# Large population limit of the spectrum of killed birth-and-death processes

J.-R. Chazottes${}^{\textup{(a)}}$, P. Collet${}^{\textup{(a)}}$, S.
Méléard${}^{\textup{(b)}}$
${}^{\textup{(a)}}$ CPHT, CNRS, Ecole polytechnique, IP Paris, Palaiseau,
France
${}^{\textup{(b)}}$ CMAP, CNRS, Ecole polytechnique, IP Paris, Palaiseau,
France

###### Abstract

We consider a general class of birth-and-death processes with state space
$\\{0,1,2,3,\ldots\\}$ which describes the size of a population going
eventually to extinction with probability one. We obtain the complete spectrum
of the generator of the process killed at $0$ in the large population limit,
that is, we scale the process by a parameter $K$, and take the limit
$K\to+\infty$. We assume that the differential equation
$\mathrm{d}x/\mathrm{d}t=b(x)-d(x)$ describing the infinite population limit
(in any finite-time interval) has a repulsive fixed point at $0$, and an
attractive fixed point $x_{*}>0$. We prove that, asymptotically, the spectrum
is the superposition of two spectra. One is the spectrum of the generator of
an Ornstein-Uhlenbeck process, which is
$n(b^{\prime}(x_{*})-d^{\prime}(x_{*}))$, $n\geq 0$. The other one is the
spectrum of a continuous-time binary branching process conditioned on non-
extinction, and is given by $n(d^{\prime}(0)-b^{\prime}(0))$, $n\geq 1$. A
major difficulty is that different scales and function spaces are involved. We
work at the level of the eigenfunctions that we split over different regions,
and study their asymptotic dependence on $K$ in each region. In particular, we
prove that the spectral gap goes to
$\min\big{\\{}b^{\prime}(0)-d^{\prime}(0),\,d^{\prime}(x_{*})-b^{\prime}(x_{*})\big{\\}}$.
This work complements a previous work of ours in which we studied the
approximation of the quasi-stationary distribution and of the mean time to
extinction.

Key-words: Ornstein-Uhlenbeck process, quantum harmonic oscillator, binary
branching process, Jacobi operators, Dirichlet form, spectral gap, Fréchet-
Kolmogorov-Riesz compactness criterion, discrete orthogonal
polynomials, quasi-eigenvectors, quasi-stationary distribution.

###### Contents

  1. 1 Introduction, heuristics, and main result
    1. 1.1 The context
    2. 1.2 Notations for basic function spaces
    3. 1.3 Heuristics
    4. 1.4 Main result
    5. 1.5 Consequences on relaxation times
    6. 1.6 Organization of the paper
  2. 2 Standing assumptions
  3. 3 Proof of Theorem 1.2
    1. 3.1 Some useful operators
    2. 3.2 Steps of the proof of Theorem 1.2
  4. 4 Properties of the eigenvectors
  5. 5 Proof of Theorems 3.5 and 3.7
    1. 5.1 Proof of Theorem 3.5
    2. 5.2 Proof of Theorem 3.7
  6. 6 Fréchet-Kolmogorov-Riesz compactness criterion and Dirichlet form
    1. 6.1 Fréchet-Kolmogorov-Riesz compactness criterion
    2. 6.2 Dirichlet form for the operator $\mathcal{L}_{{\scriptscriptstyle K}}$
  7. 7 Spectral theory of $\mathcal{M}_{0}$
  8. 8 Local maximum principle and consequences thereof
  9. A Quasi-eigenvalues and quasi-eigenvectors of self-adjoint operators

##  1 Introduction, heuristics, and main result

###  1.1 The context

We consider a class of birth-and-death processes $(X^{{\scriptscriptstyle
K}}_{t})_{t\geq 0}$ with state space $\mathds{Z}_{{\scriptscriptstyle\geq
0}}$111We denote by $\mathds{Z}_{{\scriptscriptstyle\geq 0}}$ the set of non-
negative integers, and by $\mathds{Z}_{{\scriptscriptstyle>0}}$ the set of
positive integers. which describes how the size of a single population evolves
according to birth and death rates of the form

$\lambda^{{\scriptscriptstyle(K)}}_{n}=K\,b\left(\frac{n}{K}\right)\quad\text{and}\quad\mu^{{\scriptscriptstyle(K)}}_{n}=K\,d\left(\frac{n}{K}\right)$
(1.1)

where $n\geq 1$, and $K\in\mathds{Z}_{{\scriptscriptstyle>0}}$ is a scaling
parameter, often called ‘carrying capacity’. We suppose that $b(0)=d(0)=0$,
implying that $0$ is an absorbing state for the process, modelling extinction,
and our assumptions are such that the probability to reach this state is equal
to one. The unique stationary distribution is the Dirac measure at $0$, so a
relevant distribution to look for is a quasi-stationary distribution. A
probability measure $\nu^{{\scriptscriptstyle(K)}}$ on the positive integers
is a quasi-stationary distribution if, for all $t>0$ and for all subsets
$A\subset\mathds{Z}_{{\scriptscriptstyle>0}}$, one has
${\mathbb{P}}_{\nu^{{\scriptscriptstyle(K)}}}(X^{{\scriptscriptstyle
K}}_{t}\in
A\,|\,T^{{\scriptscriptstyle(K)}}_{0}>t)=\nu^{{\scriptscriptstyle(K)}}(A)$,
where $T^{{\scriptscriptstyle(K)}}_{0}$ is the extinction time, that is, the
smallest $t>0$ such that $X^{{\scriptscriptstyle(K)}}_{t}=0$. In other words,
a quasi-stationary distribution plays the role of a stationary distribution
when conditioning upon non-extinction. We refer to [6, 9] for more
informations about quasi-stationary distributions.

