the latter
actually prevents the convergence to the aforementioned limit operator: the
spectrum is shifted by the constant $2\beta$ at the leading order as is shown
in Proposition 1.1 below. This in particular prevents the use of a bifurcation
argument. Second, this problem also presents another degeneracy from which
most of the technical difficulty stems from, since next order corrections,
instead of being polynomial in the parameter $\nu$, are actually polynomial in
$1/|\log\nu|$. We then need to refine to higher order the description of both
the inner solution at $\zeta\sim\nu$ and the outer solution at $\zeta\sim 1$.

We provide a precise description of the eigenfunctions, relating them to the
iterated kernel $(T_{i})_{i\in\mathbb{N}}$ of $\mathscr{A}_{0}$, the
linearised operator near the stationary state, a rescaled version of
$\mathscr{A}_{0}^{\zeta}$ via the change $\zeta=\nu r$, i.e.

$\mathscr{A}_{0}=\partial_{r}^{2}-\frac{1}{r}\partial_{r}+\frac{\partial_{r}(Q\cdot)}{r}\quad\textup{with}\quad
Q(r)=\frac{4r^{2}}{1+r^{2}},$ (1.13)

defined by:

$T_{j+1}(r)=-\mathscr{A}_{0}^{-1}T_{j}(r)\quad\textup{with}\quad
T_{0}(r)=\frac{r^{2}}{(1+r^{2})^{2}}=\frac{1}{8}r\partial_{r}Q,\;\;\mathscr{A}_{0}T_{0}=0.$

As $r\partial_{r}Q$ is the direction of scaling instability for the stationary
state, this description allows to understand the rescaled blow-up dynamics
(1.8). In addition, the properties of $T_{j}$ can be explicitly computed, such
as its asymptotic behavior (see Lemma 2.2)

$T_{j}(r)\sim\hat{d}_{j}r^{2j-2}\ln r,\quad\mbox{as
}r\rightarrow\infty,\quad\hat{d}_{j}\neq 0\mbox{ a constant.}$

To state our results, we use the notation $A\lesssim B$ to say that there
exists a constant $C>0$ such that $0\leq A\leq CB$. Similarly, $A\sim B$ means
that there exist constants $0<c<C$ such that $cA\leq B\leq CA$. We write
$\left<r\right>=\sqrt{1+r^{2}}$, and use the notation $D_{\zeta}^{k}$ for
$k\in\mathbb{N}$ for $k$-th adapted derivative with respect to $\zeta$ defined
by

$D_{\zeta}^{2k}=\left(\zeta\partial_{\zeta}(\frac{\partial_{\zeta}}{\zeta})\right)^{2k},\
\ D_{\zeta}^{2k+1}=\partial_{\zeta}D^{2k},$

and the weight functions

$\omega_{\nu}(\zeta)=\frac{\nu^{2}}{U_{\nu}(\zeta)}e^{-\frac{\beta\zeta^{2}}{2}}=\frac{\nu^{2}}{U_{\nu}(\zeta)}\rho_{0}(\zeta),\quad\rho_{0}(\zeta)=e^{-\frac{\beta\zeta^{2}}{2}},$
(1.14)

with corresponding weighted Lebesgue space
$L^{2}_{\frac{\omega_{\nu}}{\zeta}}$ and Sobolev spaces
$H^{k}_{\frac{\omega_{\nu}}{\zeta}}=\\{f:\;\sum\limits_{0}^{k}\|D^{k}f\|_{L^{2}_{\frac{\omega_{\nu}}{\zeta}}}<\infty\\}$.
Our first main result is to describe in details spectral properties of
$\mathscr{A}^{\zeta}$ in the regime (1.12).

######  Proposition 1.1 (Spectral properties of $\mathscr{A}^{\zeta}$).

The linear operator
$\mathscr{A}^{\zeta}:H^{2}_{\frac{\omega_{\nu}}{\zeta}}\rightarrow
L^{2}_{\frac{\omega_{\nu}}{\zeta}}$ is essentially self-adjoint with compact
resolvant. Moreover, given any $N\in\mathbb{N}$, $0<\beta_{*}<\beta^{*}$ and
$0<\delta\ll 1$, there exists a $\nu^{*}>0$ such that the following holds for
all $0<\nu\leq\nu^{*}$ and $\beta_{*}\leq\beta\leq\beta^{*}$.
$(i)$ (Eigenvalues) The first $N+1$ eigenvalues are given by

$\lambda_{n,\nu}=2\beta\Big{(}1-n+\tilde{\alpha}_{n,\nu}\Big{)},\quad\mbox{for
}n=0,1,\cdots,N,$ (1.15)

where

$\tilde{\alpha}_{n,\nu}=\frac{1}{2\ln\nu}+\bar{\alpha}_{n,\nu}\quad\mbox{with
}\quad|\bar{\alpha}_{n,\nu}|+\big{|}\nu\partial_{\nu}\tilde{\alpha}_{n,\nu}\big{|}\lesssim\frac{1}{|\ln\nu|^{2}}.$
(1.16)

In particular, we have the refinement of the first two eigenvalues with
$\gamma$ the Euler’s constant:

$\left|\tilde{\alpha}_{n,\nu}-\frac{1}{2\ln\nu}-\frac{\ln
2-\gamma-n-\ln\beta}{4|\ln\nu|^{2}}\right|\lesssim\frac{1}{|\ln\nu|^{3}},\quad\mbox{for
}n=0,1.$

$(ii)$ (Eigenfunctions) There exist eigenfunctions $\phi_{n,\nu}$ satisfying
the following. There holds the pointwise estimates for $k=0,1,2$:

$\displaystyle\left|D_{\zeta}^{k}\phi_{n,\nu}(\zeta)\right|+\left|D_{\zeta}^{k}\beta\partial_{\beta}\phi_{n,\nu}(\zeta)\right|$
$\displaystyle+\left|D_{\zeta}^{k}\nu\partial_{\nu}\phi_{n,\nu}(\zeta)\right|$
$\displaystyle\qquad\lesssim\left(\frac{\zeta}{\nu+\zeta}\right)^{2-(k\;\textup{mod}\;2)}\frac{\langle\zeta\rangle^{2n+\delta}\big{(}1+\ln\big{\langle}\frac{\zeta}{\nu}\big{\rangle}\,\mathbf{1}_{\\{n\geq
1\\}}\big{)}}{(\zeta+\nu)^{2+k}}.$ (1.17)

There holds in addition the refined identity:

$\phi_{n,\nu}(\zeta)=\sum_{j=0}^{n}c_{n,j}\beta^{j}\nu^{2j-2}T_{j}\big{(}\frac{\zeta}{\nu}\big{)}+\tilde{\phi}_{n,\nu}(\zeta),$

where the profiles $T_{j}$ and the constants $c_{n,j}$ are defined in Lemma
2.2, with for $k=0,1,2$:

$\left|D_{\zeta}^{k}\tilde{\phi}_{n,\nu}(\zeta)\right|+\left|D_{\zeta}^{k}\nu\partial_{\nu}\tilde{\phi}_{n,\nu}(\zeta)\right|+\left|D_{\zeta}^{k}\beta\partial_{\beta}\tilde{\phi}_{n,\nu}(\zeta)\right|\lesssim\min\left(\nu^{2}\langle\frac{\zeta}{\nu}\rangle^{2},\frac{1}{|\ln\nu|}\right)\left(\frac{\zeta}{\nu+\zeta}\right)^{2-(k\;\textup{mod}\;2)}\frac{\langle\zeta\rangle^{2n+\delta}}{(\zeta+\nu)^{2+k}}.$

There holds the $L^{2}_{\frac{\omega_{\nu}}{\zeta}}$ estimates for all $0\leq
m,n\leq N$:

$\langle\phi_{n,\nu},\phi_{m,\nu}\rangle^{2}_{L^{2}_{\frac{\omega_{\nu}}{\zeta}}}=c_{n}\delta_{m,n},\quad
c_{0}\sim\frac{|\ln\nu|}{8},\quad c_{1}\sim\frac{|\ln\nu|^{2}}{4},\quad
c|\ln\nu|^{2}\leq c_{n}\leq\frac{1}{c}|\ln\nu|^{2}\mbox{ for }n\geq 2,$ (1.18)

where $c$ is some positive constant.

$(iii)$ (Spectral gap estimate) For any $g\in
L^{2}_{\frac{\omega_{\nu}}{\zeta}}$ with $\langle
g,\phi_{j,\nu}\rangle_{L^{2}_{\frac{\omega_{\nu}}{\zeta}}}=0$ for $0\leq j\leq
N$, one has

$\big{\langle}g,\mathscr{A}^{\zeta}g\big{\rangle}_{L^{2}_{\frac{\omega_{\nu}}{\zeta}}}\leq\lambda_{N+1,\nu}\|g\|^{2}_{L^{2}_{\frac{\omega_{\nu}}{\zeta}}}.$
(1.19)

######  Remark 1.2.

We recover the same eigenvalues as [Dejak et al.(2012)Dejak, Lushnikov,
Ovchinnikov, and Sigal]. Though our proof is similar to [Dejak et
al.(2012)Dejak, Lushnikov, Ovchinnikov, and Sigal] since relying on matched
asymptotics, we here adopt the approach of [Collot et al.(2017)Collot, Merle,
and Raphael], yielding detailed information on the eigenfunctions and on the
variations with respect to the parameter $\nu$. We also mention that the
matching procedure performed in [Dejak et al.(2012)Dejak, Lushnikov,
Ovchinnikov, and Sigal] was formal as the analysis did not involve the
matching of derivatives. To match the derivatives, we found a degeneracy that
forces us to expand both inner and outer solutions to the next order, which
renders the analysis much more involved.

######  Remark 1.3.

Based on Proposition 1.1, we are able to construct for the problem (1.2) in
[Collot et al.(2019a)Collot, Ghoul, Masmoudi, and Nguyen] finite time blowup
solutions according to the following precise asymptotic dynamics as $t\to T$:

$u(x,t)\sim\frac{1}{\eta^{2}(t)}U\big{(}\frac{x}{\eta(t)}\Big{)},$

where the blowup rate is given by either

$\eta(t)=2e^{-\frac{\gamma+2}{2}}\sqrt{T-t}e^{-\sqrt{\frac{|\ln(T-t)|}{2}}}(1+o_{t\uparrow
T}(1)),$

or

$\eta(t)\sim
C(u_{0})(T-t)^{\frac{\ell}{2}}|\ln(T-t)|^{-\frac{\ell+1}{2(\ell-1)}}\quad\textup{for
some }\;\;\ell\in\mathbb{N},\;\;\ell\geq 2.$

It is worth saying that the rigorous analysis performed in [Collot et
al.(2019a)Collot, Ghoul, Masmoudi, and Nguyen] is greatly simplified thanks to
Proposition 1.1 in comparison with the one of [Raphaël and Schweyer(2014)].
Importantly, we believe that the precise description of the spectrum of
$\mathscr{A}^{\zeta}$ is one of the crucial steps toward the classification of
all possibilities of blowup speeds for (1.2) (at least in the radial setting)
which is a challenging problem in the analysis of blowup.

######  Remark 1.4.

The present result deals with the critical Keller-Segel system. We believe
that other critical problems can be studied with this approach, such as the
harmonic heat flow and the semilinear heat equation. Related spectral studies
were performed in the case of non-degenerate self-similar singularities for
wave type equations, see for example [Costin et al.(2017)Costin, Donninger,
and Glogić], [Costin et al.(2016)Costin, Donninger, and Xia] for the study of
stability of self-similar wave maps. It is an interesting direction to
implement the present work to the hyperbolic setting.

Our second result aims at understanding under what kind of perturbations is
Proposition 1.1 stable. This is of a particular importance for the full
nonlinear problem (1.2) analysed in [Collot et al.(2019a)Collot, Ghoul,
Masmoudi, and Nguyen], and shows the robustness of our approach. As a direct
consequence of our construction, the spectral properties of
$\mathscr{A}^{\zeta}$ stated in the previous proposition still hold true for
the following perturbed operator of the form

$\bar{\mathscr{A}}^{\zeta}=\mathscr{A}^{\zeta}+\frac{1}{\zeta}\partial_{\zeta}\left(P\cdot\right),$
(1.20)

where the perturbation $P$ satisfies

$|P(\zeta)|+|\zeta\partial_{\zeta}P(\zeta)|\lesssim\frac{\nu^{2}}{|\ln\nu|}\frac{\zeta^{2}}{(\nu^{2}+\zeta^{2})^{2}}.$
(1.21)

######  Proposition 1.5.

Assume the bound (1.21) and the same hypotheses as in Proposition 1.1. Then,
the operator
$\bar{\mathscr{A}}^{\zeta}:H^{2}_{\frac{\bar{\omega}_{\nu}}{\zeta}}\rightarrow
L^{2}_{\frac{\bar{\omega}_{\nu}}{\zeta}}$ is essentially self-adjoint with
compact resolvent, where

$\bar{\omega}_{\nu}(\zeta)=\omega_{\nu}(\zeta)\exp\left(\int_{0}^{\zeta}\frac{P(\tilde{\zeta})}{\tilde{\zeta}}d\tilde{\zeta}\right).$

The first $N+1$ eigenvalues $\\{\bar{\lambda}_{n,\nu}\\}_{0\leq n\leq N}$ of
$\bar{\mathscr{A}}^{\zeta}$ satisfy

$|\bar{\lambda}_{n,\nu}-\lambda_{n,\nu}|\leq\frac{C^{\prime}}{|\log\nu|^{2}},$
(1.22)

and there exist associated eigenfunctions $\\{\bar{\phi}_{n,\nu}\\}_{0\leq
n\leq N}$ satisfying

$\frac{\|\bar{\phi}_{n,\nu}-\phi_{n,\nu}\|_{L^{2}(\frac{\omega_{\nu}}{\zeta})}}{\|\phi_{n,\nu}\|_{L^{2}(\frac{\omega_{\nu}}{\zeta})}}\leq\frac{C^{\prime}}{\sqrt{|\log\nu|}}.$
(1.23)

######  Remark 1.6.

Note that Proposition 1.5 is not a direct consequence of Proposition 1.1 in
the sense that a standard perturbation argument does not work here. Indeed,
the potential part $\partial_{\zeta}P/\zeta$ of the perturbation in (1.20) is
of size $\nu^{-2}$ in $L^{\infty}$ (up to a logarithmic accuracy), while the
eigenvalues of the unperturbed operator $\mathscr{A}^{\zeta}$ are of order
$1$. The crucial point is that the algebraic form of the perturbation,
$\partial_{\zeta}(P\cdot)/\zeta$, ensures its orthogonality to the resonance
of the operator $\mathscr{A}_{0}$ near the origin, see Lemma 2.4 and its
proof.

######  Remark 1.7.

In [Collot et al.(2019a)Collot, Ghoul, Masmoudi, and Nguyen], the use of
Proposition 1.5 is essential to handle nonlinear terms, where the precise
control of the solution near the origin involves the rescaled stationary state
at a slightly different scale $\tilde{\nu}$, and the corresponding perturbed
linear operator is (1.20) with the perturbation potential

$P(\zeta)=\frac{Q_{\tilde{\nu}}(\zeta)-Q_{\nu}(\zeta)}{2},\quad\left|\frac{\tilde{\nu}}{\nu}-1\right|\lesssim\frac{1}{|\log\nu|},$

and the corresponding weight function
$\bar{\omega}_{\nu}(\zeta)=\frac{\nu\tilde{\nu}}{\sqrt{U_{\nu}U_{\tilde{\nu}}}}\rho(\zeta).$

Our third and last result concerns the decay of the linearised dynamics
associated to $\mathscr{L}^{z}$ for the nonradial part of the perturbation.
The work [Velázquez(2002)] provides Lyapunov functionals for the inner zone
$|z|\sim\nu$, and [Raphaël and Schweyer(2014)] uses a suitable extension to a
higher regularity level of these Lyapunov functionals, inspired from [RR].
Both results do not include the scaling term in the functionals, making the
control in the parabolic region $|z|\sim 1$ hard. We prove here a coercivity
estimate that takes the scaling term into account, for a modified version of
the linearised operator, in which the source term for the Poisson field is
localised near the origin. Note that an analogue of the radial spectral
analysis of Proposition 1.1 is not straightforward. Indeed, while the operator
$\nabla\Delta^{-1}$ is an integral operator from the origin in the radial
case, the integral involve the behaviour of the function at infinity on higher
order spherical harmonics, see (A.6) and (A.8). In particular, it is not
possible to make sense of $\nabla\Delta^{-1}$ for nonradial functions with
strong polynomial growth at infinity.

On the one hand, at the $|z|\sim\nu$ scale, there is a natural scalar product
for the linearised operator without scaling term, coming from the free energy.
The following corresponds to [Raphaël and Schweyer(2014)], Lemma 2.1 and
Proposition 2.3. The linearized operator at scale $\nu$ is written as

$\mathscr{L}_{0}u=\Delta
u-\nabla\cdot(u\nabla\Phi_{U})-\nabla\cdot(U\nabla\Phi_{u})=\nabla\cdot(U\nabla\mathscr{M}u)\quad\textup{with}\quad\mathscr{M}u=\frac{u}{U}-\Phi_{u},$
(1.24)

The quadratic form $\int u\mathscr{M}vdy$ is symmetric. There hold the
estimates if $\int udy=0$:

$\int_{\mathbb{R}^{2}}U|\mathscr{M}u|^{2}dy\lesssim\int_{\mathbb{R}^{2}}\frac{u^{2}}{U}dy,$
(1.25)

the nonnegativity $\int u\mathscr{M}u\geq 0$ and, for some $\delta_{1},C>0$,

$\int_{\mathbb{R}^{2}}u\mathscr{M}udy\geq\delta_{1}\int_{\mathbb{R}^{2}}\frac{u^{2}}{U}dy-C\Big{[}\langle
u,\Lambda U\rangle_{L^{2}}^{2}+\langle
u,\partial_{1}U\rangle_{L^{2}}^{2}+\langle
u,\partial_{2}U\rangle_{L^{2}}^{2}\Big{]},$ (1.26)

where $\Lambda$ is the scaling group infinitesimal generator:

$\Lambda u=2u+y.\nabla u.$

For functions orthogonal to $\Lambda U,\partial_{y_{1}}U,\partial_{y_{2}}U$ in
the $L^{2}$ sense, the norms defined by $\int\frac{u^{2}}{U}dy$ and $\int
u\mathscr{M}udy$ are then equivalent. On the other hand, at scale $|z|\sim 1$,
from (1.3) and as $\partial_{\zeta}\Phi_{U_{\nu}}=-4\zeta/(\nu^{2}+\zeta^{2})$
we get that $\mathscr{L}^{z}$ converges pointwise to
$\Delta+4/\zeta\partial_{\zeta}-\beta\nabla.(z\cdot)$ as $\nu\rightarrow 0$.
This operator is self adjoint in $L^{2}(\zeta^{4}\rho_{0})$. We thus introduce
the ”mixed” scalar product

$\langle
u,v\rangle_{\ast}:=\nu^{2}\int_{\mathbb{R}^{2}}u\sqrt{\rho_{0}}\mathscr{M}^{z}(v\sqrt{\rho_{0}})dz,\quad\mathscr{M}^{z}f=\frac{f}{U_{\nu}}-\Phi_{f}.$
(1.27)

It matches to leading order the first scalar product at scale $\nu$ and the
second at scale $1$, and localises the Poisson field. It is equivalent to the
$L^{2}_{\omega_{\nu}}$ scalar product under the aforementioned orthogonality
conditions. We localize the Poisson field in the linearized operator
accordingly,

$\tilde{\mathscr{L}}^{z}u:=\Delta
u-\nabla\cdot(u\nabla\Phi_{U_{\nu}})-\nabla\cdot(U_{\nu}\nabla\tilde{\Phi}_{u})-\beta\nabla\cdot(zu),\quad\tilde{\Phi}_{u}:=\frac{1}{\sqrt{\rho_{0}}}(-\Delta)^{-1}(\sqrt{\rho_{0}}u).$
(1.28)

We show that in the non-radial sector, the localised operator
$\tilde{\mathscr{L}}^{z}$ is coercive for the mixed scalar product
$\langle\cdot,\cdot\rangle_{*}$ under the natural orthogonality assumption to
$\nabla U_{\nu}$. Its proof adapts the arguments of [Raphaël and
Schweyer(2014)] for the above coercivity of $\mathscr{L}_{0}$ to the range
$0<\nu\ll 1$.

######  Proposition 1.8.

For any $0<\beta_{*}<\beta^{*}$, there exists $c,C>0$ and $\nu^{*}>0$ such
that for all $\beta_{*}<\beta<\beta^{*}$ and $0<\nu\leq\nu^{*}$, if
$u\in\dot{H}^{1}_{\omega_{\nu}}$ satisfies $\int_{|z|=\zeta}udz=0$ for almost
every $\zeta$, then:

$\langle-\tilde{\mathscr{L}}^{z}u,u\rangle_{\ast}\geq c\|\nabla
u\|_{L^{2}_{\omega_{\nu}}}^{2}-C\nu^{6}\left(\left(\int_{\mathbb{R}^{2}}u\partial_{z_{1}}U_{\nu}\sqrt{\rho_{0}}dz\right)^{2}+\left(\int_{\mathbb{R}^{2}}u\partial_{z_{2}}U_{\nu}\sqrt{\rho_{0}}dz\right)^{2}\right).$
(1.29)

######  Remark 1.9.

The above Proposition holds for $\tilde{\mathscr{L}}^{z}$ instead of
$\mathscr{L}^{z}$: a part of the Poisson field outside the origin has been
neglected. However, in the singularity formation studied in [Collot et
al.(2019a)Collot, Ghoul, Masmoudi, and Nguyen] the worst contribution to this
field from the perturbation comes from the origin, and the stationary states
decays rapidly at infinity. The difference
$\mathscr{L}^{z}-\tilde{\mathscr{L}}^{z}$ can then be controlled from other
norms, see [Collot et al.(2019a)Collot, Ghoul, Masmoudi, and Nguyen].