When $K\to\infty$, the trajectories of the rescaled process
$(K^{-1}X^{{\scriptscriptstyle(K)}}_{t})_{t\geq 0}$ converge in probability,
in any fixed time-window, to the solutions of the differential equation

$\frac{\mathrm{d}x}{\mathrm{d}t}=b(x)-d(x)$ (1.2)

if the initial condition state is for instance of the form $\lfloor
Kx_{0}\rfloor$ for a given $x_{0}>0$. We assume that the functions $b$ and $d$
only vanish at $0$, and that

$d^{\prime}(0)-b^{\prime}(0)<0$

meaning that the fixed point $0$ is repulsive. We also assume that there is a
unique attractive fixed point $x_{*}>0$, that is

$b(x_{*})=d(x_{*})\quad\text{and}\quad b^{\prime}(x_{*})-d^{\prime}(x_{*})<0.$

(We will give the complete set of assumptions on the functions $b$ and $d$
later on.)

A famous example is the so-called logistic process for which $b(x)=\lambda x$,
$d(x)=x(\mu+x)$, where $\lambda$ and $\mu$ are positive real numbers. We
assume that $\lambda>\mu$ and we have $x_{*}=\lambda-\mu$.

In [5] we obtained the precise asymptotic behaviour of the first eigenvalue of
the generator $L_{{\scriptscriptstyle K}}$ of the process killed at $0$, and
also of the law of the extinction time starting from the quasi-stationary
distribution (among other results). Here we go further and obtain the complete
spectrum of the generator of the killed process, in the limit $K\to\infty$. In
particular, the knowledge of the spectral gap allows us to obtain the time of
relaxation for the process conditioned on non-extinction to obey the quasi-
stationary distribution.

###  1.2 Notations for basic function spaces

We denote by

  * •

$\mathscr{D}$ the space of $C^{\infty}$ $\mathds{C}$-valued functions with
compact support on ;

  * •

$c_{00}$ the space of $\mathds{C}$-valued sequences with finitely many nonzero
values;

  * •

$\ell^{2}$ the space of square-summable $\mathds{C}$-valued sequences equipped
with the standard scalar product.

  * •

$L^{2}$ the space of square-integrable $\mathds{C}$-valued functions with
respect to Lebesgue measure on .

We will define several operators on $c_{00}$ and will consider their closure
on $\ell^{2}$. For simplicity, we will use the same notation for an operator
and its closure. As we will see later, there is no ambiguity on the
extensions.

###  1.3 Heuristics

The fundamental object in this paper is the spectrum of the following operator
that we momentarily define on $c_{00}$:

$\big{(}L_{{\scriptscriptstyle
K}}v\big{)}(n)=\lambda^{{\scriptscriptstyle(K)}}_{n}\big{(}v(n+1)-v(n)\big{)}+\mu^{{\scriptscriptstyle(K)}}_{n}\big{(}v(n-1){\mathds{1}}_{\\{n>1\\}}-v(n)\big{)}$
(1.3)

for $n\in\mathds{Z}_{{\scriptscriptstyle>0}}$. The idea is to ‘localize’ this
operator either around $n=\lfloor Kx_{*}\rfloor$ or $n=1$, which corresponds
in the dynamical system to the fixed point $x_{*}$ or the fixed point $0$. A
natural idea would be to ‘cut’ the operators in order to differentiate these
two dynamics. However the main difficulty is that the two different pieces
involve different scales and different function spaces. Since we don’t know
how to cope with this problem at the level of operators, we work at the level
of the eigenfunctions that we will split on different regions, and study their
asymptotic dependence on $K$ in each region.

To have an idea of the different scales involved in the problem, let us first
study the asymptotic behaviour of the birth and death rates. By Taylor
expansion around $Kx_{*}$ we have

$\lambda^{{\scriptscriptstyle(K)}}_{n}=K\,b\big{(}x_{*}\big{)}+(n-Kx_{*})\,b^{\prime}\big{(}x_{*}\big{)}+\mathcal{O}\left(\frac{(n-Kx_{*})^{2}}{K}\right)$

and

$\mu^{{\scriptscriptstyle(K)}}_{n}=K\,d\big{(}x_{*}\big{)}+(n-Kx_{*})\,d^{\prime}\big{(}x_{*}\big{)}+\mathcal{O}\left(\frac{(n-Kx_{*})^{2}}{K}\right).$

Taking $v^{{\scriptscriptstyle(K)}}(n)=u\big{(}(n-Kx_{*})/\sqrt{K}\big{)}$
with $u\in\mathscr{D}$, we get

$\big{(}L_{{\scriptscriptstyle
K}}v^{{\scriptscriptstyle(K)}}\big{)}(n)=\mathcal{OU}_{*}u\left(\frac{n-Kx_{*}}{\sqrt{K}}\right)+\mathcal{O}\left(\frac{1}{\sqrt{K}}\right)$

where

$\mathcal{OU}_{*}f(x)=b(x_{*})f^{\prime\prime}(x)+\,\big{(}b^{\prime}(x_{*})-d^{\prime}(x_{*})\big{)}\,xf^{\prime}(x)$
(1.4)

is the generator of an Ornstein-Uhlenbeck process on which satisfies the
stochastic differential equation
$\mathrm{d}X_{t}=\big{(}b^{\prime}(x_{*})-d^{\prime}(x_{*})\big{)}X_{t}\mathrm{d}t+\sqrt{2b(x_{*})}\,\mathrm{d}B_{t}$,
where $(B_{t})_{t\geq 0}$ is a one-dimensional Brownian motion. It is well
known (see Remark 3.1) that the spectrum of $\mathcal{OU}_{*}$ is $-S_{1}$
where

$S_{1}=\big{(}d^{\prime}(x_{*})-b^{\prime}(x_{*})\big{)}\mathds{Z}_{{\scriptscriptstyle\geq
0}}$ (1.5)

in the space
$L^{2}\Big{(}\sqrt{\frac{d^{\prime}(x_{*})-b^{\prime}(x_{*})}{2\pi
b(x_{*})}}\,\mathrm{e}^{-\frac{(d^{\prime}(x_{*})-b^{\prime}(x_{*})}{2b(x_{*})}x^{2}}\mathrm{d}x\Big{)}$.
Now we look at $n$ near $1$. By Taylor expansion, we have

$\lambda^{{\scriptscriptstyle(K)}}_{n}=n\left(b^{\prime}(0)+\mathcal{O}\left(\frac{n}{K}\right)\right)\quad\text{and}\quad\mu^{{\scriptscriptstyle(K)}}_{n}=n\left(b^{\prime}(0)+\mathcal{O}\left(\frac{n}{K}\right)\right).$

If $v\in c_{00}$, then we get

$\left\|L_{{\scriptscriptstyle
K}}v-\mathcal{Q}_{0}v\right\|_{\ell^{2}}\leq\frac{\mathcal{O}(1)}{K}$

where

$\big{(}\mathcal{Q}_{0}v\big{)}(n)=b^{\prime}(0)\,n\,\big{(}v(n+1)-v(n)\big{)}+d^{\prime}(0)\,n\,\big{(}v(n-1)\,{\mathds{1}}_{\\{n>1\\}}-v(n)\big{)}$

which is the generator of a (continuous-time) binary branching process killed
at $0$. We shall prove later on that, in a weighted $\ell^{2}$ space defined
below, the spectrum of $\mathcal{Q}_{0}$ is $-S_{2}$ where

$S_{2}=\big{(}b^{\prime}(0)-d^{\prime}(0)\big{)}\mathds{Z}_{{\scriptscriptstyle>0}}.$
(1.6)

The previous observations suggest that the limit of the spectrum of the
generator of the birth-and-death process
$(X^{{\scriptscriptstyle(K)}}_{t})_{t\geq 0}$, in an appropriate space, is

$\big{(}d^{\prime}(0)-b^{\prime}(0)\big{)}\mathds{Z}_{{\scriptscriptstyle>0}}\bigcup\big{(}b^{\prime}(x_{*})-d^{\prime}(x_{*})\big{)}\mathds{Z}_{{\scriptscriptstyle\geq
0}}.$

Notice that all the elements of this set are negative and this is not a
disjoint union in general. The logistic model is an example illustrating this
since
$d^{\prime}(0)-b^{\prime}(0)=b^{\prime}(x_{*})-d^{\prime}(x_{*})=\mu-\lambda$,
so we will have asymptotic double eigenvalues in this case.

We will prove that the limit of the spectrum of $L_{{\scriptscriptstyle K}}$
is obtained from the explicit spectra of the above two operators. Notice that
one is differential operator and the other one is a finite-difference
operator. This is a reverse situation with respect to numerical analysis where
the spectrum of limiting differential operators are obtained from the
knowledge of the spectrum of finite-difference operators. See for instance
[4].