The paper is organised as follows. Section 2 is devoted to the proof of
Propositions 1.1 and 1.5. The proof of Proposition 1.8 is done in Section 3.

#### Acknowledgement:

C. Collot is supported by the ERC- 2014-CoG 646650 SingWave. N. Masmoudi is
supported by NSF grant DMS-1716466. This work was supported by Tamkeen under
the NYU Abu Dhabi Research Institute grant of the center SITE. Part of this
work was done while C. Collot, T.-E. Ghoul and N. Masmoudi were visiting IHÉS
and they thank the institution.

## 2\. Proof of the spectral Propositions 1.1 and 1.5

This section is devoted to the proof of Proposition 1.1. After the change of
variable $\zeta=\nu r$, the problem (1.11) is equivalent to the following

$(\mathscr{A}_{0}-br\partial_{r})\phi=\alpha\phi,\quad r\in[0,\infty),$ (2.1)

where $\mathscr{A}_{0}$ is introduced in (1.13) and

$b=\beta\nu^{2},\quad\alpha=\lambda\nu^{2}.$

We will solve the problem (2.1) in the regime $0<b\ll 1$ by means of matched
asymptotic expansions in the following sense. Let $\zeta_{0}$ and $R_{0}$ be
fixed as

$0<\zeta_{0}\ll 1,\quad R_{0}=\frac{\zeta_{0}}{\sqrt{b}}\gg 1.$

Relying on perturbation theory, we first solve (2.1) in the inner region
$r\leq R_{0}$, and the solution is named by $\phi^{\textup{in}}$, then in the
outer region $r\geq R_{0}$ and the solution is named by $\phi^{\textup{ex}}$.
The two solutions must coincide at $r=R_{0}$ up to the first derivative from
which we determine the value of $\alpha$ by standard arguments based on the
implicit function theorem. Proposition 1.1 is a direct consequence of the
following.

######  Proposition 2.1 (Spectral properties of
$\mathscr{A}=\mathscr{A}_{0}-br\partial_{r}$).

The linear operator $\mathscr{A}$ is self-adjoint in
$L^{2}(\mathbb{R}^{+},\omega_{b}dr)$ with
$\omega_{b}^{-1}=rUe^{\frac{br^{2}}{2}}$ with compact resolvent. Given any
$N\in\mathbb{N}$, and $0<\delta\ll 1$, there exists a $b^{*}>0$ such that the
following holds for all $0<b\leq b^{*}$:
$(i)$ (Eigenvalues) We have that the first $N+1$ eigenvalues are given by:

$\alpha_{n}=2b\Big{(}1-n+\tilde{\alpha}_{n}\Big{)},\qquad n=0,...,N,$ (2.2)

where:

$\tilde{\alpha}_{n}=\frac{1}{\ln b}+\bar{\alpha}_{n}\quad\mbox{with
}\quad|\bar{\alpha}_{n}|+\big{|}b\partial_{b}\tilde{\alpha}_{n}\big{|}\lesssim\frac{1}{|\ln
b|^{2}}.$ (2.3)

In particular, we have a refinement of the first two eigenvalues, with
$\gamma$ the Euler constant:

$\left|\tilde{\alpha}_{n}-\frac{1}{\ln b}-\frac{\ln 2-\gamma-n}{|\ln
b|^{2}}\right|\lesssim\frac{1}{|\ln b|^{3}},\quad\mbox{for }n=0,1.$ (2.4)

$(ii)$ (Eigenfunctions) An eigenfunction $\phi_{n}$ is defined by (2.69) and
the following properties hold:

  * -

(Sign-changing) On the interval $[0,\infty)$, $\phi_{0}$ has constant sign,
and $\phi_{n}$ vanishes exactly $n$ times for $n\geq 1$.

  * -

(Orthogonality) For some positive constants $\bar{e}_{n}$ there holds:

$\forall(m,n)\in\mathbb{N}^{2},\quad\langle\phi_{n},\phi_{m}\rangle^{2}_{L^{2}_{\omega_{b}}}=c_{n}\delta_{m,n},\qquad
c_{n}\sim\left\\{\begin{array}[]{ll}2^{-4}|\ln
b|^{n+1}&\;\textup{for}\;n=0,1,\\\ \bar{e}_{n}|\ln
b|^{2}&\;\textup{for}\;n\geq 2.\end{array}\right.$ (2.5)

  * -

(Pointwise estimates) For $k=0,1,2$,

$\displaystyle\left|D_{r}^{k}\phi_{n}(r)\right|+\left|D_{r}^{k}b\partial_{b}\phi_{n}(r)\right|$
$\displaystyle\lesssim\left(\frac{r}{\langle
r\rangle}\right)^{2-(k\;\textup{mod}\;2)}\langle
r\rangle^{-2-k}\langle\sqrt{b}r\rangle^{2n+\delta}(1+\mathbf{1}_{\\{n\geq
1\\}}\ln\langle r\rangle).$ (2.6)

  * -

(Refined pointwise estimates) There holds the refined identity:

$\phi_{n}(r)=\sum_{j=0}^{n}c_{n,j}b^{j}T_{j}\big{(}r\big{)}+\tilde{\phi}_{n}(r),$
(2.7)

where $T_{j}$ and $c_{n,j}$ are defined in Lemma 2.2, with for $k=0,1,2$:

$\left|D^{k}_{r}\tilde{\phi}_{n}(r)\right|+\left|D^{k}_{r}b\partial_{b}\tilde{\phi}_{n}(r)\right|\lesssim\min\left(b\langle
r\rangle^{2},\frac{1}{|\ln b|}\right)\left(\frac{r}{\langle
r\rangle}\right)^{2-(k\;\textup{mod}\;2)}\langle
r\rangle^{-2-k}\langle\sqrt{b}r\rangle^{2n+\delta}$ (2.8)

$(iii)$ (Spectral gap estimate) For any $g\in
L^{2}(\mathbb{R}^{+},\omega_{b}dr)$ with $\langle
g,\phi_{j}\rangle_{L^{2}_{\omega_{b}}}=0$ for $0\leq j\leq k$, one has

$\langle
g,\mathscr{A}g\rangle_{L^{2}_{\omega_{b}}}^{2}\leq\alpha_{k+1}\|g\|^{2}_{L^{2}_{\omega_{b}}}.$
(2.9)

###### Proof.

Since the computation of $(\alpha_{n},\phi_{n})$ through the matched
asymptotic procedure is long and technical, it is left to next subsections. In
particular, the existence of the $N+1$ eigenvalues $\alpha_{0},...,\alpha_{N}$
satisfying (2.2), (2.3) and (2.4), and the refined bound (2.8), are proved in
Lemma 2.8. It then remains to use them to prove all the other results in
Proposition 2.1.

Step 1 _Self-adjointness and discreteness of $\sigma(\mathscr{A})$_: We deal
with the discreteness of $\sigma(\mathscr{A})$ and its uniqueness relying on
classical arguments for second order linear operators. The first observation
is that we can rewrite the linear operator $\mathscr{A}$ as

$\displaystyle\mathscr{A}f$
$\displaystyle=\partial_{r}^{2}f-\left(\frac{1}{r}+\Phi_{U}^{\prime}+br\right)\partial_{r}f+Uf=\frac{1}{\omega_{b}}\partial_{r}\left(\omega_{b}\partial_{r}f\right)+Uf,$

which gives the self-adjointness of $\mathscr{A}$ in
$L^{2}(\mathbb{R}^{+},\omega_{b}dr)$. For the discreteness of its spectrum, we
let

$\rho_{b}(r)=\frac{1}{r^{2}\sqrt{U}}e^{-\frac{br^{2}}{4}},$

and observe that

$\displaystyle\tilde{\mathscr{A}}f$
$\displaystyle=\rho_{b}\mathscr{A}\big{(}\rho_{b}^{-1}f\big{)}=\frac{\rho_{b}}{\omega_{b}}\partial_{r}\left(\omega_{b}\partial_{r}(\rho_{b}^{-1}f)\right)+Uf$
$\displaystyle=\partial_{r}^{2}f+\frac{\rho_{b}}{\omega_{b}}\left[\partial_{r}\left(\frac{\omega_{b}}{\rho_{b}}\right)+\omega_{b}\partial_{r}(\rho_{b}^{-1})\right]\partial_{r}f+\frac{\rho_{b}}{\omega_{b}}\partial_{r}\left(\omega_{b}\partial_{r}(\rho_{b}^{-1})\right)f+Uf$
$\displaystyle=\partial_{r}^{2}f+\frac{3}{r}\partial_{r}f+\left[-\frac{b^{2}r^{2}}{4}+\frac{2br^{2}}{1+r^{2}}\right]f+Uf.$

The linear operator $\tilde{\mathscr{A}}$ is of Schrödinger type and self-
adjoint in $L^{2}(\mathbb{R}^{+},r^{3}dr)$. Since $\tilde{\mathscr{A}}$ has
the real potential tending to infinity as $r\to\infty$, its spectrum is purely
discrete by standard arguments. This concludes the discreteness of spectrum of
$\mathscr{A}$.

Step 2 _Uniqueness of the eigenvalues:_ We first prove the sign changing
property of (ii). This is a direct consequence of lemmas 2.3 and 2.6. In
particular, we show that on the interval $(0,\frac{\zeta_{0}}{\sqrt{b}}]$,
$\phi_{0}$ does not vanish and $\phi_{n}$ has exactly one zero located at
$r_{0}\sim\frac{1}{\sqrt{nb|\log b|}}$ for $n\geq 1$ (see page 2.43 for a
detailed proof). On the interval $(\frac{\zeta_{0}}{\sqrt{b}},+\infty)$, the
eigenfunction $\phi_{n}$ is a perturbation of a Kummer’s function (see Lemma
2.6 for a proper definition) where it does not vanish for $n=0,1$ and
possesses $n-1$ zeros for $n\geq 2$.

We now rely on Sturm-Liouville theory to show that the $N+1$ first eigenvalues
of $\mathscr{A}$ are those given by (2.2). We argue by contradiction and
assume that there exists $\alpha^{*}\in(\alpha_{n+1},\alpha_{n})$ for some
$n\in\mathbb{N}$ that is an eigenvalue of $\mathscr{A}$. Denote by $\phi^{*}$
the eigenfunction corresponding to $\alpha^{*}$, and by
$\mathcal{Z}\big{[}f,(0,\infty)\big{]}$ the number of zeros of $f$ on
$(0,\infty)$. Sturm-Liouville theory asserts that

$\mathcal{Z}\big{[}\phi_{n+1},(0,\infty)\big{]}>\mathcal{Z}\big{[}\phi^{*},(0,\infty)\big{]}>\mathcal{Z}\big{[}\phi_{n},(0,\infty)\big{]},$

which is a contradiction as the term on the left is $n+1$ and that on the
right is $n$ from Step 2. The case of an eigenvalue $\alpha^{*}>\alpha_{0}$ is
ruled out similarly using that $\phi_{0}$ has no zero on $(0,\infty)$. Note
that the multiplicity of the eigenvalues of $\mathscr{A}$ has to be of one,
since from a direct check, the eigenfunction equation is an ODE that admits at
least one solution growing like $r^{C}e^{br^{2}/2}$ at infinity for some
constant $C$, and thus which is not in $L^{2}(\omega_{b})$.

Step 3 _Proof of remaining bounds:_ The spectral gap (iii) is an immediate
consequence of the fact that $\mathscr{A}$ is self-adjoint with compact
resolvent, that $\alpha_{0},...,\alpha_{N}$ have multiplicity one, and that
any other eigenvalue is smaller than $\alpha_{N}$.

To prove the pointwise estimate (2.6), we use the decomposition (2.7). The
function $\tilde{\phi}_{n}$ satisfies the bound (2.6) because of the even
sharper bound (2.8). The function
$\sum_{j=0}^{n}c_{n,j}b^{j}T_{j}\big{(}r\big{)}$ satisfies also the bound
(2.6) because of the estimates (2.15) and (2.16). Hence $\phi_{n}$ satisfies
(2.6).

To prove the $L^{2}(\omega_{b})$ estimate (2.5), we use the decomposition
(2.7) again. From the bound (2.8) we infer that, using
$U^{-1}(r)\lesssim\langle r\rangle^{3}$ and dividing the integral the two
zones $r\leq\sqrt{b}^{-1}$ and $r\leq\sqrt{b}^{-1}$:

$\displaystyle\int_{0}^{\infty}|\tilde{\phi}_{n}|^{2}\omega_{b}\lesssim\int_{0}^{\sqrt{b}^{-1}}\frac{\langle
r\rangle^{-4}}{|\ln b|^{2}}\langle
r\rangle^{3}dr+\int_{\sqrt{b}^{-1}}^{\infty}\frac{r^{4n-4+2\delta}b^{2n+\delta}}{|\ln
b|^{2}}r^{3}e^{-\frac{br^{2}}{2}}dr\lesssim\frac{1}{|\ln b|}.$

One then computes from (1.3), (2.12), (2.16) and (2.19) that for $n=0,1$ as
$b\rightarrow 0$:

$\int_{0}^{\infty}T_{0}^{2}(r)\omega_{b}(r)\sim\frac{1}{8}\int_{1}^{\infty}r^{-1}e^{-\frac{br^{2}}{2}}dr\sim\frac{|\ln
b|}{16},$
$\int_{0}^{\infty}b^{2}T_{1}^{2}(r)\omega_{b}(r)\sim\frac{1}{32}\int_{1}^{\infty}b^{2}r^{3}|\ln
r|^{2}e^{-\frac{br^{2}}{2}}dr\sim\frac{|\ln b|^{2}}{64},$

by using that $\int_{0}^{\infty}\zeta^{3}e^{-\zeta^{2}/2}d\zeta=2$. The three
above identities, with the explicit values (2.25) for $c_{i,j}$, prove (2.5)
for $n=0,1$. The general bound for $n\geq 2$ follows similarly.

∎

### 2.1. Analysis in the inner zone $r\leq R_{0}$

In this part, we solve equation (2.1) in the interval $[0,R_{0}]$ where we
consider $-br\partial_{r}-\alpha$ as a small perturbation of
$\mathscr{A}_{0}$. Let us recall some basic properties of $\mathscr{A}_{0}$ in
the following. We introduce the norms

$\|f\|_{X^{a}_{\imath}}:=\sup_{r\in[0,R_{0}]}\frac{\left<r\right>^{2-a}}{r^{2}\big{(}1+\ln\left<r\right>\big{)}^{\imath}}|f(r)|\quad\mbox{
for }\imath=0,1\mbox{ and
}\quad\|f\|_{X^{a}_{-1}}:=\sup_{r\in[0,R_{0}]}\frac{\left<r\right>^{2-a}}{r^{2}\big{(}1+\frac{2\ln\left<r\right>}{\ln
b}\big{)}}|f(r)|$

for any $a\in\mathbb{R}$, and the function spaces for $\imath=-1,0,1$,

$\mathcal{I}_{\imath}^{a}:=\left\\{f:\;\;\|f\|_{\mathcal{I}_{\imath}^{a}}\triangleq\|f\|_{X^{a}_{\imath}}+\|r\partial_{r}f\|_{X^{a}_{\imath}}+\|r^{2}\partial_{r}^{2}f\|_{X^{a}_{\imath}}<\infty\right\\}.$
(2.10)

######  Lemma 2.2 (Properties of $\mathscr{A}_{0}$).

  * (i)

(Inversion) For any $f\in\mathcal{C}(\mathbb{R}^{+},\mathbb{R})$, a solution
to $\mathscr{A}_{0}u=f$ is given by:

$\displaystyle\mathscr{A}_{0}^{-1}f(r)$
$\displaystyle:=\frac{1}{2}\psi_{0}(r)\int_{r}^{1}\frac{\zeta^{4}+4\zeta^{2}\ln\zeta-1}{\zeta}f(\zeta)d\zeta+\frac{1}{2}\tilde{\psi}_{0}(r)\int_{0}^{r}\zeta
f(\zeta)d\zeta,$ (2.11)

where $\psi_{0}$ and $\tilde{\psi}_{0}$ are the two linearly independent
solutions to $\mathscr{A}_{0}\psi=0$ given by

$\psi_{0}(r)=\frac{r^{2}}{\left<r\right>^{4}}\quad\textup{and}\quad\tilde{\psi}_{0}(r)=\frac{r^{4}+4r^{2}\ln
r-1}{\left<r\right>^{4}}.$ (2.12)

  * (ii)

(Continuity) Let $\imath\in\\{-1,0,1\\}$ and $a>-2$, then there holds the
estimate:

$\|\mathscr{A}_{0}^{-1}f\|_{\mathcal{I}_{\imath}^{a+2}}\lesssim\|f\|_{X_{\imath}^{a}}.$
(2.13)

  * (iii)

(Iterative kernel of $\mathscr{A}_{0}$) There exists a family of smooth radial
functions $\big{\\{}T_{i}\big{\\}}_{i\in\mathbb{N}}$ defined as

$\mathscr{A}_{0}T_{i+1}:=-T_{i},\quad T_{0}:=\psi_{0},$ (2.14)

which admit the asymptotic estimates

$\displaystyle\Big{|}\big{(}r\partial_{r}\big{)}^{p}T_{i}\Big{|}=\mathcal{O}(r^{2})\quad\textup{as}\quad
r\to 0,\quad\forall p\in\mathbb{N},\quad$ (2.15) $\displaystyle
T_{i}=r^{2(i-1)}\Big{(}\hat{d}_{i}\ln
r+d_{i}\Big{)}+\mathcal{O}\big{(}r^{2(i-2)}\ln^{i+1}r\big{)}\quad\textup{as}\quad
r\to\infty,$ (2.16) $\displaystyle
r\partial_{r}T_{i}=r^{2(i-1)}\left[2(i-1)\big{(}\hat{d}_{i}\ln
r+d_{i}\big{)}+\hat{d}_{i}\right]+\mathcal{O}\big{(}r^{2(i-2)}\ln^{i+1}r\big{)},$
(2.17)
$\displaystyle\Big{|}\big{(}r\partial_{r}\big{)}^{p}T_{i}\Big{|}=\mathcal{O}(r^{2(i-1)}\ln
r)\quad\textup{as}\quad r\to\infty,\quad\forall p\in\mathbb{N},$ (2.18)

where $d_{i}\in\mathbb{R}$ and

$\displaystyle\hat{d}_{1}=-\frac{1}{2},\;\;d_{1}=\frac{1}{4},\quad\hat{d}_{i+1}=-\frac{\hat{d}_{i}}{4i(i+1)},\quad
d_{i+1}=\frac{1}{8}\left(\frac{\hat{d}_{i}-2id_{i}}{i^{2}}-\frac{\hat{d}_{i}-(2i+2)d_{i}}{(i+1)^{2}}\right).$
(2.19)

###### Proof.

$(i)$ By the scaling invariance of the problem (1.2), we have
$\frac{d}{d\lambda}\left[\Delta U_{\lambda}-\nabla\cdot(\nabla
U_{\lambda})\right]_{\lambda=1}=0$, or $\mathscr{L}^{y}\Lambda U=0$. Hence
$\psi_{0}=\frac{1}{8}\int_{0}^{r}\Lambda U(x)xdx$ is the first fundamental
solution to $\mathscr{A}_{0}\psi=0$. The explicit formula of
$\tilde{\psi}_{0}$ follows from the integration of the Wronskian relation, and
the formula (2.11) is a standard way to solve linear second order ODEs.