###  1.4 Main result

For each $K\in\mathds{Z}_{{\scriptscriptstyle>0}}$, the sequence of numbers

$\pi^{{\scriptscriptstyle(K)}}_{n}:=\frac{\lambda^{{\scriptscriptstyle(K)}}_{1}\cdots\lambda^{{\scriptscriptstyle(K)}}_{n-1}}{\mu^{{\scriptscriptstyle(K)}}_{1}\cdots\mu^{{\scriptscriptstyle(K)}}_{n}},\,n\geq
2\quad\text{and}\quad\pi^{{\scriptscriptstyle(K)}}_{1}:=\frac{1}{\mu^{{\scriptscriptstyle(K)}}_{1}}$

naturally shows up in the study of birth-and-death processes. We will give
below a set of assumptions on the functions $b$ and $d$, defining the
differential equation (1.2), ensuring that the process reaches $0$ in finite
time with probability one, that the mean-time to extinction is finite, and
that the quasi-stationary distribution exists and is unique.

Let $\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$ be the space of
$\mathds{C}$-valued sequences $(v_{n})_{n\geq 1}$ such that

$\sum_{n\geq 1}|v_{n}|^{2}\pi^{{\scriptscriptstyle(K)}}_{n}<\infty\,.$

This is a Hilbert space when endowed with the scalar product $\langle
v,w\rangle_{\pi^{{\scriptscriptstyle(K)}}}:=\sum_{n\geq
1}\bar{v}_{n}w_{n}\pi^{{\scriptscriptstyle(K)}}_{n}$.

We know from [5] that the operator $L_{{\scriptscriptstyle K}}$ is closable in
$\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$. The closure (which we denote by the
same symbol) is self-adjoint and has a compact resolvent, hence its spectrum
is discrete, composed of simple eigenvalues which are negative real numbers,
and the corresponding eigenvectors are orthogonal. We normalize these
eigenvectors and we can assume that they are real (since they are defined by a
second-order real recurrence relation whose solution is determined by choosing
the first element). We write

$L_{{\scriptscriptstyle
K}}\psi^{{\scriptscriptstyle(K)}}_{j}=-\rho^{{\scriptscriptstyle(K)}}_{j}\psi^{{\scriptscriptstyle(K)}}_{j}$
(1.7)

where we order the eigenvalues $-\rho^{{\scriptscriptstyle(K)}}_{j}$ in
decreasing order as $j$ increases. To emphasize that all operators considered
in this paper are negative, we have decided to write their eigenvalues under
the form $-\rho$, with $\rho>0$.

As shown in [5], the quasi-stationary distribution exists, is unique, and
given by

$\nu^{{\scriptscriptstyle(K)}}(\\{n\\})=\frac{\pi^{{\scriptscriptstyle(K)}}_{n}\psi^{{\scriptscriptstyle(K)}}_{0}(n)}{\langle\psi^{{\scriptscriptstyle(K)}}_{0},\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle(K)}}}},\;n\in\mathds{Z}_{{\scriptscriptstyle>0}}$
(1.8)

where $\mathbf{1}=(1,1,\ldots)$. Note that it also follows from Theorem 3.2
and Lemma 9.3 in [5] that there exists $D>1$ such that for all
$K\in\mathds{Z}_{{\scriptscriptstyle>0}}$, we have
$D^{-1}\leq\psi_{0}^{{\scriptscriptstyle(K)}}\leq D$. Therefore, the Hilbert
spaces $\ell^{2}(\nu^{{\scriptscriptstyle(K)}})$ and
$\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$ are isomorphic.

Let
$S^{{\scriptscriptstyle(K)}}_{t}f(n):={\mathds{E}}_{n}\Big{(}f(X^{{\scriptscriptstyle(K)}}_{t}){\mathds{1}}_{\\{T^{{\scriptscriptstyle(K)}}_{0}>t\\}}\Big{)}$
($t\geq 0$) be the semigroup of the killed process, where
$n\in\mathds{Z}_{{\scriptscriptstyle>0}}$ and $f\in\ell^{\infty}$. The
following result justifies that we look for the spectrum of
$L_{{\scriptscriptstyle K}}$ in $\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$ .

######  Proposition 1.1.

The semigroup $(S^{{\scriptscriptstyle(K)}}_{t})_{t\geq 0}$, defined on
$\ell^{\infty}$, extends to a $C_{0}$-contraction semigroup on
$\ell^{2}(\nu^{{\scriptscriptstyle(K)}})$.

We refer to [14] for definitions and properties of $C_{0}$-contraction
semigroups.

PROOF. We follow the argument of Proposition 8.1.8 p. 162 in [3]. Since
$\nu^{{\scriptscriptstyle(K)}}$ is a quasi-stationary distribution, for $f\in
c_{00}$ and $t\geq 0$, we have

$\displaystyle\int|S^{{\scriptscriptstyle(K)}}_{t}f|^{2}\mathrm{d}\nu^{{\scriptscriptstyle(K)}}$
$\displaystyle\leq\int\mathrm{d}\nu^{{\scriptscriptstyle(K)}}(n)\,{\mathds{E}}_{n}\Big{(}\big{|}f(X^{{\scriptscriptstyle(K)}}_{t})\big{|}^{2}{\mathds{1}}_{\big{\\{}T^{{\scriptscriptstyle(K)}}_{0}>t\big{\\}}}\Big{)}$
$\displaystyle=\mathrm{e}^{-\rho^{{\scriptscriptstyle(K)}}_{0}t}\int|f|^{2}\mathrm{d}\nu^{{\scriptscriptstyle(K)}}.$