$(ii)$ We denote $u=\mathscr{A}_{0}^{-1}f$. We directly compute from (2.11)
for $r\leq 1$ that for any $a,\imath$:

$\displaystyle|u(r)|$
$\displaystyle\lesssim\left|\psi_{0}(r)\int_{r}^{1}\frac{\xi^{4}+4\xi^{2}\ln\xi-1}{\xi}f(\xi)d\xi+\tilde{\psi}_{0}(r)\int_{0}^{r}\xi
f(\xi)d\xi\right|$ $\displaystyle\quad\lesssim\left(\sup_{0\leq\xi\leq
2}\xi^{-2}|f(\xi)|\right)\left(r^{2}\int_{r}^{1}\xi
d\xi+\int_{0}^{r}\xi^{3}d\xi\right)\lesssim r^{4}\|f\|_{X^{a}_{\imath}}.$

For $1\leq r\leq R_{0}$, we use again formula (2.11) to compute for
$\imath=0,1$ and $a>-2$:

$\displaystyle|u(r)|$
$\displaystyle\lesssim|\psi_{0}(r)|\int_{1}^{r}\xi^{3}|f(\xi)|d\xi+|\tilde{\psi}_{0}(r)|\int_{0}^{r}\xi|f(\xi)|d\xi$
$\displaystyle\lesssim r^{-2}\sup_{1\leq\xi\leq
R_{0}}\frac{\xi^{-a}|f(\xi)|}{(1+\ln\langle\xi\rangle)^{\imath}}\int_{1}^{r}\xi^{3}\xi^{a}(1+\ln\langle\xi\rangle)^{\imath}d\xi+\sup_{0\leq\xi\leq
R_{0}}\frac{\langle\xi\rangle^{-a}|f(\xi)|}{(1+\ln\langle\xi\rangle)^{\imath}}\int_{0}^{r}\xi\langle\xi\rangle^{a}(1+\ln\langle\xi\rangle)^{\imath}d\xi$
$\displaystyle\lesssim\|f\|_{X_{\imath}^{a}}r^{a+2}(1+\ln\left<r\right>)^{\imath}$

For $\imath=-1$ we first notice that the function
$1+\frac{2\ln\left<r\right>}{\ln b}$ is decreasing and satisfies for any
$r\in[0,R_{0}]$:

$\frac{1}{|\ln b|}\leq\frac{|\ln\zeta_{0}|}{|\ln b|}\leq
1+\frac{2\ln\left<r\right>}{\ln b}\leq 1,$

so that for $r\in[1,R_{0}]$ and $a>-1$, with constants independent on $b$:

$\displaystyle\int_{0}^{r}\langle\xi\rangle^{a}(1+\frac{2\ln\langle\xi\rangle}{\ln
b})d\xi\lesssim 1+\left|\int_{2}^{r}\xi^{a}(1+\frac{2\ln\xi}{\ln
b})d\xi\right|$ $\displaystyle\qquad\lesssim 1+\left|r^{a+1}\left(1+\frac{2\ln
r}{\ln b}\right)-\frac{r^{a+1}}{\ln b}\right|\lesssim
r^{a+1}\left(1+\frac{2\ln\left<r\right>}{\ln b}\right).$ (2.20)

Hence for $\imath=-1$ and $a>-2$, computing as above:

$\displaystyle|u(r)|$ $\displaystyle\lesssim r^{-2}\sup_{1\leq\xi\leq
R_{0}}\frac{\xi^{-a}|f(\xi)|}{1+\frac{2\ln\langle\xi\rangle}{\ln
b}}\int_{1}^{r}\xi^{3}\xi^{a}(1+\frac{2\ln\langle\xi\rangle}{\ln
b})d\xi+\sup_{0\leq\xi\leq
R_{0}}\frac{\langle\xi\rangle^{-a}|f(\xi)|}{1+\frac{2\ln\langle\xi\rangle}{\ln
b}}\int_{0}^{r}\xi\langle\xi\rangle^{a}(1+\frac{2\ln\langle\xi\rangle}{\ln
b})d\xi$
$\displaystyle\lesssim\|f\|_{X_{\imath}^{a}}r^{a+2}(1+\frac{2\ln\left<r\right>}{\ln
b}).$

The estimates above imply for any $a>-2$ and $\imath=-1,0,1$, with a constant
independent on $b$ and $\zeta_{0}$:

$\|\mathscr{A}_{0}^{-1}f\|_{X_{\imath}^{a+2}}\lesssim\|f\|_{X_{\imath}^{a}}.$

To estimate the derivatives, we notice from (2.11) that

$\partial_{r}u=\frac{1}{2}\partial_{r}\psi_{0}\int_{r}^{1}\frac{\xi^{4}+4\xi^{2}\ln\xi-1}{\xi}f(\xi)d\xi+\frac{1}{2}\partial_{r}\tilde{\psi}_{0}\int_{0}^{r}\xi
f(\xi)d\xi.$ (2.21)

Hence, with the very same computations that we do not repeat we obtain for
$\imath=-1,0,1$ and $a>-2$:

$\|r\partial_{r}u\|_{X_{\imath}^{a+2}}\lesssim\|f\|_{X_{\imath}^{a}}.$

Next, using that $\mathscr{A}_{0}u=f$ and the definition of $\mathscr{A}_{0}$
yields

$\partial_{r}^{2}u=f+\left(\frac{1}{r}-\frac{4r}{\left<r\right>^{2}}\right)\partial_{r}u-\frac{8}{\left<r\right>^{4}}u.$

so that for $\imath=-1,0,1$ and $a>-2$, using the previous estimates for $u$
and $r\partial_{r}u$:

$\displaystyle\|r^{2}\partial_{rr}u\|_{X_{\imath}^{a+2}}$
$\displaystyle\lesssim$
$\displaystyle\|r^{2}f\|_{X_{\imath}^{a+2}}+\|\left(r-\frac{4r^{3}}{\left<r\right>^{2}}\right)\partial_{r}u\|_{X_{\imath}^{a+2}}+\|\frac{8r^{2}}{\left<r\right>^{4}}u\|_{X_{\imath}^{a+2}}$
$\displaystyle\lesssim$
$\displaystyle\|f\|_{X_{\imath}^{a}}+\|f\|_{X_{\imath}^{a}}+\|f\|_{X_{\imath}^{a}}\lesssim\|f\|_{X_{\imath}^{a}}.$

This concludes the proof of (2.13).

$(iii)$ For $r\ll 1$, we compute from (2.11)

$|T_{1}(r)|+|r\partial_{r}T_{1}(r)|=\mathcal{O}\left(r^{2}\int_{r}^{1}\xi^{-1}\xi^{2}d\xi+\int_{0}^{r}\xi\xi^{2}d\xi\right)=\mathcal{O}(r^{2})\quad\textup{as}\quad
r\to 0.$

We use $\mathscr{A}_{0}T_{1}=-\psi_{0}$ and the definition (1.13) of
$\mathscr{A}_{0}$ to estimate for $k\in\mathbb{N}$,

$|(r\partial_{r})^{k+2}T_{1}(r)|=\mathcal{O}\left(\sum_{j=0}^{k}|r^{j+1}\partial^{j+1}_{r}T_{1}|+r^{k+2}|\partial_{r}^{k}\psi_{0}|\right)=\mathcal{O}(r^{2})\quad\textup{as}\quad
r\to 0.$

Hence, the estimate (2.15) holds for $i=1$. By induction, we assume that
estimate (2.15) holds for $i\geq 1$. We compute from (2.11) and the relation
$T_{i+1}=-\mathscr{A}_{0}^{-1}T_{i}$,

$\displaystyle|T_{i+1}|+|r\partial_{r}T_{i+1}|=\mathcal{O}\left(r^{2}\int_{r}^{1}\xi^{-1}\xi^{2}d\xi+\int_{0}^{r}\xi\xi^{2}d\xi\right)=\mathcal{O}(r^{2}),$

as $r\to 0$. The estimate for higher derivative follows from the relation
$\mathscr{A}T_{i+1}=-T_{i}$ and the definition (1.13) of $\mathscr{A}_{0}$.

For $1\ll r\leq R_{0}$, we prove (2.16) by induction. For $i=1$, we compute
from (2.11) and the relation $T_{1}=-\mathscr{A}_{0}^{-1}\psi_{0}$

$\displaystyle T_{1}(r)$
$\displaystyle=\frac{1}{2}\psi_{0}\int_{r}^{1}\frac{\xi^{4}+4\xi^{2}\ln\xi-1}{\xi}\psi_{0}(\xi)d\xi+\frac{1}{2}\tilde{\psi}_{0}\int_{0}^{r}\xi\psi_{0}(\xi)d\xi$
$\displaystyle=\left(\frac{1}{2r^{2}}+\mathcal{O}(r^{-4})\right)\left(\frac{1}{2}r^{2}+\mathcal{O}(\ln^{2}r)\right)-\left(\frac{1}{2}+\mathcal{O}(r^{-2}\ln
r)\right)\left(\ln r+\mathcal{O}(r^{-2})\right)$
$\displaystyle=-\frac{1}{2}\ln
r+\frac{1}{4}+\mathcal{O}\left(\frac{\ln^{2}r}{r^{2}}\right),$

which is (2.16) for $i=1$. Assuming now that expansion (2.16) holds for some
$i\geq 1$, we use formula (2.11), the relation
$T_{i+1}=-\mathscr{A}_{0}^{-1}T_{i}$ and the elementary identity

$\int_{0}^{r}s^{k}\ln s\;ds=\frac{r^{k+1}\big{[}(k+1)\ln
r-1\big{]}}{(k+1)^{2}}\quad\textup{for all}\;\;k\in\mathbb{N},$

to compute

$\displaystyle T_{i+1}$
$\displaystyle=\frac{1}{2}\psi_{0}\int_{r}^{1}\frac{\xi^{4}+4\xi^{2}\ln\xi-1}{\xi}T_{i}(\xi)d\xi+\frac{1}{2}\tilde{\psi}_{0}\int_{0}^{r}\xi
T_{i}(\xi)d\xi$
$\displaystyle=\left(\frac{1}{2r^{2}}+\mathcal{O}(r^{-4})\right)\left[\frac{\hat{d}_{i}}{(2i+2)}\ln
r-\frac{\hat{d}_{i}-(2i+2)d_{i}}{(2i+2)^{2}}+\mathcal{O}\left(r^{-2}\ln^{i+2}r\right)\right]$
$\displaystyle\quad-\left(\frac{1}{2}+\mathcal{O}(r^{-2}\ln
r)\right)r^{2i}\left[\frac{\hat{d}_{i}}{2i}\ln
r-\frac{\hat{d}_{i}-2id_{i}}{4i^{2}}+\mathcal{O}\left(r^{-2}\ln^{i+1}r\right)\right]$
$\displaystyle=r^{2i}\left[\frac{-\hat{d}_{i}}{4i(i+1)}\ln
r+\frac{1}{8}\left(\frac{\hat{d}_{i}-2id_{i}}{i^{2}}-\frac{\hat{d}_{i}-(2i+2)d_{i}}{(i+1)^{2}}\right)\right]+\mathcal{O}\left(r^{2i-2}\ln^{i+2}r\right),$

which gives

$\hat{d}_{i+1}=-\frac{\hat{d}_{i}}{4i(i+1)},\quad
d_{i+1}=\frac{1}{8}\left(\frac{\hat{d}_{i}-2id_{i}}{i^{2}}-\frac{\hat{d}_{i}-(2i+2)d_{i}}{(i+1)^{2}}\right).$

This concludes the proof of (2.16).

The proof of (2.17) follows similarly by induction. Indeed, assuming that
(2.17) holds for $i\in\mathbb{N}$, we compute from (2.11), the relation
$T_{i+1}-=\mathscr{A}_{0}^{-1}T_{i}$ and the expansion (2.17) for $1\ll r\leq
R_{0}$:

$\displaystyle r\partial_{r}T_{i+1}$
$\displaystyle=\frac{r}{2}\partial_{r}\psi_{0}\int_{r}^{1}\frac{\xi^{4}+4\xi^{2}\ln\xi-1}{\xi}T_{i}(\xi)d\xi+\frac{r}{2}\partial_{r}\tilde{\psi}_{0}\int_{0}^{r}\xi
T_{i}(\xi)d\xi$
$\displaystyle=\left(-r^{2i}+\mathcal{O}(r^{2i-2})\right)\left[\frac{\hat{d}_{i}}{(2i+2)}\ln
r-\frac{\hat{d}_{i}-(2i+2)d_{i}}{(2i+2)^{2}}+\mathcal{O}\left(r^{-2}\ln^{i+2}r\right)\right]+\mathcal{O}\left(\frac{\ln^{2}r}{r^{2-2i}}\right)$
$\displaystyle=r^{2i}\left[\frac{-\hat{d}_{i}}{2(i+1)}\ln
r+\frac{\hat{d}_{i}-2(i+1)d_{i}}{4(i+1)^{2}}\right]+\mathcal{O}\left(r^{2i-2}\ln^{i+2}r\right).$

Using the recursive definition of $\hat{d}_{i}$ and $d_{i}$, i.e,

$\hat{d}_{i+1}=-\frac{\hat{d}_{i}}{4i(i+1)},\quad
d_{i}=-4i(i+1)d_{i+1}-2(2i+1)\hat{d}_{i+1},$ (2.22)

we have the simplification $\frac{-\hat{d}_{i}}{2(i+1)}=2i\hat{d}_{i+1}$ and

$\displaystyle\frac{\hat{d}_{i}-2(i+1)d_{i}}{4(i+1)^{2}}$
$\displaystyle=\frac{\hat{d}_{i}}{4i^{2}}-\frac{d_{i}}{2i}-2d_{i+1}=-\frac{i+1}{i}\hat{d}_{i+1}+\frac{2i+1}{i}\hat{d}_{i+1}+2id_{i+1}=\hat{d}_{i+1}+2id_{i+1}.$

This concludes the proof of (2.17). The estimate (2.18) follows by induction
from the definition of $\mathscr{A}_{0}$, the relation
$\mathscr{A}_{0}T_{i+1}=-T_{i}$ and the Leibniz rule. This completes the proof
of Lemma 2.2. ∎

In the following we show that the profiles $T_{j}$ given in Lemma 2.2 are
actually the building blocks of the eigenfunction of the linear operator
$\mathscr{A}=\mathscr{A}_{0}-br\partial_{r}$ on $[0,R_{0}]$. In particular, we
have the following.

######  Lemma 2.3 (Inner eigenfunctions for the radial mode).

Let $n\in\mathbb{N}$, $0<\zeta_{0}\ll 1$ and $0<b\ll 1$ be small enough. Then
for any $|\bar{\alpha}|\lesssim|\ln b|^{-2}$ there exists a smooth function
$\phi_{n}^{\textup{in}}\in\mathcal{C}^{\infty}([0,R_{0}],\mathbb{R})$
satisfying

$\mathscr{A}\phi_{n}^{\textup{in}}=2b\big{(}1-n+\tilde{\alpha}\big{)}\phi_{n}^{\textup{in}}\quad\textup{with}\quad\tilde{\alpha}=\frac{1}{\ln
b}+\bar{\alpha},$ (2.23)

where $\phi_{n}^{\textup{in}}$ is of the form

$\phi_{n}^{\textup{in}}(r)=\sum_{j=0}^{n}c_{n,j}b^{j}T_{j}+b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)+2\bar{\alpha}\sum_{j=0}^{n}b^{j+1}\Big{(}-c_{n,j}T_{j+1}+S_{j}\Big{)}+b\mathcal{R}_{n},$
(2.24)

and the constants $\big{(}c_{n,j}\big{)}_{0\leq j\leq n}$ are given by

$c_{n,j}=2^{j}\frac{n!}{(n-j)!},\quad c_{n,j+1}=2(n-j)c_{n,j},\quad
c_{n,0}=1.$ (2.25)

The corrective functions $R_{n}$, $S_{j}$ satisfy the following estimates for
any $n\geq 0$:

$\displaystyle\|S_{j}\|_{\mathcal{I}_{1}^{2j}+\|b\partial_{b}S_{j}\|_{\mathcal{I}_{1}^{2j}}}+\|\partial_{\bar{\alpha}}S_{j}\|_{\mathcal{I}_{1}^{2j}}\lesssim\zeta_{0}^{2},$
(2.26)
$\displaystyle\|\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{0}}+\|b\partial_{b}\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{0}}+\|\partial_{\bar{\alpha}}\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{0}}\lesssim
1,$ (2.27)

with the following refinements for $n=0$:

$S_{0}=\frac{1}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i}\log(r+1)+\tilde{S}_{0},\quad\|\tilde{S}_{0}\|_{\mathcal{I}_{0}^{2}}+\|b\partial_{b}\tilde{S}_{0}\|_{\mathcal{I}_{0}^{2}}+\|\partial_{\bar{\alpha}}\tilde{S}_{0}\|_{\mathcal{I}_{0}^{2}}\lesssim
b,$ (2.28)
$\mathcal{R}_{0}=-\frac{1}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i}\left\\{\frac{1}{\log
b}\left[2\ln(r+1)-\Psi(i+2)-\gamma\right]+1\right\\}+\tilde{\mathcal{R}}_{0},$
$\|\tilde{\mathcal{R}}_{0}\|_{\mathcal{I}_{-1}^{0}}+\|b\partial_{b}\tilde{\mathcal{R}}_{0}\|_{\mathcal{I}_{-1}^{0}}\lesssim|\log
b|^{-1},\qquad\|\partial_{\bar{\alpha}}\tilde{\mathcal{R}}_{0}\|_{\mathcal{I}_{-1}^{0}}\lesssim
1,$ (2.29)

and for $n=1$:

$\mathcal{R}_{1}=-\frac{1}{2}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i}r^{2i}\left\\{\frac{1}{\log
b}\left[2\ln(r+1)-\frac{1}{i}-\Psi(i+2)-\gamma\right]+1-\frac{1}{\log
b}\right\\}+\tilde{\mathcal{R}}_{1},$
$\displaystyle\|\tilde{\mathcal{R}}_{1}\|_{\mathcal{I}_{-1}^{0}}+\|b\partial_{b}\tilde{\mathcal{R}}_{1}\|_{\mathcal{I}_{-1}^{0}}\lesssim|\log
b|^{-1},\quad\|\partial_{\bar{\alpha}}\tilde{\mathcal{R}}_{1}\|_{\mathcal{I}_{-1}^{0}}\lesssim
1.$ (2.30)
$S_{1}=-\frac{1}{2}\sum_{i=2}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i-1}r^{2i}\ln(r+1)+\tilde{S}_{1},$
$\|S_{0}\|_{\mathcal{I}_{0}^{2}}+\|b\partial_{b}S_{0}\|_{\mathcal{I}_{0}^{2}}+\|\partial_{\bar{\alpha}}S_{0}\|_{\mathcal{I}_{1}^{2}}\lesssim
b,\ \
\|\tilde{S}_{1}\|_{\mathcal{I}_{0}^{2}}+\|b\partial_{b}\tilde{S}_{1}\|_{\mathcal{I}_{0}^{2}}+\|\partial_{\bar{\alpha}}\tilde{S}_{1}\|_{\mathcal{I}_{1}^{2}}\lesssim
1.$ (2.31)

Finally, on the interval $(0,R_{0}]$, $\phi^{\textup{in}}_{0}$ does not vanish
and $\phi^{\textup{in}}_{n}$ has exactly one zero for $n\geq 1$.

###### Proof.

The proof mainly relies on classical arguments based on the Banach fixed point
theorem to construct the corrective profiles $\mathcal{R}_{n}$ and $S_{j}$ for
$0\leq j\leq n$.

Step 1 _Preliminary results:_ For $j\in\mathbb{N}$, we let

$\Theta_{j}=r\partial_{r}T_{j}-2(j-1)T_{j},$ (2.32)

which admits the following slowly growing tail from (2.16) and (2.17),

$|\Theta_{0}(r)|=\mathcal{O}(r^{-4}),\quad|\Theta_{j}(r)|=\mathcal{O}(r^{2(j-1)})\;\;\textup{for}\;\;j\geq
1,\quad\textup{as}\quad r\to\infty.$ (2.33)

and for $j\geq 1$:

$\left|\Theta_{j}(r)+\frac{2}{\ln b}T_{j}(r)\right|\lesssim r^{2}\langle
r\rangle^{2(j-2)}\left(1+\frac{2\ln(r+1)}{\ln b}\right).$ (2.34)

We compute the following integral by integrating by parts:

$\displaystyle\int_{0}^{\infty}r\Theta_{0}(r)dr$ $\displaystyle=$
$\displaystyle\lim_{R\rightarrow\infty}\int_{0}^{R}r\Theta_{0}(r)dr=\lim_{R\rightarrow\infty}\left(2\int_{0}^{R}rT_{0}(r)dr+\int_{0}^{R}r^{2}\partial_{r}T_{0}(r)dr\right)$
$\displaystyle=$ $\displaystyle\lim_{r\rightarrow\infty}R^{2}T_{0}(R)=1.$

From this and (2.11), as $|r\partial_{r}T_{0}+2T_{0}|\lesssim(1+r)^{-4}$ the
following corrective term satisfies as $r\rightarrow\infty$:

$\displaystyle\mathscr{A}_{0}^{-1}\Theta_{0}(r)$ $\displaystyle=$
$\displaystyle\mathcal{O}(\psi_{0}(r)\ln(r))+\mathcal{O}(r^{-2})+\frac{1}{2}\tilde{\psi}_{0}(r)\int_{0}^{\infty}\zeta
f(\zeta)d\zeta=\frac{1}{2}+\mathcal{O}(r^{-2}\ln r),$

and hence:

$\left|-\frac{2}{\ln b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right|\lesssim
r^{2}\langle r\rangle^{-2}\left(1+\frac{2\ln(r+1)}{\ln b}\right).$ (2.35)

These estimates show that

$\displaystyle\|b^{j-1}\left(\Theta_{j}(r)+\frac{2}{\ln
b}T_{j}(r)\right)\|_{X^{0}_{-1}}\lesssim\zeta_{0}^{2(j-1)}\|\Theta_{j}(r)+\frac{2}{\ln
b}T_{j}(r)\|_{X^{2(j-1)}_{-1}}\lesssim 1,$ $\displaystyle\quad\|-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\|_{\mathcal{I}^{0}_{-1}}\lesssim 1.$
(2.36)

Step 2 _Equations satisfied by $S_{j}$ and $\mathcal{R}_{n}$:_ Plugging the
decomposition (2.24) into (2.23) and using $\mathscr{A}_{0}T_{j}=-T_{j-1}$
with the convention $T_{-1}=0$ yields

$\displaystyle\Big{[}\mathscr{A}_{0}-br\partial_{r}-2b\big{(}1-n+\frac{1}{\ln
b}+\bar{\alpha}\big{)}\Big{]}\sum_{j=0}^{n}c_{n,j}b^{j}T_{j}$
$\displaystyle\quad=-\sum_{j=0}^{n-1}c_{n,j}b^{j+1}T_{j}\Big{[}2(n-j)+2(j-1)-2(n-1)\Big{]}-\sum_{j=0}^{n}c_{n,j}b^{j+1}\Theta_{j}-2\left(\frac{1}{\ln
b}+\bar{\alpha}\right)\sum_{j=0}^{n}c_{n,j}b^{j+1}T_{j}$
$\displaystyle\quad=-\sum_{j=1}^{n}c_{n,j}b^{j+1}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)-b\Theta_{0}-\frac{2b}{\ln
b}T_{0}-2\bar{\alpha}\sum_{j=0}^{n}c_{n,j}b^{j+1}T_{j},$

and

$\displaystyle\Big{[}\mathscr{A}_{0}-br\partial_{r}-2b\big{(}1-n+\frac{1}{\ln
b}+\bar{\alpha}\big{)}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)$ $\displaystyle=$ $\displaystyle
b\Theta_{0}+\frac{2b}{\ln
b}T_{0}-b\Big{[}r\partial_{r}+2\big{(}1-n+\frac{1}{\ln
b}+\bar{\alpha}\big{)}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}^{-1}\Theta_{0}\right)$

and

$\displaystyle\Big{[}\mathscr{A}_{0}-br\partial_{r}-2b\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\left(2\bar{\alpha}\sum_{j=0}^{n}b^{j+1}\Big{[}-c_{n,j}T_{j+1}+S_{j}\Big{]}\right)$
$\displaystyle\quad=2\bar{\alpha}\sum_{j=0}^{n}b^{j+1}\left\\{\mathscr{A}_{0}S_{j}-\Big{[}br\partial_{r}+2b\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\Big{(}-c_{n,j}T_{j+1}+S_{j}\Big{)}\right\\}+2\bar{\alpha}\sum_{j=0}^{n}c_{n,j}b^{j+1}T_{j}.$

We then rewrite equation (2.23) as

$\displaystyle 0=$
$\displaystyle\Big{[}\mathscr{A}_{0}-br\partial_{r}-2b\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\phi_{n}^{\textup{in}}$
$\displaystyle=$
$\displaystyle\bar{\alpha}\sum_{j=0}^{n}b^{j+1}\left\\{\mathscr{A}_{0}S_{j}-b\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\Big{(}-c_{n,j}T_{j+1}+S_{j}\Big{)}\right\\}$
$\displaystyle+b\Bigl{\\{}\mathscr{A}_{0}\mathcal{R}_{n}-b\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\mathcal{R}_{n}-\sum_{j=1}^{n}c_{n,j}b^{j}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)$
$\displaystyle-\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)\Bigr{\\}}$ (2.37)

Step 3 _Computation of $\big{(}S_{j}\big{)}_{0\leq j\leq n}$:_ From equation
(2.37), we choose $S_{j}$ to be the solution of the equation

$\mathscr{A}_{0}S_{j}=b\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\Big{(}-c_{n,j}T_{j+1}+S_{j}\Big{)}.$
(2.38)

Note from part $(iii)$ of Lemma 2.2 that $T_{j+1}\in\mathcal{I}_{1}^{2j}$ for
$j\geq 0$. We aim at proving that for $b$ and $\zeta_{0}$ small enough, there
exists a unique solution $S_{j}\in\mathcal{I}_{1}^{2j}$ to equation (2.38) via
the Banach fixed point theorem. Let $\Gamma$ be the affine mapping acting on
$f\in\mathcal{I}_{1}^{2j}$ defined as

$\displaystyle\Gamma(f)$
$\displaystyle=\mathscr{A}_{0}^{-1}\Big{[}b\big{(}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\big{)}\big{(}-c_{n,j}T_{j+1}+f\big{)}\Big{]}=\Gamma(0)+D\Gamma(f),$

where $\mathscr{A}_{0}^{-1}$ is defined as in (2.11) and

$\displaystyle\Gamma(0)$
$\displaystyle=bc_{n,j}\mathscr{A}_{0}^{-1}\left(\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}T_{j+1}\right),\quad
D\Gamma(f)=b\mathscr{A}_{0}^{-1}\left(\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}f\right).$

We estimate from (2.13),

$\displaystyle\|\Gamma(0)\|_{\mathcal{I}_{1}^{2j}}\lesssim
R_{0}^{2}\|\Gamma(0)\|_{\mathcal{I}_{1}^{2j+2}}\lesssim
R_{0}^{2}b\|T_{j+1}\|_{\mathcal{I}_{1}^{2j}}\lesssim
R_{0}^{2}b\lesssim\zeta_{0}^{2},$

and for all $f\in\mathcal{I}_{1}^{a}$ with $a=2j$ or $a=2j+2$,

$\displaystyle\|D\Gamma(f)\|_{\mathcal{I}_{1}^{a}}$ $\displaystyle\lesssim
R_{0}^{-2}\|D\Gamma(f)\|_{\mathcal{I}_{1}^{a+2}}=R_{0}^{2}b\|\mathscr{A}_{0}^{-1}\left(\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}f\right)\|_{\mathcal{I}_{1}^{a+2}}$
$\displaystyle\lesssim\zeta_{0}^{2}\|\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}f\|_{X_{1}^{a}}\lesssim\zeta_{0}^{2}\|f\|_{\mathcal{I}_{1}^{a}}$
(2.39)

Since $0<\zeta_{0}\ll 1$ and $\Gamma$ is an affine mapping, the above
estimates imply that $\Gamma$ is a contraction on
$B_{\mathcal{I}_{1}^{2j}}(0,C\zeta_{0}^{2})$ for some constant $C>0$
independent of the problem. Therefore, there exists a unique fixed point
$S_{j}=\Gamma(S_{j})$ such that
$\|S_{j}\|_{\mathcal{I}_{1}^{2j}}\lesssim\zeta^{2}_{0}$ so that the first
estimate in (2.26) holds. For the estimates of
$\partial_{\tilde{\alpha}}S_{j}$ and $\partial_{b}S_{j}$, we differentiate the
relation $S_{j}=\Gamma(S_{j})$ to obtain

$\displaystyle\partial_{b}S_{j}$
$\displaystyle=D\Gamma(\partial_{b}S_{j})+\big{(}\partial_{b}\Gamma\big{)}(S_{j}),\quad\partial_{\bar{\alpha}}S_{j}=D\Gamma(\partial_{\bar{\alpha}}S_{j})+\big{(}\partial_{\bar{\alpha}}\Gamma\big{)}(S_{j}),$

where we have the identities since $b\partial_{b}\tilde{\alpha}=-1/(\ln
b)^{2}$ and $\partial_{\bar{\alpha}}\tilde{\alpha}=1$:

$\displaystyle\partial_{b}\Gamma(f)$
$\displaystyle=\mathscr{A}_{0}^{-1}\Big{[}\big{(}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}-\frac{1}{|\ln
b|^{2}}\big{)}\big{)}\big{(}-c_{n,j}T_{j+1}+f\big{)}\Big{]},$ (2.40)
$\displaystyle\partial_{\bar{\alpha}}\Gamma(f)$
$\displaystyle=b\mathscr{A}_{0}^{-1}\Big{[}-c_{n,j}T_{j+1}+f\Big{]}.$ (2.41)

From (2.39), we see that
$\|D\Gamma\|_{\mathcal{I}_{1}^{a}\to\mathcal{I}_{1}^{a}}\lesssim\zeta_{0}^{2}$
with $a=2j+2$ or $a=2j$. Hence $Id-D\Gamma$ is invertible and:

$\displaystyle\big{\|}\partial_{b}S_{j}\big{\|}_{\mathcal{I}_{1}^{2j+2}}=\big{\|}(\textup{Id}-D\Gamma)^{-1}(\partial_{b}\Gamma)(S_{j})\big{\|}_{\mathcal{I}_{1}^{2j+2}}\lesssim\big{\|}(\partial_{b}\Gamma)(S_{j})\big{\|}_{\mathcal{I}_{1}^{2j+2}},$
$\displaystyle\big{\|}\partial_{\bar{\alpha}}S_{j}\big{\|}_{\mathcal{I}_{1}^{2j}}=\big{\|}(\textup{Id}-D\Gamma)^{-1}(\partial_{\bar{\alpha}}\Gamma)(S_{j})\big{\|}_{\mathcal{I}_{1}^{2j}}\lesssim\big{\|}(\partial_{\bar{\alpha}}\Gamma)(S_{j})\big{\|}_{\mathcal{I}_{1}^{2j}}.$

We estimate from (2.13) and (2.40),

$\displaystyle\big{\|}(\partial_{b}\Gamma)(S_{j})\big{\|}_{\mathcal{I}_{1}^{2j+2}}\quad\lesssim\|T_{j+1}\|_{\mathcal{I}_{1}^{2j}}+\|S_{j+1}\|_{\mathcal{I}_{1}^{2j}}\lesssim
1.$

Similarly, we estimate from (2.13) and (2.41),

$\displaystyle\big{\|}(\partial_{\bar{\alpha}}\Gamma)(S_{j})\big{\|}_{\mathcal{I}_{1}^{2j}}\lesssim
bR_{0}^{2}\Big{(}\|T_{j+1}\|_{\mathcal{I}_{1}^{2j}}+\|S_{j+1}\|_{\mathcal{I}_{1}^{2j}}\Big{)}\lesssim\zeta_{0}^{2},$

which concludes the proof of (2.26).

Refinement for $n=1$. We do not give technical details are these are the very
same ones as above for the general case. For $n=1$ the $S_{0}$ equation is:

$\mathscr{A}_{0}S_{0}=b\Big{[}r\partial_{r}+2\tilde{\alpha}\Big{]}\Big{(}-T_{1}+S_{0}\Big{)}.$

As $\|b(r\partial_{r}+2\tilde{\alpha})T_{1}\|_{\mathcal{I}^{0}_{0}}\lesssim b$
from (2.16) and (2.23) we get
$\|S_{0}\|_{\mathcal{I}_{0}^{2}}+\|b\partial_{b}S_{0}\|_{\mathcal{I}_{0}^{2}}+\|b\partial_{\tilde{\alpha}}S_{0}\|_{\mathcal{I}_{0}^{2}}\lesssim\zeta_{0}^{2}b$
by the same strategy as above. The $S_{1}$ equation is:

$\mathscr{A}_{0}S_{1}=b\Big{[}r\partial_{r}+2\tilde{\alpha}\Big{]}\Big{(}-2T_{2}+S_{1}\Big{)}.$

Let
$\hat{S}_{1}=-\frac{1}{2}\sum_{i=2}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i}r^{2i}\ln
r$, which produces
$\left(\partial_{rr}+\frac{3}{r}-br\partial_{r}\right)\hat{S}_{1}=-\frac{br^{2}\ln
r}{4}$. Looking for a solution $S_{1}=\hat{S}_{1}+\tilde{S}_{1}$ produces

$\mathscr{A}_{0}\tilde{S}_{1}=b\Big{[}r\partial_{r}+2\tilde{\alpha}\Big{]}\tilde{S}_{1}+b\left(\frac{r^{2}\ln
r}{4}-2r\partial_{r}T_{2}\right)-4\tilde{\alpha}T_{2}+b2\tilde{\alpha}\hat{S}_{1}-\left(\left(\frac{4r}{\langle
r\rangle^{2}}-\frac{4}{r}\right)\partial_{r}+\frac{8}{\langle
r\rangle^{4}}\right)\hat{S}_{1}.$

The source term above is of size $1$ in $\mathcal{I}_{0}^{0}$ from (2.16) and
(2.23) so that from the strategy used above one obtains
$\|\tilde{S}_{1}\|_{\mathcal{I}^{2}_{0}}+\|b\partial_{b}\tilde{S}_{1}\|_{\mathcal{I}^{2}_{0}}+\|\partial_{\bar{\alpha}}\tilde{S}_{1}\|_{\mathcal{I}_{1}^{2}}\lesssim\zeta_{0}^{2}$.

Refinement for $n=0$. For $n=0$ the $S_{0}$ equation is:

$\mathscr{A}_{0}S_{0}=b\Big{[}r\partial_{r}+2\big{(}1+\tilde{\alpha}\big{)}\Big{]}\Big{(}-T_{1}+S_{0}\Big{)}.$

We look for a solution $S_{0}=\hat{S}_{0}(r+1)+\tilde{S}_{0}$ with
$\hat{S}_{0}=\frac{1}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i}\log(r)$.
As
$(\partial_{rr}+\frac{3}{r}\partial_{r}-b(r\partial_{r}+2))\hat{S}_{0}=b\log
r$, $\tilde{S}_{0}$ solves

$\displaystyle\mathscr{A}_{0}\tilde{S}_{0}$ $\displaystyle=$ $\displaystyle
b\Big{[}r\partial_{r}+2\big{(}1+\tilde{\alpha}\big{)}\Big{]}\tilde{S}_{0}$
$\displaystyle-b(2T_{1}+\log(r+1))-b(r\partial_{r}+2\tilde{\alpha})T_{1}-\left(\left(\frac{4r}{\langle
r\rangle^{4}}-\frac{1}{r}-\frac{3}{r+1}\right)\partial_{r}+\frac{8}{\langle
r\rangle^{4}}+b\partial_{r}-2b\tilde{\alpha}\right)\hat{S}_{0}(r+1).$

The source term above is of size $b$ in $\mathcal{I}_{0}^{0}$ from (2.16) and
(2.23) so that from the strategy used above
$\|\tilde{S}_{0}\|_{\mathcal{I}^{2}_{0}}+\|b\partial_{b}\tilde{S}_{0}\|_{\mathcal{I}^{2}_{0}}+\|\partial_{\bar{\alpha}}\tilde{S}_{0}\|_{\mathcal{I}^{2}_{1}}\lesssim
b\zeta_{0}^{2}$.

Step 4 _Computation of $\mathcal{R}_{n}$_: From (2.37), we choose
$\mathcal{R}_{n}$ to be the solution of the equation

$\displaystyle\mathscr{A}_{0}\mathcal{R}_{n}$ $\displaystyle=$ $\displaystyle
b\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\mathcal{R}_{n}+\sum_{j=1}^{n}c_{n,j}b^{j}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right),$
$\displaystyle+\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right).$

where $\Theta_{j}$ is introduced in (2.32). The computation is similar to that
for $S_{j}$. We let $\Gamma$ be the affine mapping
$\Gamma(f)=\Gamma(0)+D\Gamma(f)$, where

$\displaystyle\Gamma(0)$
$\displaystyle=-b\mathscr{A}_{0}^{-1}\left[\sum_{j=1}^{n}c_{n,j}b^{j-1}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)-\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\Big{]}\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)\right],$
$D\Gamma(f)=b\mathscr{A}_{0}^{-1}\Big{[}\big{(}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\big{)}f\Big{]}.$

From (2.36) and (2.13) we obtain:

$\displaystyle\|\Gamma(0)\|_{\mathcal{I}_{-1}^{2}}$ $\displaystyle\lesssim
b\sum_{j=1}^{n}\left\|\mathscr{A}_{0}^{-1}b^{j-1}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)\right\|_{\mathcal{I}_{-1}^{2}}+b\left\|\mathscr{A}^{-1}_{0}(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta)\right\|_{\mathcal{I}_{-1}^{2}}$
$\displaystyle\lesssim
b\sum_{j=1}^{n}\left\|b^{j-1}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)\right\|_{\mathcal{I}_{-1}^{0}}+b\left\|-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta\right\|_{\mathcal{I}_{-1}^{0}}\lesssim b.$

Using (2.13), we estimate for all $f\in\mathcal{I}_{-1}^{2}$,

$\displaystyle\|D\Gamma(f)\|_{\mathcal{I}_{-1}^{2}}\lesssim
b\|\big{(}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}\big{)}\big{)}f\|_{X_{-1}^{0}}\lesssim
bR_{0}^{2}\|f\|_{\mathcal{I}_{-1}^{2}}\lesssim\zeta_{0}^{2}\|f\|_{\mathcal{I}_{-1}^{2}}.$
(2.42)

We then deduce that $\Gamma(f)$ is contraction on
$B_{\mathcal{I}_{-1}^{2}}(0,bC)$ for some constant $C>0$, hence, there exists
a unique fixed point $\mathcal{R}_{n}=\Gamma(\mathcal{R}_{n})$ satisfying
$\|\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{2}}\lesssim b$. As
$\|\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{0}}\lesssim
b^{-1}\|\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{2}}\lesssim 1$ the first estimate
in (2.27) holds. For the estimates of $\partial_{b}\mathcal{R}_{n}$ and
$\partial_{\bar{\alpha}}\mathcal{R}_{n}$, we differentiate the relation
$\mathcal{R}_{n}=\Gamma(\mathcal{R}_{n})$:

$\displaystyle\partial_{b}\mathcal{R}_{n}$
$\displaystyle=D\Gamma(\partial_{b}\mathcal{R}_{n})+\big{(}\partial_{b}\Gamma\big{)}(\mathcal{R}_{n}),\quad\partial_{\bar{\alpha}}\mathcal{R}_{n}=D\Gamma(\partial_{\bar{\alpha}}\mathcal{R}_{n})+\big{(}\partial_{\bar{\alpha}}\Gamma\big{)}(\mathcal{R}_{n}),$

where we have the identities since $b\partial_{b}\tilde{\alpha}=-1/(\ln
b)^{2}$ and $\partial_{\bar{\alpha}}\tilde{\alpha}=1$,

$\displaystyle\partial_{b}\Gamma(f)$ $\displaystyle=$
$\displaystyle\mathscr{A}_{0}^{-1}\sum_{j=1}^{n}c_{n,j}b^{j-1}\left(j\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)-\frac{2}{|\ln b|^{2}}T_{j}\right)$
$\displaystyle-\mathscr{A}_{0}^{-1}\Big{[}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}-\frac{1}{|\ln
b|^{2}}\big{)}\Big{]}\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}+\frac{2}{|\ln b|^{2}}T_{j}\right)$
$\displaystyle+\mathscr{A}_{0}^{-1}\Big{[}\big{(}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}-\frac{1}{|\ln
b|^{2}}\big{)}\big{)}f\Big{]},$
$\partial_{\bar{\alpha}}\Gamma(f)=b\mathscr{A}_{0}^{-1}f+b\mathscr{A}_{0}^{-1}\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)$

We have derived from (2.42) that
$\|D\Gamma\|_{\mathcal{I}_{-1}^{2}\to\mathcal{I}_{-1}^{2}}\lesssim\zeta_{0}^{2}$,
hence, $\textup{Id}-D\Gamma$ is invertible on $\mathcal{I}_{-1}^{2}$. In
particular, we have the estimates

$\displaystyle\|\partial_{b}\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{2}}=\big{\|}(\textup{Id}-D\Gamma)^{-1}\big{(}\partial_{b}\Gamma\big{)}(\mathcal{R}_{n})\big{\|}_{\mathcal{I}_{-1}^{2}}\lesssim\big{\|}\big{(}\partial_{b}\Gamma\big{)}(\mathcal{R}_{n})\big{\|}_{\mathcal{I}_{-1}^{2}},$
$\displaystyle\|\partial_{\bar{\alpha}}\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{2}}=\big{\|}(\textup{Id}-D\Gamma)^{-1}\big{(}\partial_{\bar{\alpha}}\Gamma\big{)}(\mathcal{R}_{n})\big{\|}_{\mathcal{I}_{-1}^{2}}\lesssim\big{\|}\big{(}\partial_{\bar{\alpha}}\Gamma\big{)}(\mathcal{R}_{n})\big{\|}_{\mathcal{I}_{-1}^{2}}.$

Using (2.36), $\|\frac{b^{j-1}}{|\log
b|^{2}}T_{j}\|_{\mathcal{I}^{0}_{-1}}\lesssim 1$, the estimate on
$\mathcal{R}_{n}$, we have by (2.13):

$\displaystyle\big{\|}\big{(}\partial_{b}\Gamma\big{)}(\mathcal{R}_{n})\big{\|}_{\mathcal{I}_{-1}^{2}}\leq\sum_{j=1}^{n}\left(b^{j-1}\|\mathscr{A}_{0}^{-1}\left(\Theta_{j}+\frac{2}{\ln
b}T_{j}\right)\|_{\mathcal{I}^{2}_{-1}}+\|\mathscr{A}_{0}^{-1}b^{j-1}\frac{1}{|\ln
b|^{2}}T_{j}\|_{\mathcal{I}^{2}_{-1}}\right)$
$\displaystyle+\|\mathscr{A}_{0}^{-1}\left(r\partial_{r}+2\big{(}1-n+\tilde{\alpha}-\frac{1}{|\ln
b|^{2}}\big{)}\Big{]}\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}+\frac{2}{|\ln
b|^{2}}T_{j}\right)\right)\|_{\mathcal{I}^{2}_{-1}}$
$\displaystyle+\|\mathscr{A}_{0}^{-1}\Big{[}\big{(}r\partial_{r}+2\big{(}1-n+\tilde{\alpha}-\frac{1}{|\ln
b|^{2}}\big{)}\big{)}\mathcal{R}_{n}\|_{\mathcal{I}^{2}_{-1}}$
$\displaystyle\quad\lesssim
1+\|\mathcal{R}_{n}\|_{\mathcal{I}^{0}_{-1}}\lesssim
1+b^{-1}\|\mathcal{R}_{n}\|_{\mathcal{I}^{2}_{-1}}\lesssim 1.$

Similarly, we have by (2.13),

$\displaystyle\big{\|}\big{(}\partial_{\bar{\alpha}}\Gamma\big{)}(\mathcal{R}_{n})\big{\|}_{\mathcal{I}_{-1}^{2}}$
$\displaystyle\leq\|b\mathscr{A}_{0}^{-1}\mathcal{R}_{n}\|_{\mathcal{I}^{2}_{-1}}+b\|\mathscr{A}^{-1}_{0}\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)\|_{\mathcal{I}_{-1}^{2}}$
$\displaystyle\lesssim\|b\mathcal{R}_{n}\|_{\mathcal{I}^{0}_{-1}}+b\|-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\|_{\mathcal{I}_{-1}^{0}}\lesssim\|\mathcal{R}_{n}\|_{\mathcal{I}^{2}_{-1}}+b\lesssim
b.$

Hence
$\|\partial_{\bar{\alpha}}\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{0}}\lesssim
b^{-1}\|\partial_{\bar{\alpha}}\mathcal{R}_{n}\|_{\mathcal{I}_{-1}^{2}}\lesssim
1$.