Therefore

$\|S^{{\scriptscriptstyle(K)}}_{t}f\|_{\ell^{2}(\nu^{{\scriptscriptstyle(K)}})}\leq\mathrm{e}^{-\frac{\rho^{{\scriptscriptstyle(K)}}_{0}{t}}{2}}\|f\|_{\ell^{2}(\nu^{{\scriptscriptstyle(K)}})},\;t\geq
0.$

Since $c_{00}$ is dense in $\ell^{2}(\nu^{{\scriptscriptstyle(K)}})$, we get

$\|S^{{\scriptscriptstyle(K)}}_{t}\|_{\ell^{2}(\nu^{{\scriptscriptstyle(K)}})}\leq\mathrm{e}^{-\frac{\rho^{{\scriptscriptstyle(K)}}_{0}{t}}{2}},\;t\geq
0.$

This implies that $(S^{{\scriptscriptstyle(K)}}_{t})_{t\geq 0}$ extends to a
contraction semigroup in $\ell^{2}(\nu^{{\scriptscriptstyle(K)}})$. Now, since
${\mathbb{P}}_{n}(X^{{\scriptscriptstyle(K)}}_{t}=m,T^{{\scriptscriptstyle(K)}}_{0}>t)\to\delta_{n,m}$,
as $t\to 0$, then for any $f\in c_{00}$, $S^{{\scriptscriptstyle(K)}}_{t}f\to
f$ pointwise, hence by dominated convergence we obtain
$S^{{\scriptscriptstyle(K)}}_{t}f\to f$ in
$\ell^{2}(\nu^{{\scriptscriptstyle(K)}})$ as $t\to 0$. The proposition follows
from the contraction property obtained above and the fact that $c_{00}$ is
dense in $\ell^{2}(\nu^{{\scriptscriptstyle(K)}})$. $\;\blacksquare$
Observe that the same result holds in fact for any $1\leq p<\infty$ (with a
similar proof).

Recall from [5] that we also have

${\mathbb{P}}_{\nu^{{\scriptscriptstyle(K)}}}(T^{{\scriptscriptstyle(K)}}_{0}>t)=\mathrm{e}^{-\rho^{{\scriptscriptstyle(K)}}_{0}t},\;t>0$

and the mean-time to extinction starting from $\nu^{{\scriptscriptstyle(K)}}$
is

$\displaystyle{\mathds{E}}_{\nu^{{\scriptscriptstyle(K)}}}\big{[}T^{{\scriptscriptstyle(K)}}_{0}\big{]}$
$\displaystyle=\frac{1}{\rho^{{\scriptscriptstyle(K)}}_{0}}=\frac{\sqrt{2\pi}\,\exp\bigg{(}K{\int}_{0}^{x_{*}}\log\frac{b(x)}{d(x)}\,\mathrm{d}x\bigg{)}}{b(x_{*})\left(\sqrt{\frac{b(1/{\scriptscriptstyle
K})}{d(1/{\scriptscriptstyle K})}}-\sqrt{\frac{d(1/{\scriptscriptstyle
K})}{b(1/{\scriptscriptstyle
K})}}\right)\\!\sqrt{KH^{\prime\prime}(x_{*})}}\\!\left(\\!1\\!+\\!\mathcal{O}\\!\left(\frac{(\log
K)^{3}}{\sqrt{K}}\right)\\!\\!\right).$

(See the next section for the definition of $H$.) In the logistic model this
gives (recall that $x_{*}=\lambda-\mu$)

${\mathds{E}}_{\nu^{{\scriptscriptstyle(K)}}}\big{[}T^{{\scriptscriptstyle(K)}}_{0}\big{]}=\frac{\sqrt{2\pi}\,\mu\,\exp\Big{(}K\big{(}\lambda-\mu+\mu\log\frac{\mu}{\lambda}\big{)}\Big{)}}{(\lambda-\mu)^{2}\sqrt{K}}\\!\left(\\!1\\!+\\!\mathcal{O}\\!\left(\frac{(\log
K)^{3}}{\sqrt{K}}\right)\\!\\!\right).$

Thus $\rho^{{\scriptscriptstyle(K)}}_{0}$ is exponentially small in $K$, and
we also proved in [5] that the ‘spectral gap’ satisfies (see Theorem 3.3 in
[5])

$\rho^{{\scriptscriptstyle(K)}}_{1}-\rho^{{\scriptscriptstyle(K)}}_{0}\geq\frac{\mathcal{O}(1)}{\log
K},\;K\in\mathds{Z}_{{\scriptscriptstyle>1}}.$ (1.9)

This lower bound goes to $0$ as $K\to+\infty$. A noteworthy consequence of the
main results of this paper (see Corollary 1.3) is that the spectral gap does
not close when $K$ tends to infinity, contrary to what could have been
suspected from the lower bound in (1.9).