Computation of $\mathcal{R}_{1}$: For $n=1$ a refinement is necessary. The
equation for $\mathcal{R}_{1}$ is:

$\displaystyle\mathscr{A}_{0}\mathcal{R}_{1}$ $\displaystyle=$ $\displaystyle
b\Big{[}r\partial_{r}+2\tilde{\alpha}\Big{]}\mathcal{R}_{1}+2b\left(r\partial_{r}T_{1}+\frac{2}{\ln
b}T_{1}\right)+\Big{[}r\partial_{r}+2\tilde{\alpha}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right).$

We look for a solution under the form
$\mathcal{R}_{1}(r)=\mathcal{R}_{1,1}(r)+\mathcal{R}_{1,2}(r+1)+\tilde{\mathcal{R}}$
where

$\mathcal{R}_{1,1}=-(\frac{1}{2}-\frac{1}{2\ln
b})\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i}r^{2i},\ \
(\partial_{rr}+\frac{3}{r}\partial_{r}-br\partial_{r})\mathcal{R}_{1,1}=-(1-\frac{1}{\ln
b})b.$ $\mathcal{R}_{1,2}=-\frac{1}{2\log
b}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i}r^{2i}\left[2\ln(r)-\frac{1}{i}-\Psi(i+2)-\gamma\right],$
$\big{(}\partial_{r}^{2}+\frac{3}{r}\partial_{r}-br\partial_{r}\big{)}\mathcal{R}_{1,2}=-\frac{b}{\log
b}(2\log r-1),$

so that

$\displaystyle\mathscr{A}_{0}\tilde{\mathcal{R}}_{1}-b(r\partial_{r}+2\tilde{\alpha})\tilde{\mathcal{R}}_{1}$
$\displaystyle=$ $\displaystyle\left(1-\frac{1}{\log b}\right)b+\frac{b}{\log
b}(2\log(r+1)-1)+2b(r\partial_{r}T_{1}+\frac{2}{\log
b}T_{1})+\Big{[}r\partial_{r}+2\tilde{\alpha}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)$
$\displaystyle-\left(\left(\frac{4r}{\langle
r\rangle^{4}}-\frac{4}{r}\right)\partial_{r}+\frac{8}{\langle
r\rangle^{4}}\right)\mathcal{R}_{1,1}-\left(\left(\frac{4r}{\langle
r\rangle^{4}}-\frac{1}{r}-\frac{3}{r+1}\right)\partial_{r}+\frac{8}{\langle
r\rangle^{4}}+b\partial_{r}\right)\mathcal{R}_{1,2}(r+1)$
$\displaystyle+2b\tilde{\alpha}(\mathcal{R}_{1,1}+\mathcal{R}_{1,2}(r+1))$

Each line in the right hand side above contains cancellations as
$r\rightarrow\infty$: the first is $\mathcal{O}(br^{-1}\log r)$ from (2.16),
so is the second from the definition of $\mathcal{R}_{1,1}$ and
$\mathcal{R}_{1,2}$. For the last line,
$\|\mathcal{R}_{1,1}+\mathcal{R}_{1,2}\|_{\mathcal{I}_{-1}^{0}}\lesssim 1$ and
$|\tilde{\alpha}|\lesssim|\log b|^{-1}$. This shows that the right hand side
is of size $|\log b|^{-1}$ in $\mathcal{I}_{-1}^{0}$. So that
$\|\tilde{\mathcal{R}}_{1}\|_{\mathcal{I}_{-1}^{0}}\lesssim|\log b|^{-1}$,
$\|b\partial_{b}\tilde{\mathcal{R}}_{1}\|_{\mathcal{I}_{-1}^{0}}\lesssim|\log
b|^{-1}$ and
$\|\partial_{\bar{\alpha}}\tilde{\mathcal{R}}_{1}\|_{\mathcal{I}_{-1}^{0}}\lesssim
1$.

Computation of $\mathcal{R}_{0}$: For $n=0$ a refinement is also necessary.
The equation for $\mathcal{R}_{0}$ is:

$\mathscr{A}_{0}\mathcal{R}_{0}=b\Big{[}r\partial_{r}+2\big{(}1+\tilde{\alpha}\big{)}\Big{]}\mathcal{R}_{0}+\Big{[}r\partial_{r}+2\big{(}1+\tilde{\alpha}\big{)}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right).$

We look for a solution under the form
$\mathcal{R}_{0}(r)=\mathcal{R}_{0,1}(r)+\mathcal{R}_{0,2}(r+1)+\tilde{\mathcal{R}}_{0}$
where

$\mathcal{R}_{0,1}=\frac{1}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i},\
\
(\partial_{rr}+\frac{3}{r}\partial_{r}-b(r\partial_{r}+2))\mathcal{R}_{0,1}=b$
$\mathcal{R}_{0,2}=\frac{1}{2\log
b}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i}[2\log(r)-\Psi(i+2)-\gamma],\
\
(\partial_{rr}+\frac{3}{r}\partial_{r}-b(r\partial_{r}+2))\mathcal{R}_{0,2}=\frac{2b}{\log
b}\log r$

so that

$\displaystyle\mathscr{A}_{0}\tilde{\mathcal{R}}_{0}-b(r\partial_{r}+2\tilde{\alpha})\tilde{\mathcal{R}}_{0}$
$\displaystyle=$ $\displaystyle-b-\frac{2b}{\log
b}\log(r+1)+\Big{[}r\partial_{r}+2+2\tilde{\alpha}\Big{]}b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)$
$\displaystyle-\left(\left(\frac{4r}{\langle
r\rangle^{4}}-\frac{4}{r}\right)\partial_{r}+\frac{8}{\langle
r\rangle^{4}}\right)\mathcal{R}_{1,1}-\left(\left(\frac{4r}{\langle
r\rangle^{4}}-\frac{1}{r}-\frac{3}{r+1}\right)\partial_{r}+\frac{8}{\langle
r\rangle^{4}}+b\partial_{r}\right)\mathcal{R}_{1,2}(r+1)$
$\displaystyle+2b\tilde{\alpha}(\mathcal{R}_{0,1}+\mathcal{R}_{0,2}(r+1))$

In the right hand side, the first line is $\mathcal{O}(br^{-1}\log r)$ from
(2.16), and so is the second from the definition of $\mathcal{R}_{0,1}$ and
$\mathcal{R}_{0,2}$. For the last line,
$\|\mathcal{R}_{0,1}+\mathcal{R}_{0,2}\|_{\mathcal{I}_{-1}^{0}}\lesssim 1$ and
$|\tilde{\alpha}|\lesssim|\log b|^{-1}$. Therefore the right hand side is of
size $|\log b|^{-1}$ in $\mathcal{I}_{-1}^{0}$, and we get
$\|\tilde{\mathcal{R}}_{0}\|_{\mathcal{I}_{-1}^{0}}+\|b\partial_{b}\tilde{\mathcal{R}}_{0}\|_{\mathcal{I}_{-1}^{0}}\lesssim|\log
b|^{-1}$ and
$\|\partial_{\bar{\alpha}}\tilde{\mathcal{R}}_{0}\|_{\mathcal{I}_{-1}^{0}}\lesssim
1$.

Step 5 _Number of zeros_ : For the case $n=0$, the identity (2.24) gives with
(2.25), (2.14) and (2.12):

$\phi_{0}^{\textup{in}}(r)=\frac{r^{2}}{\langle
r\rangle^{4}}+b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)+2\bar{\alpha}b\Big{(}-T_{1}+S_{0}\Big{)}+b\mathcal{R}_{0}.$
(2.43)

From the pointwise bounds (2.35), (2.16), (2.28) and (2.29), and
$|\bar{\alpha}|\lesssim|\ln b|^{-2}$ we infer:

$\left|b\left(-\frac{2}{\ln
b}T_{1}+\mathscr{A}_{0}^{-1}\Theta_{0}\right)+2\bar{\alpha}b\Big{(}-T_{1}+S_{0}\Big{)}+b\mathcal{R}_{0}\right|<\frac{r^{2}}{\langle
r\rangle^{4}}$

on $(0,R_{0}]$ for $\zeta_{0}$ small enough and $b$ small enough, so
$\phi_{0}^{\textup{in}}(r)$ has no zero. For $n\geq 1$ one has that, from the
identities (2.24) and (2.25), the bounds (2.35), (2.16), (2.26), (2.27),
(2.30), (2.31) and $|\bar{\alpha}|\lesssim|\ln b|^{-2}$:

$\phi^{\textup{in}}_{n}=T_{0}(r)+2nbT_{1}(r)+\tilde{\phi}^{\textup{in}}_{n}(r),\quad\mbox{
with
}\quad|\tilde{\phi}^{\textup{in}}_{n}(r)|+r|\partial_{r}\tilde{\phi}^{\textup{in}}_{n}(r)|\leq
b\zeta_{0}^{2}r^{2}\langle r\rangle^{-2}\langle\ln\langle r\rangle\rangle,$
(2.44)

where the bound is valid on $[0,R_{0}]$. We recall from (2.15):

$T_{0}(r)+2nbT_{1}(r)=\frac{1}{r^{2}}-bn\log(r)+\mathcal{O}(b+r^{-3})\quad\mbox{as
}r\rightarrow\infty,$ (2.45)
$\partial_{r}T_{0}(r)+2nb\partial_{r}T_{1}(r)=\frac{-2}{r^{3}}-\frac{bn}{r}+\mathcal{O}(b|\ln
r|r^{-2}+r^{-4})\quad\mbox{as }r\rightarrow\infty.$ (2.46)

From the above identities, we obtain that $\phi^{\textup{in}}$ vanishes
exactly once on $[0,R_{0}]$ at the point $r_{0}$,

$r_{0}=\frac{1}{\sqrt{b}\sqrt{n|\log b|}}(1+\mathcal{O}(\zeta_{0}^{2}),$
(2.47)

and that there exists a constant $c>0$ such that

$\frac{c(r_{0}-r)}{r_{0}r^{2}}\leq\phi^{\textup{in}}_{n}(r)\leq\frac{(r_{0}-r)}{cr_{0}r^{2}}\mbox{
on
}[1,r_{0}],\quad\frac{(r_{0}-r)}{cr_{0}r^{2}}\leq\phi^{\textup{in}}_{n}(r)\leq\frac{c(r_{0}-r)}{r_{0}r^{2}}\mbox{
on }[r_{0},R_{0}].$ (2.48)

∎

######  Lemma 2.4.

Let $V$ be a smooth function satisfying $|\partial_{r}^{k}V|\lesssim|\ln
b|^{-1}r^{2-k}\langle r\rangle^{-4}$ for $k=0,1$. Then for any fixed $n$, for
$\zeta_{0}$ small enough, there exists $b^{*}>0$ such that for all $0<b<b^{*}$
and $\tilde{\alpha}=\mathcal{O}(|\ln b|^{-1})$, there exists a solution
$\phi^{\textup{in},V}_{n}$ to

$\mathscr{A}_{0}\phi^{\textup{in},V}_{n}-b\big{[}r\partial_{r}+2(1-n+\tilde{\alpha})\big{]}\phi^{\textup{in},V}_{n}+r^{-1}\partial_{r}(V\phi^{\textup{in},V}_{n})=0$

on $[0,R_{0}]$ which satisfies

$\|\phi^{\textup{in},V}_{n}-\phi^{\textup{in}}_{n}\|_{\mathcal{I}^{-2}_{0}}\lesssim\frac{1}{|\ln
b|}.$ (2.49)

###### Proof.

We only treat the case $n\geq 1$. Indeed, from Lemma 2.3, $\phi^{\textup{in}}$
vanishes once on $[0,R_{0}]$ for $n\geq 1$ at the point $r_{0}$ defined by
(2.47), whereas for $n=0$ it does not. Reintegrating the Wronskian relation is
then harder in the case $n=1$, and the case $n=0$ can be treated with the very
same ideas but simpler computations. We shall use results on
$\phi^{\textup{in}}$ proved in the ” _Number of zeros_ ” part of the proof of
Lemma 2.3.

Step 1 _Uniform asymptotic for the second fundamental solution_ : We claim
that there exists $\Gamma$ another linearly independent solution to

$\mathscr{A}_{0}\Gamma-b(r\partial_{r}+2(1-n+\tilde{\alpha}))\Gamma=0$

on $[0,R_{0}]$ such that:

$|\Gamma(r)|\leq C\mbox{ and }|\partial_{r}\Gamma(r)|\leq Cr|\ln r|\langle
r\rangle^{-2}\langle\ln r\rangle^{-1}\mbox{ on }[0,R_{0}]$ (2.50)

with a constant $C$ that is independent of $b$ and $\tilde{\alpha}$. Indeed,
from standard arguments, the Wronskian
$W=\Gamma^{\prime}\phi_{n}^{\textup{in}}-\Gamma\phi_{n}^{\textup{in}^{\prime}}$
is (fixing the integration constant without loss of generality):

$W=\frac{r}{(1+r^{2})^{2}}e^{b\frac{r^{2}}{2}}$ (2.51)

so the second fundamental solution is given by, reintegrating the Wronskian
relation (we again fix here an integration constant without loss of
generality):

$\Gamma(r)=\phi^{\textup{in}}(r)\int_{1}^{r}\frac{W(\xi)}{|\phi^{\textup{in}}(\xi)|^{2}}d\xi=\phi^{\textup{in}}(r)\int_{1}^{r}\frac{\xi
e^{b\frac{\xi^{2}}{2}}}{(1+\xi^{2})^{2}|\phi^{\textup{in}}(\xi)|^{2}}d\xi.$

The asymptotic near the origin follows from (2.24), (2.44) and (2.15), and
direct computations, so we only focus on the asymptotic of $\Gamma$ for $r$
large. For $1\leq r\leq r_{0}$ from (2.48):

$|\Gamma(r)|\lesssim\frac{(r_{0}-r)r_{0}}{r^{2}}\int_{1}^{r}\frac{\xi}{(r_{0}-\xi)^{2}}d\xi\lesssim
1.$

Next, for $r\geq r_{0}$, we avoid the singularity in the integral by noticing
that there exists a constant $C$ such that

$\Gamma(r)=C\phi^{\textup{in}}(r)+\phi^{\textup{in}}(r)\int_{R_{0}}^{r}\frac{W(\xi)}{|\phi^{\textup{in}}(\xi)|^{2}}d\xi.$

To estimate $C$, one computes from the first formula for $\Gamma$ and the
asymptotic (2.48) near $r_{0}$ of $\phi^{\textup{in}}$:

$\Gamma^{\prime}(r_{0})=\lim_{r\uparrow
r_{0}}\left(\partial_{r}\phi^{\textup{in}}(r)\int_{1}^{r}\frac{W(\xi)}{|\phi^{\textup{in}}(\xi)|^{2}}d\xi+\frac{W(r)}{\phi^{\textup{in}}(r)}\right)=\mathcal{O}(r_{0}^{-1}).$

Similarly, we have

$\Gamma^{\prime}(r_{0})=C(\phi^{\textup{in}}(r_{0}))^{\prime}+\lim_{r\downarrow
r_{0}}\left(\partial_{r}\phi^{\textup{in}}(r)\int_{R_{0}}^{r}\frac{W(\xi)}{|\phi^{\textup{in}}(\xi)|^{2}}d\xi+\frac{W(r)}{\phi^{\textup{in}}(r)}\right)=C(\phi^{\textup{in}}(r_{0}))^{\prime}+\mathcal{O}(r_{0}^{-1}).$

As $\partial_{r}\phi^{\textup{in}}(r_{0})=-2r_{0}^{-3}(1+\mathcal{O}(1))$ we
obtain $C=\mathcal{O}(r_{0}^{2})=\mathcal{O}(b^{-1}|\log b|^{-1})$. For all
$r_{0}\leq r\leq R_{0}$ we find from (2.48):

$\left|\phi^{\textup{in}}(r)\int_{R_{0}}^{r}\frac{W(\xi)}{|\phi^{\textup{in}}(\xi)|^{2}}d\xi\right|\lesssim\frac{(r-r_{0})r_{0}^{2}}{r}\int_{r}^{R_{0}}\frac{d\xi}{(\xi-
r_{0})^{2}\xi}\lesssim 1,$

and:

$|C\phi^{\textup{in}}(r)|\lesssim
r_{0}^{2}\frac{(r-r_{0})}{r_{0}^{2}r^{-1}}\lesssim 1.$

Hence $|\Gamma(r)|\lesssim 1$ for $r_{0}<r\leq R_{0}$ as well. This proves
(2.50) for $\Gamma$. The proof for $\partial_{r}\Gamma$ is verbatim the same
so that we skip it.

Step 2 _Bound for the resolvant under orthogonality condition_ : Let a
solution to $\mathscr{A}_{b,\tilde{\alpha}}u=r^{-1}\partial_{r}(Vf)$ be given
by

$u(r)=\phi^{\textup{in}}(r)\int_{r}^{R_{0}}\frac{\Gamma(\xi)}{W(\xi)}\xi^{-1}\partial_{\xi}(Vf)(\xi)d\xi+\Gamma(r)\int_{0}^{r}\frac{\phi^{\textup{in}}(\xi)}{W(\xi)}\xi^{-1}\partial_{\xi}(Vf)(\xi)d\xi,$

then we claim the resolvent bound:

$\|u\|_{\mathcal{I}_{0}^{-2}}\lesssim\frac{1}{|\ln
b|}\left(\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}\right).$ (2.52)

We now prove this claim. From the hypothesis on $V$, (2.44) and (2.50), the
first term can be bounded by

$\displaystyle\left|\phi^{\textup{in}}(r)\int_{r}^{R_{0}}\frac{\Gamma(\xi)}{W(\xi)}\xi^{-1}\partial_{\xi}(Vf)(\xi)d\xi\right|\lesssim\frac{1}{|\ln
b|}r^{2}\left(\langle r\rangle^{-4}+b\langle r\rangle^{-2}\langle\ln\langle
r\rangle\rangle\right)\int_{r}^{R_{0}}\left(|\xi|^{-1}|f(\xi)|+|\partial_{\xi}f(\xi)|\right)d\xi$
$\displaystyle\lesssim\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}r^{2}\left(\langle r\rangle^{-4}+b\langle r\rangle^{-2}\langle\ln\langle
r\rangle\rangle\right)\int_{r}^{R_{0}}\xi\langle\xi\rangle^{-4}\langle\ln\langle\xi\rangle\rangle
d\xi$
$\displaystyle\lesssim\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}r^{2}\left(\langle r\rangle^{-6}\langle\ln\langle\xi\rangle\rangle+b\langle
r\rangle^{-4}\langle\ln\langle
r\rangle\rangle^{2}\right)\lesssim\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}r^{2}\langle r\rangle^{-4}.$

For the second term, we use the decomposition (2.44), the identities (2.12)
and (2.51), the bound (2.50) and the bounds on $V$ to get

$\displaystyle\left|\Gamma^{\textup{in}}(r)\int_{0}^{r}\frac{\phi^{\textup{in}}(\xi)}{W(\xi)}\xi^{-1}\partial_{\xi}(Vf)(\xi)d\xi\right|=\left|\Gamma^{\textup{in}}(r)\int_{0}^{r}\frac{\langle\xi\rangle^{4}e^{-b\frac{r^{2}}{2}}}{\xi^{2}}\left(\frac{\xi^{2}}{\langle\xi\rangle^{4}}+bT_{1}+\tilde{\phi}^{\textup{in}}\right)\partial_{\xi}(Vf)(\xi)d\xi\right|$
$\displaystyle=$
$\displaystyle\left|\Gamma^{\textup{in}}(r)\left(V(r)f(r)e^{-\frac{br^{2}}{2}}-\int_{0}^{r}b\xi
Vfe^{-\frac{b\xi^{2}}{2}}d\xi+\int_{0}^{r}\frac{\langle\xi\rangle^{4}e^{-b\frac{r^{2}}{2}}}{\xi^{2}}\left(bT_{1}+\tilde{\phi}^{\textup{in}}\right)\partial_{\xi}(Vf)(\xi)d\xi\right)\right|$
$\displaystyle\lesssim$ $\displaystyle\frac{1}{|\ln b|}\left(r^{-2}\langle
r\rangle^{-4}|f(r)|+b\int_{0}^{r}\xi^{3}\langle\xi\rangle^{-4}|f|d\xi+b\int_{0}^{r}\xi\langle\xi\rangle^{-2}\langle\ln\langle\xi\rangle\rangle(|f|+\xi|\partial_{\xi}f|)d\xi\right)$
$\displaystyle\lesssim$
$\displaystyle\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}\left(r^{4}\langle
r\rangle^{-8}+b\int_{0}^{r}\xi^{5}\langle\xi\rangle^{-8}d\xi+b\int_{0}^{r}\xi^{3}\langle\xi\rangle^{-6}\langle\ln\langle\xi\rangle\rangle
d\xi\right)$ $\displaystyle\lesssim$
$\displaystyle\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}r^{2}\langle r\rangle^{-4}$

because $r\lesssim b^{-1}$. Combining the above two bounds yields the
following estimate on $[0,R_{0}]$,

$|u(r)|\lesssim\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}r^{2}\langle r\rangle^{-4}.$

Differentiating the identity satisfied by $u$ yields

$\partial_{r}u=\partial_{r}\phi^{\textup{in}}(r)\int_{r}^{R_{0}}\frac{\Gamma(\xi)}{W(\xi)}\xi^{-1}\partial_{\xi}(Vf)(\xi)d\xi+\partial_{r}\Gamma(r)\int_{0}^{r}\frac{\phi^{\textup{in}}(\xi)}{W(\xi)}\xi^{-1}\partial_{\xi}(Vf)(\xi)d\xi.$

Hence, computing the same way the integral terms as we just did, and using
(2.44), (2.46) and (2.50) we get

$|\partial_{r}u(r)|\lesssim\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}r\langle r\rangle^{-4}.$

Using the definition of $\mathscr{A}_{0}$, we write

$\partial_{r}^{2}u=\left(\frac{1}{r}-\frac{Q}{r}\right)\partial_{r}u-\frac{\partial_{r}Q}{r}u+b(r\partial_{r}+2(1-n+\tilde{\alpha}))u+r^{-1}\partial_{r}(Vf),$

from which and the hypotheses on $V$ and the bounds on $u$ and
$\partial_{r}u$, we obtain

$|\partial_{r}^{2}u|\lesssim\frac{\|f\|_{X^{-2}_{1}}+\|r\partial_{r}f\|_{X^{-2}_{1}}}{|\ln
b|}\langle r\rangle^{-4}.$

The bounds on $u$, $\partial_{r}u$ and $\partial_{r}^{2}u$ imply (2.52).

Step 3 _Fixed point_ : We look for a solution to
$\big{[}\mathscr{A}_{0}-b(r\partial_{r}+2(1-n+\tilde{\alpha}))-r^{-1}\partial_{r}(V\cdot)\big{]}\phi^{\textup{in},V}=0$
under the form

$\phi^{\textup{in},V}=\phi^{\textup{in}}+\tilde{\phi}^{\textup{in},V}.$

Then, $\tilde{\phi}^{\textup{in},V}$ solves

$\big{[}\mathscr{A}_{0}-br\partial_{r}+2b(1-n+\tilde{\alpha})\big{]}\tilde{\phi}^{\textup{in},V}=r^{-1}\partial_{r}(V\phi^{\textup{in}})+r^{-1}\partial_{r}(V\tilde{\phi}^{\textup{in},V}).$

We solve this using a fixed point argument in $\mathcal{I}^{-2}_{0}$. As
$\|\phi^{\textup{in}}\|_{\mathcal{I}^{-2}_{1}}\lesssim 1$ from Lemma 2.3, as
$\|\cdot\|_{\mathcal{I}^{-2}_{1}}\lesssim\|\cdot\|_{\mathcal{I}^{-2}_{0}}$
from the very definition of these spaces, the bound (2.52) implies

$\displaystyle\|\big{[}\mathscr{A}_{0}-br\partial_{r}+2b(1-n+\tilde{\alpha})\big{]}^{-1}(r^{-1}\partial_{r}(V\tilde{\phi}^{\textup{in}}))\|_{\mathcal{I}^{-2}_{0}}$
$\displaystyle\qquad\lesssim\frac{1}{|\ln
b|}(\|\phi^{\textup{in}}\|_{X^{-2}_{1}}+\|r\partial_{r}\phi^{\textup{in}}\|_{X^{-2}_{1}})\lesssim\frac{1}{|\ln
b|}\|\phi^{\textup{in}}\|_{\mathcal{I}^{-2}_{1}}\lesssim\frac{1}{|\ln b|},$
$\left\|\mathscr{A}_{0}-br\partial_{r}+2b(1-n+\tilde{\alpha})\big{]}^{-1}(r^{-1}\partial_{r}(V\tilde{\phi}^{\textup{in},V}))\right\|_{\mathcal{I}^{-2}_{0}}\lesssim\frac{\|\tilde{\phi}^{\textup{in},V}\|_{\mathcal{I}^{-2}_{1}}}{|\ln
b|}\lesssim\frac{\|\tilde{\phi}^{\textup{in},V}\|_{\mathcal{I}^{-2}_{0}}}{|\ln
b|}.$

Hence, the mapping which to $\tilde{\phi}^{\textup{in},V}$ assigns

$\big{[}\mathscr{A}_{0}-br\partial_{r}+2b(1-n+\tilde{\alpha})\big{]}^{-1}\left(r^{-1}\partial_{r}(V\phi^{\textup{in}})+r^{-1}\partial_{r}(V\tilde{\phi}^{\textup{in},V})\right)$

is a contraction in $B_{\mathcal{I}^{-2}_{0}}(0,C|\ln b|^{-1})$ for $C$ large
enough and then for $b$ small enough. Its unique fixed point is the desired
solution, and satisfies the conclusion of the lemma. ∎

### 2.2. Analysis in the outer zone $r\geq R_{0}$

In this part we solve problem (2.1) in the interval $[R_{0},\infty)$ where the
potential term can be treated as a small perturbation. To this end, we rewrite
equation (2.1) as

$\partial_{r}^{2}\phi+\frac{3}{r}\partial_{r}\phi-
br\partial_{r}\phi-\alpha\phi-\frac{4}{r(1+r^{2})}\partial_{r}\phi+\frac{8}{(1+r^{2})^{2}}\phi=0.$
(2.53)

Introducing the change of variable

$\phi^{\textup{ex}}(r)=q(z)\quad\textup{with}\;\;z=\frac{br^{2}}{2},$ (2.54)

yields the equation satisfied by $q$,

$\big{(}\mathcal{K}_{\theta}+P_{0}\big{)}q(z)=0,\quad z\geq
z_{0}=\frac{\zeta_{0}^{2}}{2},\quad\theta=\frac{\alpha}{2b},$ (2.55)

where $\mathcal{K}_{\theta}$ is a Kummer type operator defined by

$\mathcal{K}_{\theta}=z\partial_{z}^{2}+(2-z)\partial_{z}-\theta,$ (2.56)

and $P_{0}$ is the potential

$P_{0}=-\frac{2b}{(b+2z)}\partial_{z}+\frac{4b}{(b+2z)^{2}}.$ (2.57)

We will treat the differential operator $P_{0}$ as a perturbation of
$\mathcal{K}_{\theta}$ in the outer zone. We first claim the following.

######  Lemma 2.5 (Properties of $\mathcal{K}_{\theta}$).

  * (i)

(Inversion) Assume that $-\theta\not\in\mathbb{N}$, then an explicit inversion
of $\mathcal{K}_{\theta}$ is given by

$\mathcal{K}_{\theta}^{-1}f:=h_{\theta}(z)\int_{z_{0}}^{z}\tilde{h}_{\theta}(\xi)f(\xi)\xi
e^{-\xi}d\xi+\tilde{h}_{\theta}(z)\int_{z}^{\infty}h_{\theta}(\xi)f(\xi)\xi
e^{-\xi}d\xi,$ (2.58)

where $h_{\theta}$ and $\tilde{h}_{\theta}$ are the two linearly independent
solutions to Kummer’s equation $\mathcal{K}_{\theta}h=0$:

$\displaystyle
h_{\theta}(z)=\frac{1}{z\Gamma(\theta)}+\frac{1}{\Gamma(\theta-1)}\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\Big{]},$ (2.59)
$\displaystyle\qquad\qquad\tilde{h}_{\theta}(z)=\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i},$
(2.60)

where $(a)_{i}=\frac{\Gamma(a+i)}{\Gamma(a)}$, $\Gamma$ is the Gamma function,
and $\Psi=\Gamma^{\prime}/\Gamma$ is the digamma function. Moreover, we have
the asymptotic behavior as $z\to\infty$,

$h_{\theta}(z)=z^{-\theta}\big{(}1+\mathcal{O}(z^{-1})\big{)},\quad\tilde{h}_{\theta}(z)=\frac{\Gamma(2)}{\Gamma(\theta)}e^{z}z^{\theta-2}\big{(}1+\mathcal{O}(z^{-1})\big{)}.$
(2.61)

and for $z_{0}\leq z\leq 2$, for $C$ dependent of $z_{0}$ if $n=0$ and
independent if $n\geq 1$:

$|h_{\theta}(z)|+|\tilde{h}_{\theta}(z)|\lesssim C$ (2.62)

  * (ii)

(Continuity) Let $a\in\mathbb{R}$, and $\mathcal{E}_{0}^{p,a}$ be the Banach
space of functions $f:[z_{0},\infty)\to\mathbb{R}$ equipped with the norm

$\displaystyle\|f\|_{\mathcal{E}^{a}}$ $\displaystyle:=\sup_{z\geq
z_{0}}\langle
z\rangle^{-a}\big{(}|f(z)|+|z\partial_{z}f(z)|+|z^{2}\partial_{z}^{2}f(z)|\big{)}.$

Then for any continuous function $f:[z_{0},\infty)\to\mathbb{R}$, we have the
estimate for $a>-\theta$:

$\|\mathcal{K}_{\theta}^{-1}f\|_{\mathcal{E}^{a}}\lesssim
C(z_{0})\sup_{z_{0}\leq z<\infty}\left<z\right>^{-a}|f(z)|.$ (2.63)

###### Proof.

$(i)$ See formulas 13.1.2, 13.1.6 and 13.1.22 in [Abramowitz and Stegun(1992)]
for the definition of $h_{\theta}$, $\tilde{h}_{\theta}$ and the Wronskian
$W(h_{\theta},\tilde{h}_{\theta})$ respectively. For the bound for $z_{0}\leq
z\leq 2$, notice that from the Gamma function’s recurrence relation and the
bound on $\bar{\alpha}$:

$\displaystyle\Gamma(\theta)=\frac{\Gamma(\theta+n)}{\theta(\theta+1)...(\theta+n-1)}$
$\displaystyle=\frac{\Gamma(1+\tilde{\alpha})}{(1-n+\tilde{\alpha})(2-n+\tilde{\alpha})...(-1+\tilde{\alpha})\tilde{\alpha}}$
$\displaystyle\quad\sim\frac{(-1)^{n}}{(n-1)!\tilde{\alpha}}+\mathcal{O}(1)=\mathcal{O}(|\ln
b|),$ (2.64)

for $n\geq 1$, and $\Gamma(\theta)=\Gamma(1+\tilde{\alpha})=\mathcal{O}(1)$
for $n=0$.

$(ii)$ The proof follows from straightforward computations. Let

$D=\sup_{z_{0}\leq z<\infty}\left<z\right>^{-a}|f(z)|,$

From (2.61), we compute for $z\geq 2$,

$\displaystyle\left|\tilde{h}_{\theta}(z)\int_{z}^{\infty}fh_{\theta}\xi
e^{-\xi}d\xi\right|\lesssim
Dz^{\theta-2}e^{z}\int_{z}^{\infty}\xi^{a}\xi^{-\theta}\xi
e^{-\xi}d\xi\lesssim Dz^{a-1},$

and from (2.59), we compute for $z\in[z_{0},2]$,

$\displaystyle\left|\tilde{h}_{\theta}(z)\int_{z}^{\infty}fh_{\theta}\xi
e^{-\xi}d\xi\right|$
$\displaystyle\lesssim\left|\tilde{h}_{\theta}(z)\int_{z}^{2}fh_{\theta}\xi
e^{-\xi}d\xi+\tilde{h}_{\theta}(z)\int_{2}^{\infty}fh_{\theta}\xi
e^{-\xi}d\xi\right|$ $\displaystyle\lesssim D\int_{z}^{2}\xi
d\xi+D\int_{2}^{\infty}\xi^{a-\theta+1}e^{-\xi}d\xi\lesssim D.$

Similarly, we have for $z\geq 2$, as $a>-\theta$

$\displaystyle\left|h_{\theta}(z)\int_{z_{0}}^{z}f\tilde{h}_{\theta}\xi
e^{-\xi}d\xi\right|$ $\displaystyle\lesssim Dz^{-\theta}\int_{z_{0}}^{2}\xi
d\xi+Dz^{-\theta}\int_{2}^{z}\xi^{a}\xi^{\theta-2}\xi^{1}d\xi\lesssim Dz^{a}.$

and for $z_{0}\leq z\leq 2$,

$\left|\int_{z_{0}}^{z}f\tilde{h}_{\theta}\xi e^{-\xi}d\xi\right|\lesssim
D\int_{z_{0}}^{z}\xi d\xi\lesssim D.$

This proves the continuity bound (2.63) for $\mathcal{K}_{\theta}^{-1}f$. We
now take derivatives. For $z\geq 2$, we estimate from (2.58), (2.61):

$\displaystyle\Big{|}z\partial_{z}\mathcal{K}_{\theta}^{-1}f(z)\Big{|}$
$\displaystyle\lesssim\left|z\partial_{z}\tilde{h}_{\theta}\int_{z}^{\infty}fh_{\theta}\xi
e^{-\xi}d\xi\right|+\left|z\partial_{z}h_{\theta}\int_{z_{0}}^{z}f\tilde{h}_{\theta}\xi
e^{-\xi}d\xi\right|$ $\displaystyle\lesssim
D\left(e^{z}z^{\theta-1}z^{a-\theta+1}e^{-z}+z^{-\theta}z^{a+\theta}\right)\lesssim
Dz^{a}.$

For $z\in[z_{0},2]$, we estimate from (2.60):

$\displaystyle\Big{|}z\partial_{z}\mathcal{K}_{\theta}^{-1}f(z)\Big{|}$
$\displaystyle\lesssim\left|z\partial_{z}\tilde{h}_{\theta}\int_{z}^{\infty}fh_{\theta}\xi
e^{-\xi}d\xi\right|+\left|z\partial_{z}h_{\theta}\int_{z_{0}}^{z}f\tilde{h}_{\theta}\xi
e^{-\xi}d\xi\right|$ $\displaystyle\lesssim D\big{(}\int_{z}^{2}\xi
d\xi+\int_{2}^{\infty}\xi^{a-\theta+1}e^{-\xi}d\xi\big{)}+D\int_{z_{0}}^{z}\xi
d\xi\lesssim D.$

Using $\mathcal{K}_{\theta}\mathcal{K}_{\theta}^{-1}f=f$ and the definition of
$\mathcal{K}_{\theta}$, we have the estimate for $z\geq 2$,

$\displaystyle|z\partial_{z}^{2}\mathcal{K}_{\theta}^{-1}f(z)|\lesssim|z\partial_{z}\mathcal{K}^{-1}_{\theta}f(z)|+|\mathcal{K}_{\theta}^{-1}f(z)|\lesssim
Dz^{a},$

and for $z\in[z_{0},2]$,

$\displaystyle|z^{2}\partial_{z}^{2}\mathcal{K}_{\theta}^{-1}f(z)|\lesssim|z\partial_{z}\mathcal{K}_{\theta}^{-1}f(z)|+|z\mathcal{K}_{\theta}^{-1}f(z)|\lesssim
D.$

Collecting the above estimates yields the estimate (2.63). This concludes the
proof of Lemma 2.5. ∎

We are now in the position of computing the solution $q$ to equation (2.55) by
a perturbation argument.

######  Lemma 2.6 (Outer eigenfunctions for the radial mode).

Fix $n\in\mathbb{N}$, and $\theta=1-n+1/\ln b+\bar{\alpha}$. For
$0<\zeta_{0}\ll 1$ and any small $0<\delta\ll 1$, there exist $b^{*}>0$ such
that for all $0<b\leq b^{*}$, for all $\bar{\alpha}=\mathcal{O}(|\ln b|^{-2})$
there exists a smooth solution

$q(b,\bar{\alpha},z)=\Gamma(\theta)h_{\theta}(z)+\mathcal{G}(b,\bar{\alpha},z)$
(2.65)

to (2.55) on $[z_{0},\infty)$, where $h_{\theta}$ is introduced in Lemma 2.5
and $\mathcal{G}$ satisfies the following estimates for some universal $C>0$:

$\|\mathcal{G}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim b|\ln
b|^{C},\quad\|b\partial_{b}\mathcal{G}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
b|\ln
b|^{C},\quad\|\partial_{\theta}\mathcal{G}\|_{\mathcal{E}_{0}^{-\theta+\delta}}\lesssim
b|\ln b|^{C}.$ (2.66)

where the constants in the estimates depend on $z_{0}$. Finally, on the
interval $[z_{0},\infty)$, $q$ does not vanish for $n=0,1$, while for $n\geq
2$ it possesses $n-1$ zeros.

######  Lemma 2.7.

Assume $P_{0}$ is replaced by $P_{0}(q)+\frac{1}{2}\partial_{z}(\tilde{V}q)/z$
where $\tilde{V}$ satisfies $|\tilde{V}|+|z\partial_{z}\tilde{V}|\lesssim
b|\ln b|^{-1}z^{-1}$ on $[z_{0},\infty)$. Then existence result of Lemma 2.6
of a solution
$q^{V}=\Gamma(\theta)h_{\theta}(z)+\mathcal{G}^{V}(b,\bar{\alpha},z)$ and the
first bound in (2.66) still hold true.

###### Proof of Lemma 2.6.

From the bound on the Gamma function (2.64), we will simply consider a
solution of the form $q(z)=h_{\theta}(z)+\mathcal{G}(b,\bar{\alpha},z)$ (with
the abuse of notation of keeping the notation $\mathcal{G}$), and prove the
estimate (2.66) for $\mathcal{G}(b,\bar{\alpha},z)$, which will prove the
Lemma upon multiplication by $\Gamma(\theta)$.

Step 1 _Existence and bounds_ : Note that $P_{0}$ has the form:

$P_{0}(q)=V_{1}q+V_{2}\partial_{z}q,\quad\mbox{with }|V_{1}|+|zV_{1}|\lesssim
bz^{-1}.$ (2.67)

Let us write from (2.65) the equation satisfied by $\mathcal{G}$,

$\displaystyle\mathcal{K}_{\theta}\mathcal{G}+P_{0}\mathcal{G}+P_{0}h_{\theta}=0.$

Let $\Gamma$ the affine mapping defined as

$\Gamma(f)=-\mathcal{K}_{\theta}^{-1}\big{[}P_{0}f+P_{0}h_{\theta}\big{]}\equiv
D\Gamma(f)+D\Gamma(h_{\theta}),$

where

$D\Gamma(f)=-\mathcal{K}_{\theta}^{-1}\big{[}P_{0}f\big{]},$

and $\mathcal{K}_{\theta}^{-1}$ is given by (2.58). We estimate from the
definition (2.57) of $P_{0}$ and (2.63),

$\displaystyle\|D\Gamma(h_{\theta})\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\|\mathcal{K}_{\theta}^{-1}P_{0}h_{\theta}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\sup_{z\in[z_{0},\infty)}\left<z\right>^{\theta-\delta}|P_{0}h_{\theta}|\lesssim
b.$

From (2.63), we estimate for all $f\in\mathcal{E}^{0,-\theta+\delta}$,

$\|D\Gamma(f)\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\sup_{z\in[z_{0},\infty)}\left<z\right>^{\theta-\delta}|P_{0}f(z)|\lesssim
b\|f\|_{\mathcal{E}^{-\theta+\delta}}.$ (2.68)

It follows that $\Gamma$ is a contraction mapping on
$B_{\mathcal{E}^{-\theta+\delta}}(0,Mb)$ for some $M=M(\zeta_{0})>0$ large
enough. Hence, there exists a unique fixed point $\mathcal{G}$ with

$\mathcal{G}=\Gamma(\mathcal{G})\quad\textup{with}\quad\|\mathcal{G}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
b.$

Differentiating the above fixed point relation yields:

$\displaystyle\partial_{\theta}\mathcal{G}=D\Gamma(\partial_{\theta}\mathcal{G})+(\partial_{\theta}\Gamma)(\mathcal{G}),\quad\partial_{b}\mathcal{G}=D\Gamma(\partial_{b}\mathcal{G})+(\partial_{b}\Gamma)(\mathcal{G}).$

Since $P_{0}$ depends on $b$ and not on $\theta$, whereas $h_{\theta}$,
$h^{\prime}_{\theta}$ and $\mathcal{K}_{\theta}$ depend on $\theta$ and not on
$b$, we have the identities:

$\displaystyle(\partial_{\theta}\Gamma)(\mathcal{G})=-\partial_{\theta}(\mathcal{K}_{\theta}^{-1})(P_{0}(\mathcal{G}+h_{\theta}))-\mathcal{K}^{-1}(P_{0}\partial_{\theta}h_{\theta}),\quad(\partial_{b}\Gamma)(\mathcal{G})=-\mathcal{K}_{\theta}^{-1}(\partial_{b}P_{0}(\mathcal{G}+h_{\theta})).$

We compute from (2.59) that:

$\displaystyle\partial_{\theta}h_{\theta}(z)$ $\displaystyle=$
$\displaystyle-\frac{\Psi(\theta)}{z\Gamma(\theta)}-\frac{\Psi(\theta-1)}{\Gamma(\theta-1)}\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\Big{]}$
$\displaystyle+\frac{1}{\Gamma(\theta-1)}\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}\ln
z+\Psi^{\prime}(\theta+i)-\Psi(1+i)-\Psi(2+i)\Big{]}$
$\displaystyle+\frac{1}{\Gamma(\theta-1)}\sum_{i=0}^{\infty}\frac{(\theta)_{i}(\Psi(\theta+i)-\Psi(\theta))}{(2)_{i}i!}z^{i}\Big{[}\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\Big{]}.$

Hence, we infer from
$|\Psi(\theta)|+|\Psi(\theta-1)|\lesssim|\tilde{\alpha}|^{-1}\lesssim|\ln b|$,
$|\Psi^{\prime}(\theta+i)|\lesssim|\tilde{\alpha}|^{-2}\lesssim|\ln b|^{2}$
and $|(\theta)_{i}(\Psi(\theta+i)-\Psi(\theta))|\lesssim 1$ the rough upper
bound on $[z_{0},\infty)$:

$|\partial_{\theta}h_{\theta}(z)|\lesssim|\ln b|^{2}z^{-1}\ln\langle
z\rangle\langle z\rangle^{-\theta},$

which extends to derivatives. Similarly, we have from (2.60)

$\partial_{\theta}\tilde{h}_{\theta}(z)=\sum_{i=0}^{\infty}\frac{(\theta)_{i}(\Psi(\theta+i)-\Psi(\theta))}{(2)_{i}i!}z^{i},$

satisfies the rough upper bound
$|\partial_{\theta}\tilde{h}_{\theta}(z)|\lesssim\ln\langle z\rangle\langle
z\rangle^{\theta-2}e^{z}$ on $[z_{0},\infty)$. We get from (2.58):

$\displaystyle(\partial_{\theta}\mathcal{K}_{\theta}^{-1})f$ $\displaystyle=$
$\displaystyle(\partial_{\theta}h_{\theta})(z)\int_{z_{0}}^{z}\tilde{h}_{\theta}(\xi)f(\xi)\xi^{2}e^{-\xi}d\xi+h_{\theta}(z)\int_{z_{0}}^{z}\partial_{\theta}\tilde{h}_{\theta}(\xi)f(\xi)\xi^{2}e^{-\xi}d\xi$
$\displaystyle+\partial_{\theta}(\tilde{h}_{\theta})(z)\int_{z}^{\infty}h_{\theta}(\xi)f(\xi)\xi^{2}e^{-\xi}d\xi+\tilde{h}_{\theta}(z)\int_{z}^{\infty}\partial_{\theta}(h_{\theta})(\xi)f(\xi)\xi^{2}e^{-\xi}d\xi.$

Hence, as from the above, the bounds for $h_{\theta}$ and $\tilde{h}_{\theta}$
still hold up to a logarithmic loss in $z$ and $b$ and $\delta>0$, using the
same argument as in the proof of Lemma 2.5 we get:

$\|(\partial_{\theta}\mathcal{K}_{\theta}^{-1})f\|_{\mathcal{E}^{-\theta+\delta}}\lesssim|\ln
b|^{2}\sup_{z_{0}\leq z<\infty}\left<z\right>^{\theta-\frac{\delta}{2}}|f(z)|$

and from (2.63):

$\|\mathcal{K}^{-1}(P_{0}\partial_{\theta}h_{\theta})\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\|P_{0}\partial_{\theta}h_{\theta}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
b|\ln b|^{2}.$

Thus, as $\delta$ is small, from the definition of $P_{0}$:

$\|\partial_{\theta}(\mathcal{K}_{\theta}^{-1})(P_{0}(\mathcal{G}+h_{\theta}))\|_{\mathcal{E}^{-\theta+\delta}}\lesssim|\ln
b|^{2}\|P_{0}(\mathcal{G}+h_{\theta})\|_{\mathcal{E}^{-\theta+\frac{\delta}{2}}}\lesssim
b|\ln b|^{2}\|\mathcal{G}+h_{\theta}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
b|\ln b|^{2}.$

We proved above the continuity bound
$\|D\Gamma\|_{C(\mathcal{E}^{-\theta+\delta})}\lesssim b$ and the identity,

$\partial_{\theta}\mathcal{G}=D\Gamma(\partial_{\theta}\mathcal{G})-\partial_{\theta}(\mathcal{K}_{\theta}^{-1})(P_{0}(\mathcal{G}+h_{\theta}))-\mathcal{K}^{-1}(P_{0}\partial_{\theta}h_{\theta}).$

Hence one can invert the operator $Id-D\Gamma$ for $b$ small enough, with
$\|Id+D\Gamma\|_{C(\mathcal{E}^{-\theta+\delta})}\lesssim 1$ and the above
identity gives:

$\|\partial_{\theta}\mathcal{G}\|_{\mathcal{E}^{-\theta+\delta}}=\|(Id-D\Gamma)^{-1}\left(\partial_{\theta}(\mathcal{K}_{\theta}^{-1})(P_{0}(\mathcal{G}+h_{\theta}))+\mathcal{K}^{-1}(P_{0}\partial_{\theta}h_{\theta})\right)\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
b|\ln b|^{2}.$

From the definition of $P_{0}$ and (2.63) we find:

$\|\mathcal{K}_{\theta}^{-1}(\partial_{b}P_{0}(\mathcal{G}+h_{\theta}))\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\|\partial_{b}P_{0}(\mathcal{G}+h_{\theta})\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\|\mathcal{G}+h_{\theta}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
1.$

Hence we obtain similarly from the relation
$\partial_{b}\mathcal{G}=D\Gamma(\partial_{b}\mathcal{G})-\mathcal{K}_{\theta}^{-1}(\partial_{b}P_{0}(\mathcal{G}+h_{\theta}))$
the bound:

$\|\partial_{b}\mathcal{G}\|_{\mathcal{E}^{-\theta+\delta}}\lesssim\|(Id-D\Gamma)^{-1}\mathcal{K}_{\theta}^{-1}(\partial_{b}P_{0}(\mathcal{G}+h_{\theta}))\|_{\mathcal{E}^{-\theta+\delta}}\lesssim
1.$

Step 2 _Number of zeros_ : This is a consequence of the well-known properties
of Kummer’s function $h_{\theta}$ (see[Abramowitz and Stegun(1992)]). Since
$\theta=1-n+\frac{1}{\ln b}+\mathcal{O}\left(|\ln b|^{-2}\right)$,
$h_{\theta}$ has no positive zeros for $n=0$ and possesses
$\lfloor\theta\rfloor=n-1$ zeros on the interval $(0,+\infty)$. The estimate
(2.66) and the asymptotic behavior (2.61) ensure that there exists $z_{*}>0$
such that $|q(z)|\neq 0$ and $z\partial_{z}q(z)=-\theta
z^{-\theta}(1+\mathcal{O}(b))\neq 0$ for $z\geq z_{*}$. Thus, $q(z)$ does not
change sign for $z\geq z_{*}$. It remains to show that on the interval
$z\in(z_{0},z_{*})$, $q$ has the same number of zeros than $h_{\theta}$. We
consider two cases.
\- If $|h_{\theta}(z)|\geq c_{0}$ for all $z\in(z_{0},z_{*})$ for some
$c_{0}>0$, then the estimate (2.66) implies that $|q(z)|>0$ as well on
$(z_{0},z_{*})$ for $b$ sufficiently small.
\- If $h_{\theta}(z)$ has $n-1$ zeros on $(z_{0},z_{*})$, say
$h(z_{1})=h(z_{2})=\cdots=h(z_{n-1})=0$ with $z_{1}<z_{2}<\cdots<z_{n-1}$. By
definition, we have $|h_{\theta}(z)|\geq\delta_{0}$ on
$(z_{0},z_{*})\setminus\cup_{j=1}^{n-1}B_{z_{j}}(\epsilon_{0})$ for a fixed
small constant $0<\epsilon_{0}\ll 1$ and
$\delta_{0}=\delta_{0}(\epsilon_{0})>0$. Using (2.66) yields $|q(z)|>0$ for
$z\in(z_{0},z_{*})\setminus\cup_{j=1}^{n-1}B_{\epsilon_{0}}(z_{j})$ for $b$
small enough. Consider $z_{j}$ a zero of $h_{\theta}$, namely that
$h_{\theta}(z_{j})=0$. Since $h_{\theta}$ is a non-zero solution of a second
order differential equation, necessarily $|h_{\theta}^{\prime}(z_{j})|>0$. We
may assume that $h_{\theta}^{\prime}(z_{j})>0$, which infers that there are
$z_{j}^{-}\in(z_{j}-\epsilon_{0},z_{j})$ and
$z_{j}^{+}\in(z_{j},z_{j}+\epsilon_{0})$ such that $h(z_{j}^{-})<0$ and
$h(z_{j}^{+})>0$. We then use (2.66) and the intermediate value Theorem to
conclude that there is $\tilde{z}_{j}\in(z_{j}^{-},z_{j}^{+})$ for which
$q(\tilde{z}_{j})=0$. We also note that $|h^{\prime}_{\theta}(z_{j})|\geq
c_{1}>0$ for $z\in B_{\epsilon_{0}}(z_{j})$, from which and (2.66), we deduce
that $|q^{\prime}(z)|\neq 0$ for $z\in B_{\epsilon_{0}}(z_{j})$. Hence,
$\tilde{z}_{j}$ is the only zero of $q(z)$ in $B_{\epsilon_{0}}(z_{j})$. This
concludes the proof of Lemma 2.6. ∎

###### Proof of Lemma 2.7.

The decomposition (2.67) and the associated bounds still hold for
$P_{0}+\frac{1}{2}\partial_{z}(\tilde{V}\cdot)/z$. This was the only
information used on $P_{0}$ in the proof of Lemma 2.6, so the very same proof
applies. ∎

### 2.3. Conclusion via matched asymptotic expansions, proof of Proposition
1.5

From Lemmas 2.3 and 2.6, we are now able to derive the full solution to the
eigenproblem (2.1). In particular we claim the following.

######  Lemma 2.8 (Matched eigenfunction for the radial mode).

Fix $n\in\mathbb{N}$. Then there exists $C>0$, such that for $\zeta_{0}$ small
enough, there exists $0<b^{*}\ll 1$ such that for all $0<b\leq b^{*}$, there
exists $|\bar{\alpha}_{n}|\leq C|\ln b|^{-2}$ such that the following holds
for the function

$\phi_{n}(r):=\left\\{\begin{array}[]{ll}\;\;\;\;\phi_{n}^{\textup{in}}(r)&\textup{for}\;\;r\leq
R_{0},\\\ \beta_{0}\phi_{n}^{\textup{ex}}(r)&\textup{for}\;\;r\geq
R_{0},\end{array}\right.\quad\beta_{0}=\frac{\phi_{n}^{\textup{in}}(R_{0})}{\phi_{n}^{\textup{ex}}(R_{0})},\quad
R_{0}=\frac{\zeta_{0}}{\sqrt{b}},$ (2.69)

where $\phi_{n}^{\textup{in}}=\phi_{n}^{\textup{in}}[b,\bar{\alpha}]$ and
$\phi_{n}^{\textup{ex}}(r)=\phi_{n}^{\textup{ex}}[b,\bar{\alpha}](r)=q[b,\bar{\alpha}]\left(\frac{br^{2}}{2}\right)=q\left(z\right)$
are described in Lemmas 2.3 and 2.6 respectively.

  * (i)

The function $\phi_{n}$ is a smooth solution to the equation

$\big{(}\mathscr{A}_{0}-br\partial_{r}\big{)}\phi_{n}=2b\big{(}1-n+\frac{1}{\ln
b}+\bar{\alpha}_{n}\big{)}\phi_{n}.$ (2.70)

  * (ii)

The estimates (2.3) and (2.4) for $\alpha_{n}$ hold true. The estimate (2.6)
for $\phi_{n}$ holds true.

######  Corollary 2.9.

For the perturbed operator
$\mathscr{A}_{0}\phi_{n}-br\partial_{r}\phi_{n}+r^{-1}\partial_{r}(V\cdot)$
where $V$ satisfies $|\partial_{r}^{k}V|\lesssim|\ln b|^{-1}r^{2-k}\langle
r\rangle^{-4}$ for $k=0,1$, then item (i) of Lemma 2.8 holds true if the inner
and outer eigenfunctions are those associated to the perturbed problems
described by Lemma 2.4 and 2.7 respectively.

###### Proof of Lemma 2.8.

Recall from (2.56) the relation

$\theta=1-n+\tilde{\alpha},\quad\tilde{\alpha}=\frac{1}{\ln b}+\bar{\alpha}.$
(2.71)

Since the equation (2.70) is a second order ODE with smooth coefficients
outside the origin, it suffices to prove that the two functions and their
first order derivatives agree on both sides of $R_{0}$, and (2.69) will then
provide a global solution to (2.70) on $(0,\infty)$. From the special choice
of $\beta_{0}$ this is equivalent to:

$\frac{\partial_{r}\phi_{n}^{\textup{in}}(R_{0})}{\partial_{r}\phi_{n}^{\textup{ex}}(R_{0})}=\beta_{0}\Longleftrightarrow\Theta(b,\bar{\alpha})=\frac{(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0})}{2\phi_{n}^{\textup{in}}(R_{0})}-\frac{(z\partial_{z})q(z_{0})}{q(z_{0})}=0.$
(2.72)

Therefore, to prove the Lemma it suffices to prove that for $b$ small enough
there exists $\bar{\alpha}=\bar{\alpha}_{n}(b)$ such that
$\Theta(b,\bar{\alpha})=0$, and such that item (ii) holds true. We rely in a
standard way on implicit function theorem. The estimate for
$\partial_{b}\bar{\alpha}_{n}$ then follows by

$\partial_{b}\bar{\alpha}_{n}=-\frac{(\partial_{b}\Theta)(b,\bar{\alpha}_{n})}{(\partial_{\bar{\alpha}_{n}}\Theta)(b,\bar{\alpha}_{n})}.$
(2.73)

To ease the writing, we mention only the dependence in $b$ and $\bar{\alpha}$
at key locations in what follows.

Step 1 _The interior term:_ It is convenient to rewrite from (2.24) the
expression of $\phi_{n}^{\textup{in}}$ as

$\phi^{\textup{in}}_{n}[b,\bar{\alpha}](r)=F_{n}[b](r)+\bar{\alpha}bG_{n}[b,\bar{\alpha}](r)+E_{n}[b,\bar{\alpha}](r),$
(2.74)

where $F_{n}$ and $\bar{\alpha}G_{n}$ are leading order terms and $E_{n}$ is a
remainder:

$\displaystyle F_{n}[b](r)=\sum_{j=0}^{n}c_{n,j}b^{j}T_{j}(r),\quad
G_{n}[b,\bar{\alpha}]=\sum_{j=0}^{n}b^{j}\Big{(}-c_{n,j}T_{j+1}(r)+S_{j}[b,\bar{\alpha}](r)\Big{)},$
(2.75) $E_{n}[b,\bar{\alpha}](r)=b\left(-\frac{2}{\ln
b}T_{1}(r)+\mathscr{A}^{-1}\Theta_{0}(r)\right)+b\mathcal{R}_{n}[b,\bar{\alpha}](r).$

We have the following estimates from (2.16), (2.36), (2.27), and assuming
$|\bar{\alpha}|\lesssim|\ln b|^{-2}$:

$\sum_{0\leq k\leq 2,\ 0\leq\ell+\ell^{\prime}\leq
1}^{2}|((r\partial_{r})^{k}(b\partial_{b})^{\ell}\partial_{\bar{\alpha}}^{\ell^{\prime}}E_{n})(R_{0})|\leq
C(\zeta_{0})\frac{b}{|\ln b|},\\\ $ (2.76) $\displaystyle
F_{n}(R_{0})=b\left(-\frac{\ln
b}{2}H_{n}(\zeta_{0})+K_{n}(\zeta_{0})\right)+\mathcal{O}(b^{\frac{3}{2}}),$
(2.77) $\displaystyle(r\partial_{r}F_{n}(R_{0}))(R_{0})=b\left(-\frac{\ln
b}{2}\zeta\partial_{\zeta}H_{n}(\zeta_{0})+\zeta\partial_{\zeta}K_{n}(\zeta_{0})\right)+\mathcal{O}(b^{\frac{3}{2}}),$

where $H_{n}$ and $G_{n}$ are defined by:

$H_{n}(\zeta_{0})=\sum_{i=1}^{n}c_{n,i}\hat{d}_{i}\zeta_{0}^{2(i-1)},\ \
K_{n}(\zeta_{0})=\frac{1}{\zeta_{0}^{2}}+\sum_{i=1}^{n}c_{n,i}\zeta_{0}^{2(i-1)}\left(\hat{d}_{i}\ln\zeta_{0}+d_{i}\right).$
(2.78)

Notice for $0<\zeta_{0}\ll 1$ small that $|H_{n}(\zeta_{0})|\neq 0$. Gathering
all these estimates and (2.26) we arrive at

$\phi^{\textup{in}}_{n}(R_{0})=b\left(-\frac{\ln
b}{2}H_{n}(\zeta_{0})+K_{n}(\zeta_{0})+\bar{\alpha}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|})\right),$ (2.79) $r\partial
r\phi^{\textup{in}}_{n}(R_{0})=b\left(-\frac{\ln
b}{2}\zeta\partial_{\zeta}H_{n}(\zeta_{0})+\zeta\partial_{\zeta}K_{n}(\zeta_{0})+\bar{\alpha}r\partial_{r}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|})\right),$ $\displaystyle\partial_{b}\left(\frac{1}{b\ln
b}\phi^{\textup{in}}_{n}(R_{0})\right)$ $\displaystyle=-\frac{1}{b|\ln
b|^{2}}(\zeta\partial_{\zeta}K_{n}(\zeta_{0})+\bar{\alpha}r\partial_{r}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|}))$ $\displaystyle\quad+\frac{1}{b\ln
b}\left(\bar{\alpha}\partial_{b}G_{n}(R_{0})+\partial_{b}E(R_{0})\right)=\mathcal{O}\left(\frac{1}{b|\ln
b|^{2}}\right),$ $\displaystyle\partial_{b}\left(\frac{1}{b\ln b}r\partial
r\phi^{\textup{in}}_{n}(R_{0})\right)$ $\displaystyle=-\frac{1}{b|\ln
b|^{2}}(\zeta\partial_{\zeta}K_{n}(\zeta_{0})+\bar{\alpha}r\partial_{r}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|}))$ $\displaystyle\quad+\frac{1}{b\ln
b}\left(\bar{\alpha}\partial_{b}r\partial_{r}G_{n}(R_{0})+\partial_{b}r\partial_{r}E(R_{0})\right)=\mathcal{O}\left(\frac{1}{b|\ln
b|^{2}}\right),$
$\displaystyle\partial_{\bar{\alpha}}(\phi^{\textup{in}}_{n}(R_{0}))$
$\displaystyle=bG_{n}(R_{0})+\bar{\alpha}b\partial_{\bar{\alpha}}G_{n}(R_{0})+b\partial_{\bar{\alpha}}E_{n}(R_{0})$
$\displaystyle=bG_{n}(R_{0})+b\mathcal{O}(|\ln b|^{-2})\mathcal{O}(\ln
b)+b\mathcal{O}(|\ln b|^{-1})=b\left(G_{n}(R_{0})+\mathcal{O}(|\ln
b|^{-1})\right),$
$\displaystyle\partial_{\bar{\alpha}}(r\partial_{r}\phi^{\textup{in}}_{n}(R_{0}))$
$\displaystyle=br\partial_{r}G_{n}(R_{0})+\bar{\alpha}b\partial_{\bar{\alpha}}r\partial_{r}G_{n}(R_{0})+b\partial_{\bar{\alpha}}r\partial_{r}E_{n}(R_{0})$
$\displaystyle=br\partial_{r}G_{n}(R_{0})+b\mathcal{O}(|\ln
b|^{-2})\mathcal{O}(\ln b)+b\mathcal{O}(|\ln
b|^{-1})=b\left(r\partial_{r}G_{n}(R_{0})+\mathcal{O}(|\ln b|^{-1})\right).$

We compute that, from (2.26):

$|r\partial_{r}G_{n}(R_{0})|+|G_{n}(R_{0})|\leq C(n)|\ln b|,\quad\mbox{with
}C(n)\mbox{ independent of }\zeta_{0}.$

The collection of the above identities gives us the following leading order
expression for the quantity involving the inner solution in (2.72):

$\displaystyle\frac{(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0})}{\phi_{n}^{\textup{in}}(R_{0})}=\frac{-\frac{\ln
b}{2}\zeta\partial_{\zeta}H_{n}(\zeta_{0})+\zeta\partial_{\zeta}K_{n}(\zeta_{0})+\bar{\alpha}r\partial_{r}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|})}{-\frac{\ln
b}{2}H_{n}(\zeta_{0})+K_{n}(\zeta_{0})+\bar{\alpha}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|})}$ $\displaystyle=\frac{\zeta\partial_{\zeta}H_{n}(\zeta_{0})-\frac{2}{\ln
b}\zeta\partial_{\zeta}K_{n}(\zeta_{0})-\frac{2}{\ln
b}\bar{\alpha}r\partial_{r}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|^{2}})}{H_{n}(\zeta_{0})-\frac{2}{\ln b}K_{n}(\zeta_{0})-\frac{2}{\ln
b}\bar{\alpha}G_{n}(R_{0})+\mathcal{O}(\frac{1}{|\ln b|^{2}})}$
$\displaystyle=\frac{\zeta\partial_{\zeta}H_{n}(\zeta_{0})}{H_{n}(\zeta_{0})}+\frac{2}{\ln
b}\frac{K_{n}(\zeta_{0})\zeta\partial_{\zeta}H_{n}(\zeta_{0})-H_{n}(\zeta_{0})\zeta\partial_{\zeta}K_{n}(\zeta_{0})}{H_{n}^{2}(\zeta_{0})}$
$\displaystyle\qquad+\frac{2}{\ln
b}\bar{\alpha}\frac{G_{n}\zeta\partial_{\zeta}H_{n}(\zeta_{0})-H_{n}(\zeta_{0})\zeta\partial_{\zeta}G_{n}}{H_{n}(\zeta_{0})^{2}}+\mathcal{O}(|\ln
b|^{-2})$
$\displaystyle=\frac{\zeta\partial_{\zeta}H_{n}(\zeta_{0})}{H_{n}(\zeta_{0})}+\frac{2}{\ln
b}\frac{K_{n}(\zeta_{0})\zeta\partial_{\zeta}H_{n}(\zeta_{0})-H_{n}(\zeta_{0})\zeta\partial_{\zeta}K_{n}(\zeta_{0})}{H_{n}^{2}(\zeta_{0})}+\bar{\alpha}\frac{\mathcal{O}(1)}{H_{n}(\zeta_{0})^{2}}+\mathcal{O}(|\ln
b|^{-2}),$ (2.80)

and

$\displaystyle\partial_{b}\left(\frac{(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0})}{\phi_{n}^{\textup{in}}(R_{0})}\right)=\partial_{b}\left(\frac{(b\ln
b)^{-1}(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0})}{(b\ln
b)^{-1}\phi_{n}^{\textup{in}}(R_{0})}\right)$
$\displaystyle=\frac{\partial_{b}((b\ln
b)^{-1}(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0}))(b\ln
b)^{-1}\phi_{n}^{\textup{in}}(R_{0})-\partial_{b}(b\ln
b)^{-1}\phi_{n}^{\textup{in}}(R_{0})b\ln
b)^{-1}(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0})}{((b\ln
b)^{-1}\phi_{n}^{\textup{in}}(R_{0}))^{2}}$
$\displaystyle=\frac{\mathcal{O}(b^{-1}|\ln b|^{-2})}{((b\ln
b)^{-1}\phi_{n}^{\textup{in}}(R_{0}))^{2}}=\mathcal{O}\left(\frac{1}{b|\ln
b|^{2}}\right),$ (2.81)

and

$\displaystyle\partial_{\bar{\alpha}}\left(\frac{(r\partial_{r})\phi_{n}^{\textup{in}}(R_{0})}{\phi_{n}^{\textup{in}}(R_{0})}\right)=\frac{\partial_{\bar{\alpha}}r\partial_{r}\phi_{n}^{\textup{in}}(R_{0})\phi_{n}^{\textup{in}}(R_{0})-\partial_{\bar{\alpha}}\phi_{n}^{\textup{in}}(R_{0})\partial_{r}\phi_{n}^{\textup{in}}(R_{0})}{|\phi_{n}^{\textup{in}}(R_{0})|^{2}}$
$\displaystyle=\frac{\left(\zeta\partial_{\zeta}G_{n}(R_{0})+\mathcal{O}(|\ln
b|^{-1})\right)\left(-\frac{\ln
b}{2}H_{n}(\zeta_{0})+\mathcal{O}(1)\right)-\left(G_{n}(R_{0})+\mathcal{O}(|\ln
b|^{-1})\right)\left(-\frac{\ln
b}{2}\zeta\partial_{\zeta}H_{n}(\zeta_{0})+\mathcal{O}(1)\right)}{\left(-\frac{\ln
b}{2}H_{n}(\zeta_{0})+\mathcal{O}(1)\right)^{2}}$ $\displaystyle=\frac{2}{\ln
b}\frac{G_{n}\zeta\partial_{\zeta}H_{n}(\zeta_{0})-\zeta\partial_{\zeta}G_{n}H_{n}(\zeta_{0})}{H_{n}^{2}(\zeta_{0})}=\mathcal{O}(1)$
(2.82)

where the constant in the two $\mathcal{O}(1)$ above are independent of
$\zeta_{0}$.