In fact we will fully describe the asymptotics of all eigenvalues, which is
the content of our main theorem. To state it, we need to order $S_{1}\cup
S_{2}$ to take care of possible multiplicities. Recall that $S_{1}$ and
$S_{2}$ have been defined in (1.5) and (1.6) and that they are positive
sequences. This is done through the definition of a non-decreasing infinite
(positive) sequence $(\eta_{n})_{n\geq 0}$. Let $\eta_{0}=0$. We construct
this sequence recursively as follows.

  * $\diamond$

If $\eta_{n}\in S_{1}\Delta S_{2}$, then $\eta_{n+1}=\min\\{\eta:\eta\in
S_{1}\cup S_{2}:\eta>\eta_{n}\\}$.

  * $\diamond$

If $\eta_{n}\in S_{1}\cap S_{2}$, then

    * $\bullet$

If $\eta_{n-1}=\eta_{n}$, then $\eta_{n+1}=\min\\{\eta:\eta\in S_{1}\cup
S_{2}:\eta>\eta_{n}\\}$.

    * $\bullet$

If $\eta_{n-1}<\eta_{n}$, then $\eta_{n+1}=\eta_{n}$.

The main result of this paper, whose assumptions will be stated in Section 2,
is the following.

######  Theorem 1.2 (Convergence of the spectrum).

The spectrum of $L_{{\scriptscriptstyle K}}$ in
$\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$ converges pointwise to
$(-\eta_{n})_{n\geq 0}$ when $K$ tends to infinity. In other words

$\lim_{K\to+\infty}\rho^{{\scriptscriptstyle(K)}}_{j}=\eta_{j},\;\forall
j\in\mathds{Z}_{{\scriptscriptstyle\geq 0}}.$

######  Corollary 1.3.

The spectral gap
$\rho^{{\scriptscriptstyle(K)}}_{1}-\rho^{{\scriptscriptstyle(K)}}_{0}$
converges to

$\min\big{\\{}b^{\prime}(0)-d^{\prime}(0),\,d^{\prime}(x_{*})-b^{\prime}(x_{*})\big{\\}}\;.$

Let us give some examples. In the logistic model, we have
$d^{\prime}(0)-b^{\prime}(0)=b^{\prime}(x_{*})-d^{\prime}(x_{*})=\mu-\lambda$
(asymptotic double eigenvalues), and the spectral gap is equal to
$\lambda-\mu$. Another example is the Ayala-Gilpin-Ehrenfeld model [1] defined
by $b(x)=\lambda x$, $d(x)=x(\mu+x^{\theta})$ where $\theta\in(0,1)$ is a
parameter, and $\lambda>\mu$. In this case,
$d^{\prime}(0)-b^{\prime}(0)=\mu-\lambda$, $x_{*}=(\lambda-\mu)^{1/\theta}$,
and $b^{\prime}(x_{*})-d^{\prime}(x_{*})=\theta(\mu-\lambda)$, so the spectral
gap is $\theta(\lambda-\mu)$. Yet another example is Smith’s model [13]
defined by $b(x)=\lambda x/(1+x)$, $d(x)=(x(\mu+x))/(1+x)$, where
$\lambda>\mu$. One easily finds $d^{\prime}(0)-b^{\prime}(0)=\mu-\lambda$,
$x_{*}=\lambda-\mu$, and
$b^{\prime}(x_{*})-d^{\prime}(x_{*})=(\mu-\lambda)/(1+\lambda-\mu)$, so the
spectral gap is $(\lambda-\mu)/(1+\lambda-\mu)$.

###  1.5 Consequences on relaxation times

Recall that the spectral gap is the inverse of the relaxation time to the
quasi-stationary distribution, namely for
$\psi\in\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$, $t>0$ and
$K\in\mathds{Z}_{{\scriptscriptstyle>0}}$ we have

$\left\|\mathrm{e}^{\rho^{{\scriptscriptstyle(K)}}_{0}t}\,S^{{\scriptscriptstyle(K)}}_{t}\psi-\psi_{0}^{{\scriptscriptstyle(K)}}\langle\psi_{0}^{{\scriptscriptstyle(K)}},\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle(K)}}}\nu^{{\scriptscriptstyle(K)}}(\psi)\right\|_{\ell^{2}(\pi^{{\scriptscriptstyle(K)}})}\leq\|\psi\|_{\ell^{2}(\pi^{{\scriptscriptstyle(K)}})}\,\mathrm{e}^{-\big{(}\rho^{{\scriptscriptstyle(K)}}_{1}-\rho^{{\scriptscriptstyle(K)}}_{0}\big{)}t}.$

From Corollary 1.3, it turns out that the relaxation time converges to a
finite limit as $K$ tends to infinity.

We can also characterize the decay of correlations for the so-called
$Q$-process, namely the birth-and-death process conditioned on survival.
Recall that the $Q$-process is the irreducible Markov process with state space
$\mathds{Z}_{{\scriptscriptstyle>0}}$, defined by the semigroup

$R^{{\scriptscriptstyle(K)}}_{t}g=\mathrm{e}^{\rho_{0}^{{\scriptscriptstyle(K)}}t}\frac{1}{\psi_{0}^{{\scriptscriptstyle(K)}}}\,S^{{\scriptscriptstyle(K)}}_{t}\big{(}g\psi_{0}^{{\scriptscriptstyle(K)}}\big{)}.$

It satisfies $R^{{\scriptscriptstyle(K)}}_{t}{\mathds{1}}_{\\{n\geq
1\\}}={\mathds{1}}_{\\{n\geq 1\\}}$, and its unique invariant distribution
$\mathfrak{m}^{{\scriptscriptstyle(K)}}$, defined by
${\big{(}R_{t}^{{\scriptscriptstyle(K)}}\big{)}}^{\dagger}\mathfrak{m}^{{\scriptscriptstyle(K)}}=\mathfrak{m}^{{\scriptscriptstyle(K)}}$,
is related to $\nu^{{\scriptscriptstyle(K)}}$ by
$\mathfrak{m}^{{\scriptscriptstyle(K)}}(g)=\nu^{{\scriptscriptstyle(K)}}(\psi_{0}^{{\scriptscriptstyle(K)}}g)$
where $\nu^{{\scriptscriptstyle(K)}}$ has been defined in (1.8). Indeed we
have $g\in\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})$ if and only if
$\psi_{0}^{{\scriptscriptstyle(K)}}g\in\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$,
and

$\|g\|^{2}_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})}=\frac{\|\psi_{0}^{{\scriptscriptstyle(K)}}g\|^{2}_{\ell^{2}(\pi^{{\scriptscriptstyle(K)}})}}{\langle\psi_{0}^{{\scriptscriptstyle(K)}},\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle(K)}}}}.$

Hence, we get the following result.

######  Proposition 1.4.

Let $g\in\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})$. Then for all $t>0$

$\big{\|}R^{{\scriptscriptstyle(K)}}_{t}g-\langle\psi_{0}^{{\scriptscriptstyle(K)}},\mathbf{1}\rangle_{\pi^{{\scriptscriptstyle(K)}}}\mathfrak{m}^{{\scriptscriptstyle(K)}}(g)\big{\|}_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})}\leq\|g\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})}\,\mathrm{e}^{-(\rho^{{\scriptscriptstyle(K)}}_{1}-\rho^{{\scriptscriptstyle(K)}}_{0})\,t}.$

Furthermore, for
$g_{1},g_{2}\in\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})$ and for all
$t>0$, we have

$\displaystyle\bigg{|}\int R^{{\scriptscriptstyle(K)}}_{t}g_{1}\cdot
g_{2}\,\mathrm{d}\mathfrak{m}^{{\scriptscriptstyle(K)}}-\int
g_{1}\,\mathrm{d}\mathfrak{m}^{{\scriptscriptstyle(K)}}\int
g_{2}\,\mathrm{d}\mathfrak{m}^{{\scriptscriptstyle(K)}}\bigg{|}$
$\displaystyle\leq\|g_{1}\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})}\|g_{2}\|_{\ell^{2}(\mathfrak{m}^{{\scriptscriptstyle(K)}})}\,\mathrm{e}^{-\big{(}\rho^{{\scriptscriptstyle(K)}}_{1}-\rho^{{\scriptscriptstyle(K)}}_{0}\big{)}t}.$

As before, the rate of decay of correlations converges when $K$ goes to
infinity.

###  1.6 Organization of the paper

The proof of the Theorem 1.2 relies on two results stated in Section 3.
Theorem 3.5 ensures that the set $S_{1}\cup S_{2}$ is contained in the set of
accumulation points of the eigenvalues of $\L_{{\scriptscriptstyle K}}$ when
$K$ tends to infinity. The proof is based on the construction of quasi-
eigenvectors and is given in Section 5. The second result is Theorem 3.7 which
ensures that all the previous accumulation points are contained in $S_{1}\cup
S_{2}$ taking care of eventual multiplicities. Its proof, given in section 5,
relies on two propositions. The first one (Proposition 4.1) is the splitting
of the eigenvectors of $\mathcal{L}_{{\scriptscriptstyle K}}$ into two
dominant parts, one localised near the origin, the other one near $\lfloor
Kx_{*}\rfloor$. The second (Proposition 4.5) relies on compactness arguments
of each piece of the previous splitting. Section 6 collects various auxiliary
results (some of more general nature).

One of the main difficulties of the proof is that the two pieces of the
spectrum correspond to limiting operators which are obtained at different
scales and leave in different function spaces.

##  2 Standing assumptions

We work under the assumptions of [5] which we recall for convenience.

The functions $b,d:\mathds{R}_{+}\to\mathds{R}_{+}$ defining the differential
equation (1.2) are supposed to be such that

$b(0)=d(0)=0$

and the functions $x\mapsto b(x)/x$ and $x\mapsto d(x)/x$ are defined on
$\mathds{R}_{+}$ and assumed to be positive, twice differentiable and
increasing (in particular the sequences
$(\lambda^{{\scriptscriptstyle(K)}}_{n})_{n}$ and
$(\mu^{{\scriptscriptstyle(K)}}_{n})_{n}$ defined in (1.1) are increasing for
each $K$).