The case $n=1$: Injecting $\bar{\alpha}=e_{1}/|\log b|^{2}+\hat{\alpha}$,
$|\hat{\alpha}|\lesssim|\ln b|^{-3}$ in the refined asymptotics (2.30) and
(2.31) gives

$\phi^{\textup{in}}_{1}(r)=F_{1}(r)+\hat{\alpha}bG_{1}(r)+E_{1}(r),$

where

$\displaystyle F_{1}(r)$
$\displaystyle=T_{0}(r)+2bT_{1}(r)+b\left(-\frac{2}{\ln
b}T_{1}(r)+\mathscr{A}_{0}^{-1}\Theta_{0}\right)$
$\displaystyle\quad+\frac{2e_{1}}{|\ln
b|^{2}}(-bT_{1}(r)-2b^{2}T_{2}(r)-\frac{b^{2}}{2}\sum_{i=2}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i-1}r^{2i}\ln(r+1))$
$\displaystyle\quad-\frac{b}{2}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}b^{i}r^{2i}\left\\{\frac{1}{\ln
b}\left[2\ln(r+1)-\frac{1}{i}-\Psi(i+2)-\gamma\right]+1-\frac{1}{\ln
b}\right\\},$ $G_{1}(r)=2(-T_{1}(r)+S_{0}(r)-2bT_{2}+bS_{1}(r)),\quad
E_{1}(r)=b\tilde{\mathcal{R}}_{1}(r)+\frac{2e_{1}}{|\ln
b|^{2}}(bS_{0}(r)+b^{2}\tilde{S}_{1}(r)).$

One has from (2.16), as $\hat{d}_{1}=-1/2$, $d_{1}=1/4$ and
$\hat{d}_{2}=1/16$, $e_{1}=\ln 2-\gamma-1$ and $R_{0}=\zeta_{0}/\sqrt{b}$:

$\displaystyle F_{1}(R_{0})$
$\displaystyle=\frac{b}{\zeta_{0}^{2}}+2b\left(-\frac{\ln\zeta_{0}-\frac{\ln
b}{2}}{2}+\frac{1}{4}\right)+b\left(-\frac{2}{\ln
b}\left(-\frac{\ln\zeta_{0}-\frac{\ln b}{2}}{2}\right)+\frac{1}{2}\right)$
$\displaystyle+\frac{2e_{1}}{|\ln b|^{2}}\left(-\frac{b\ln
b}{4}+\frac{b\zeta_{0}^{2}\ln b}{16}+\frac{b\ln
b}{4}\sum_{i=2}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}\right)$
$\displaystyle-\frac{b}{2}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}\left\\{\frac{1}{\ln
b}\left[2\ln\zeta_{0}-\ln b-\frac{1}{i}-\Psi(i+2)-\gamma\right]+1-\frac{1}{\ln
b}\right\\}+\mathcal{O}\left(\frac{b}{|\ln b|^{2}}\right)$ $\displaystyle
F_{1}(R_{0})$ $\displaystyle=b\Bigr{[}\frac{\ln
b}{2}+\frac{1}{\zeta_{0}^{2}}-\ln\zeta_{0}+\frac{1}{2}+\frac{\ln\zeta_{0}}{\ln
b}+\frac{e_{1}}{2\ln
b}\left(-1+\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}\right)$
$\displaystyle-\frac{1}{2\ln
b}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}\left[2\ln\zeta_{0}-\frac{1}{i}-\Psi(i+2)-\gamma-1\right]\Bigr{]}$
$\displaystyle=b\left\\{\frac{\ln
b}{2}+\frac{1}{\zeta_{0}^{2}}-\ln\zeta_{0}+\frac{1}{2}+\frac{\ln\zeta_{0}}{\ln
b}-\frac{e_{1}}{2\ln b}-\frac{1}{2\ln
b}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}\left[2\ln\zeta_{0}-\ln
2-\frac{1}{i}-\Psi(i+2)\right]\right\\}$
$\displaystyle+\mathcal{O}\left(\frac{b}{|\ln b|^{2}}\right),$ (2.83)

and similarly, we have

$(r\partial_{r}F_{1})(R_{0})=\frac{-2b}{\zeta_{0}^{2}}-b+\frac{b}{\ln
b}-\frac{b}{2\ln
b}\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}2i\left[2\ln\zeta_{0}-\Psi(i+2)-\ln
2\right]+\mathcal{O}\left(\frac{b}{|\ln b|^{2}}\right).$

From (2.30) and (2.31), we obtain

$\sum_{0\leq k\leq 2}((r\partial_{r})^{k}E_{1})(R_{0})|\leq
C(\zeta_{0})\frac{b}{|\ln b|^{2}},$

Hence, as $G_{1}(R_{0})=\mathcal{O}(|\ln b|)$ and
$r\partial_{r}G_{1}(R_{0})=\mathcal{O}(|\ln b|)$, we obtain from the above
identities

$\displaystyle\phi_{1}^{\textup{in}}(R_{0})=b\Bigr{[}-\frac{\ln
b}{2}H_{1}(\zeta_{0})+K_{1}(\zeta_{0})+\frac{1}{2\ln
b}J_{1}(\zeta_{0})+\hat{\alpha}bG_{1}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|^{2}})\Bigr{]},$ $\displaystyle
r\partial_{r}\phi_{1}^{\textup{in}}(R_{0})=b\left[\zeta\partial_{\zeta}K_{1}+\frac{1}{2\ln
b}\zeta\partial_{\zeta}J_{1}+\hat{\alpha}r\partial_{r}G_{1}(R_{0})+\mathcal{O}\left(\frac{1}{|\ln
b|^{2}}\right)\right].$

where we used (2.78), so that $H_{1}(\zeta)=1$ and
$K_{1}(\zeta)=\zeta^{-2}-\ln\zeta+1/2$ and

$J_{1}(\zeta_{0})=2\ln\zeta_{0}-e_{1}-\sum_{i=1}^{\infty}\frac{(1)_{i-1}}{(2)_{i}i!2^{i}}\zeta_{0}^{2i}\left[2\ln\zeta_{0}-\ln
2-\frac{1}{i}-\Psi(i+2)\right].$ (2.84)

We finally obtain

$\displaystyle\frac{r\partial_{r}\phi^{\textup{in}}_{1}(R_{0})}{\phi^{\textup{in}}_{1}(R_{0})}$
$\displaystyle=\frac{\zeta\partial_{\zeta}K_{1}+\frac{1}{2\ln
b}\zeta\partial_{\zeta}J_{1}+\hat{\alpha}r\partial_{r}G_{1}(R_{0})+\mathcal{O}\left(\frac{1}{|\ln
b|^{2}}\right)}{-\frac{\ln
b}{2}H_{1}(\zeta_{0})+K_{1}(\zeta_{0})+\frac{1}{2\ln
b}J_{1}(\zeta_{0})+\hat{\alpha}bG_{1}(R_{0})+\mathcal{O}(\frac{1}{|\ln
b|^{2}})}$ $\displaystyle=-\frac{2}{\ln
b}\frac{\zeta\partial_{\zeta}K_{1}+\frac{1}{2\ln
b}\zeta\partial_{\zeta}J_{1}+\hat{\alpha}r\partial_{r}G_{1}(R_{0})+O\left(\frac{1}{|\ln
b|^{2}}\right)}{H_{1}(\zeta_{0})-\frac{2}{\ln b}K_{1}(\zeta_{0})-\frac{1}{|\ln
b|^{2}}J_{1}(\zeta_{0})-\frac{2}{\ln
b}\hat{\alpha}bG_{1}(R_{0})+\mathcal{O}(\frac{1}{|\ln b|^{3}})}$
$\displaystyle=-\frac{2}{\ln
b}\left\\{\frac{\zeta\partial_{\zeta}K_{1}}{H_{1}}+\frac{1}{\ln
b}\frac{\zeta\partial_{\zeta}J_{1}H_{1}+2K_{1}\zeta\partial_{\zeta}K_{1}}{H_{1}^{2}}\right.$
$\displaystyle\qquad\quad\left.+\tilde{\alpha}\frac{r\partial_{r}G_{1}H_{1}+\frac{2}{\ln
b}G_{1}\zeta\partial_{\zeta}K_{1}}{H_{1}^{2}}+\mathcal{O}(|\ln
b|^{-2})\right\\}$ $\displaystyle=-\frac{2}{\ln
b}\frac{\zeta\partial_{\zeta}K_{1}}{H_{1}}-\frac{2}{\ln
b^{2}}\frac{\zeta\partial_{\zeta}J_{1}H_{1}+2K_{1}\zeta\partial_{\zeta}K_{1}}{H_{1}^{2}}+\tilde{\alpha}\frac{\mathcal{O}(1)}{H_{1}^{2}}+\mathcal{O}(|\ln
b|^{-3})$ (2.85)

where the constant in the $\mathcal{O}(1)$ is independent of $\zeta_{0}$.

The case $n=0$: We first use the refined asymptotics (2.29) and (2.28) to
obtain:

$\phi^{\textup{in}}_{0}(r)=F_{0}(r)+\bar{\alpha}bG_{0}(r)+E_{0}(r),$

where:

$\displaystyle F_{0}(r)$ $\displaystyle=T_{0}(r)+b\left(-\frac{2}{\ln
b}T_{1}(r)+\mathscr{A}_{0}^{-1}\Theta_{0}\right)+\frac{b}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i}\left\\{\frac{1}{\ln
b}\left[2\ln(r+1)-\Psi(i+2)-\gamma\right]+1\right\\},$
$G_{0}(r)=2\left(-T_{1}(r)+\frac{1}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}b^{i}r^{2i}\ln(r+1)\right),\quad
E_{0}(r)=b\tilde{\mathcal{R}}_{0}(r)+2\bar{\alpha}b\tilde{S}_{0}.$

One has from (2.16), as $\hat{d}_{1}=-1/2$, $d_{1}=1/4$:

$\displaystyle F_{0}(R_{0})$
$\displaystyle=\frac{b}{\zeta_{0}^{2}}+b\left(-\frac{2}{\ln
b}\left(-\frac{\ln\zeta_{0}-\frac{\ln
b}{2}}{2}+\frac{1}{4}\right)+\frac{1}{2}\right)$
$\displaystyle\quad+\frac{b}{2}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}\zeta_{0}^{2i}\left\\{\frac{1}{\ln
b}\left[2\ln\zeta_{0}-\Psi(i+2)-\gamma\right]\right\\}+\mathcal{O}(b^{\frac{3}{2}})$
$\displaystyle=\frac{b}{\zeta_{0}^{2}}+\frac{b\ln\zeta_{0}}{\ln
b}-\frac{b}{2\ln b}+\frac{b}{2\ln
b}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}\zeta_{0}^{2i}\left\\{2\ln\zeta_{0}-\Psi(i+2)-\gamma\right\\}+\mathcal{O}(b^{\frac{3}{2}}),$

and similarly, we have

$(r\partial_{r}F_{0})(R_{0})=\frac{-2b}{\zeta_{0}^{2}}+\frac{b}{\ln
b}+\frac{b}{2\ln
b}\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}\zeta_{0}^{2i}2i\left[2\ln\zeta_{0}+\frac{1}{i}-\Psi(i+2)-\gamma\right]+\mathcal{O}\left(b^{\frac{3}{2}}\right)$
$\partial_{b}(b^{-1}F_{0}(R_{0}))=O\left(\frac{1}{b|\ln b|^{2}}\right),\ \
\partial_{b}(b^{-1}r\partial_{r}F_{0}(R_{0}))=\mathcal{O}\left(\frac{1}{b|\ln
b|^{2}}\right).$

From (2.29), we obtain

$\displaystyle\sum_{0\leq k\leq 2,\ 0\leq\ell+\ell^{\prime}\leq
1}((b\partial_{b})^{\ell}\partial_{\bar{\alpha}}^{\ell^{\prime}}(r\partial_{r})^{k}E_{0})(R_{0})|\leq
C(\zeta_{0})\frac{b}{|\ln b|^{2}}.$

One also has

$G_{0}(R_{0})=2\left(-\frac{1}{4}\ln b-\frac{1}{4}\ln
b\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}\zeta_{0}^{2i}\right)+\mathcal{O}(1)=-\frac{\ln
b}{2}\tilde{G}_{0}(\zeta_{0})+\mathcal{O}(1),\quad\partial_{b}G_{0}(R_{0})=\mathcal{O}\left(\frac{1}{b}\right),$

where

$\tilde{G}_{0}(\zeta_{0})=\sum_{i=0}^{\infty}\frac{1}{(2)_{i}2^{i}}\zeta_{0}^{2i},$
(2.86)

so that

$r\partial_{r}G_{0}(R_{0})=-\frac{\ln
b}{2}\zeta\partial_{\zeta}\tilde{G}_{0}(\zeta_{0})+\mathcal{O}(1),\quad\partial_{b}r\partial_{r}\tilde{G}_{0}(R_{0})=\mathcal{O}\left(\frac{1}{b}\right).$

We obtain from the above identities

$\displaystyle\phi_{0}^{\textup{in}}(R_{0})$
$\displaystyle=b\Bigr{[}\frac{1}{\zeta_{0}^{2}}+\frac{1}{2\ln
b}J_{0}(\zeta_{0})-\frac{\ln
b}{2}\bar{\alpha}\tilde{G}_{0}(\zeta_{0})+\mathcal{O}(|\ln b|^{-2})\Bigr{]},$
$\displaystyle r\partial_{r}\phi_{0}^{\textup{in}}(R_{0})$
$\displaystyle=b\Bigr{[}\frac{-2}{\zeta_{0}^{2}}+\frac{1}{\ln
b}\zeta\partial_{\zeta}J_{0}(\zeta_{0})-\frac{\ln
b}{2}\bar{\alpha}r\partial_{r}\tilde{G}_{0}(\zeta_{0})+\mathcal{O}(|\ln
b|^{-2})\Bigr{]},$

where

$J_{0}(\zeta_{0})=2\ln\zeta_{0}-1+\sum_{i=1}^{\infty}\frac{1}{(2)_{i}2^{i}}\zeta_{0}^{2i}\left[2\ln\zeta_{0}-\Psi(i+2)-\gamma\right],$
(2.87)

and for $\bar{\alpha}=\mathcal{O}(|\ln b|^{-2})$,

$\partial_{b}\left(b^{-1}\phi_{0}^{\textup{in}}(R_{0})\right)=O\left(\frac{1}{b|\ln
b|^{2}}\right),\quad\partial_{b}\left(b^{-1}r\partial_{r}\phi_{0}^{\textup{in}}(R_{0})\right)=\mathcal{O}\left(\frac{1}{b|\ln
b|^{2}}\right),$
$\partial_{\bar{\alpha}}\left(\phi_{0}^{\textup{in}}(R_{0})\right)=-\frac{b\ln
b}{2}\tilde{G}_{0}(\zeta_{0})+\mathcal{O}\left(b\right),\quad\partial_{\bar{\alpha}}\left(r\partial_{r}\phi_{0}^{\textup{in}}(R_{0})\right)=-\frac{b\ln
b}{2}r\partial_{r}\tilde{G}_{0}(\zeta_{0})+\mathcal{O}\left(b\right)$

We finally obtain

$\displaystyle\frac{r\partial_{r}\phi^{\textup{in}}_{0}(R_{0})}{\phi^{\textup{in}}_{0}(R_{0})}=\frac{\frac{-2}{\zeta_{0}^{2}}+\frac{1}{2\ln
b}\zeta\partial_{\zeta}J_{0}(\zeta_{0})+\bar{\alpha}r\partial_{r}G_{0}(R_{0})+\mathcal{O}(|\ln
b|^{-2})}{\frac{1}{\zeta_{0}^{2}}+\frac{1}{2\ln
b}J_{0}(\zeta_{0})+\bar{\alpha}G_{0}(R_{0})+\mathcal{O}(|\ln b|^{-2})}$
$\displaystyle\quad=\frac{-2+\frac{\zeta_{0}^{2}}{\ln
b}\zeta\partial_{\zeta}J_{0}(\zeta_{0})+\zeta_{0}^{2}\bar{\alpha}r\partial_{r}G_{0}(R_{0})+\mathcal{O}(|\ln
b|^{-2})}{1+\frac{\zeta_{0}^{2}}{2\ln
b}J_{0}(\zeta_{0})+\bar{\alpha}\zeta_{0}^{2}G_{0}(R_{0})+\mathcal{O}(|\ln
b|^{-2})}$ $\displaystyle\quad=-2+\frac{1}{\ln
b}\zeta_{0}^{2}(\frac{1}{2}\zeta\partial_{\zeta}J_{0}+J_{0})-\frac{\ln
b}{2}\bar{\alpha}\zeta_{0}^{2}(\zeta\partial_{\zeta}\tilde{G}_{1}(\zeta_{0})+2\tilde{G}_{1}(\zeta_{0})+\mathcal{O}(|\ln
b|^{-1}))+\mathcal{O}(|\ln b|^{-2}),$ (2.88)

and

$\displaystyle\partial_{b}\left(\frac{r\partial_{r}\phi^{\textup{in}}_{0}(R_{0})}{\phi^{\textup{in}}_{0}(R_{0})}\right)=\partial_{b}\left(\frac{b^{-1}r\partial_{r}\phi^{\textup{in}}_{0}(R_{0})}{b^{-1}\phi^{\textup{in}}_{0}(R_{0})}\right)$
$\displaystyle\quad=\frac{\partial_{b}(b^{-1}r\partial_{r}\phi^{\textup{in}}_{0}(R_{0}))b^{-1}\phi^{\textup{in}}_{0}(R_{0})-\partial_{b}(b^{-1}\phi^{\textup{in}}_{0}(R_{0}))b^{-1}r\partial_{r}\phi^{\textup{in}}_{0}(R_{0})}{b^{-1}\phi^{\textup{in}}_{0}(R_{0})}=\mathcal{O}(\frac{1}{b|\ln
b|^{2}}),$ (2.89)

and

$\displaystyle\partial_{\bar{\alpha}}\left(\frac{r\partial_{r}\phi^{\textup{in}}_{0}(R_{0})}{\phi^{\textup{in}}_{0}(R_{0})}\right)$
$\displaystyle=-\frac{\ln
b}{2}\zeta_{0}^{2}(\zeta\partial_{\zeta}\tilde{G}_{1}(\zeta_{0})+2\tilde{G}_{1}(\zeta_{0})+\mathcal{O}(|\ln
b|^{-1})),$ (2.90)

where the constant in the $\mathcal{O}(|\ln b|^{-2})$ is independent of
$\bar{\alpha}$.

Step 2: _The exterior term_. Recall the decomposition
$q[b,\bar{\alpha}](z)=\Gamma(\theta)h_{\theta}(z)+\mathcal{G}[b,\bar{\alpha}](z)$
from (2.65). From the estimates (2.66) the second term is of lower order and
satisfies:

$\sum_{0\leq k+\ell\leq
1}|(b\partial_{b})^{k}\partial_{\bar{\alpha}}^{\ell}(\mathcal{G}(z_{0}))|+|(b\partial_{b})^{k}\partial_{\bar{\alpha}}^{\ell}(z\partial_{z}\mathcal{G}(z_{0}))|\lesssim
b^{\frac{1}{2}}.$ (2.91)

We now investigate the formula giving $h_{\theta}$. From the recurrence
relation of the Gamma function and the identity
$\partial_{\theta}(\theta)_{i}=(\theta)_{i}(\Psi(\theta+i)-\Psi(\theta))$:

$\displaystyle\Gamma(\theta)h_{\theta}(z)$ $\displaystyle=$
$\displaystyle\frac{1}{z}+(\theta-1)\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\Big{]},$ $\displaystyle
z\partial_{z}\Gamma(\theta)h_{\theta}(z)$ $\displaystyle=$
$\displaystyle-\frac{1}{z}+(\theta-1)\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}i\left(\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\right)+1\Big{]},$
$\displaystyle\partial_{\theta}\Gamma(\theta)h_{\theta}(z)$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}\big{(}\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\big{)}\big{(}1+(\theta-1)(\Psi(\theta+i)-\Psi(\theta))\big{)}$
$\displaystyle\hskip
284.52756pt+(\theta-1)\partial_{\theta}\Psi(\theta+i)\Big{]},$
$\displaystyle\partial_{\theta}z\partial_{z}\Gamma(\theta)h_{\theta}(z)$
$\displaystyle\quad=\sum_{i=0}^{\infty}\frac{(\theta)_{i}}{(2)_{i}i!}z^{i}\Big{[}i\big{\\{}\left(\ln
z+\Psi(\theta+i)-\Psi(1+i)-\Psi(2+i)\big{)}\big{(}1+(\theta-1)(\Psi(\theta+i)-\Psi(\theta))\big{\\}}\right)$
$\displaystyle\hskip
284.52756pt+i\left((\theta-1)\partial_{\theta}\Psi(\theta+i)\right)+1\Big{]}.$

We now decompose all above expressions into leading order and lower terms. We
first collect some estimates on the coefficients. Note that for $i\geq n$ one