We start by the biologically relevant assumptions:

  * $\centerdot$

$\lim_{x\to+\infty}\frac{b(x)}{d(x)}=0$ (deaths prevail over births for very
large densities).

  * $\centerdot$

$b^{\prime}(0)>d^{\prime}(0)>0$ (at low density births prevail).

  * $\centerdot$

There is a unique $x_{*}>0$ such that $b(x_{*})=d(x_{*})$, so $x_{*}$ is the
only positive fixed point of (1.2).

We assume that $b^{\prime}(x_{*})\neq d^{\prime}(x_{*})$ (genericity
condition). The remaining (technical) assumptions are the following:

  * $\centerdot$

${\int}_{\frac{x_{*}}{2}}^{+\infty}\frac{\mathrm{d}x}{d(x)}<+\infty$ and
$\sup_{x\in\mathds{R}_{+}}\big{(}\frac{d^{\prime}(x)}{d(x)}-\frac{1}{x}\big{)}<+\infty$.

  * $\centerdot$

The function $x\mapsto\log\frac{d(x)}{b(x)}$ is increasing on
$\mathds{R}_{+}$.

  * $\centerdot$

The function $H:\mathds{R}_{+}\to\mathds{R}$ defined by
$H(x)=\int_{x_{*}}^{x}\log\frac{d(s)}{b(s)}\mathrm{d}s$ is three times
differentiable, and
$\sup_{x\in\mathds{R}_{+}}(1+x^{2})|H^{\prime\prime\prime}(x)|<+\infty$.

The assumptions imply that $0$ is a repulsive (or unstable) fixed point of
(1.2), whereas $x_{*}$ is an attractive (or stable) one, that is,
$b^{\prime}(x_{*})<d^{\prime}(x_{*})$. It also follows that
$H^{\prime\prime}(x_{*})>0$. These assumptions are satisfied for many
classical examples.

As explained in [5], the above conditions imply the following properties:

  * $\centerdot$

$\sum_{n\geq
1}(\lambda^{{\scriptscriptstyle(K)}}_{n}\pi^{{\scriptscriptstyle(K)}}_{n})^{-1}=+\infty$,
which implies that the process reaches $0$ in finite time with probability
one.

  * $\centerdot$

$\sum_{n\geq 1}\pi^{{\scriptscriptstyle(K)}}_{n}<+\infty$, which implies
finiteness of the mean time to extinction.

  * $\centerdot$

$\sum_{n\geq
1}(\lambda^{{\scriptscriptstyle(K)}}_{n}\pi^{{\scriptscriptstyle(K)}}_{n})^{-1}\sum_{i\geq
n+1}\pi^{{\scriptscriptstyle(K)}}_{i}<+\infty$, which is a necessary and
sufficient condition for existence and uniqueness of the quasi-stationary
distribution.

We add a last condition to the previous ones, namely

$\lim_{x\to\infty}\frac{\log b(x)}{x}=\lim_{x\to\infty}\frac{\log d(x)}{x}=0.$
(2.1)

We could avoid it makes our life easier and we don’t have any natural example
which does not satisfy it.

##  3 Proof of Theorem 1.2

###  3.1 Some useful operators

Instead of working with $L_{{\scriptscriptstyle K}}$ on the weighted Hilbert
space $\ell^{2}(\pi^{{\scriptscriptstyle(K)}})$, we find more convenient to
work on the ‘flat’ Hilbert space $\ell^{2}$. We introduce the conjugated
operator

$\mathcal{L}_{{\scriptscriptstyle
K}}=(\Pi^{{\scriptscriptstyle(K)}})^{\frac{1}{2}}L_{{\scriptscriptstyle
K}}\,(\Pi^{{\scriptscriptstyle(K)}})^{-\frac{1}{2}}$

where $\Pi^{{\scriptscriptstyle(K)}}$ denotes the mutiplication operator

$\Pi^{{\scriptscriptstyle(K)}}v(n)=\pi^{{\scriptscriptstyle(K)}}_{n}v(n)$

for $v\in c_{00}$ and $n\in\mathds{Z}_{{\scriptscriptstyle>0}}$. One can check
that

$\displaystyle\big{(}\mathcal{L}_{{\scriptscriptstyle K}}v\big{)}(n)$
$\displaystyle=\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\,\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,v(n+1)+\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,v(n-1)\,{\mathds{1}}_{\\{n>1\\}}-\big{(}\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}\big{)}\,v(n)$

for $n\in\mathds{Z}_{{\scriptscriptstyle>0}}$. We denote also by
$\mathcal{L}_{{\scriptscriptstyle K}}$ its closure in $\ell^{2}$ and by
$\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ its domain, and we have

$\mathcal{L}_{{\scriptscriptstyle
K}}\phi^{{\scriptscriptstyle(K)}}_{j}=-\rho^{{\scriptscriptstyle(K)}}_{j}\phi^{{\scriptscriptstyle(K)}}_{j}$

where the eigenvalues $-\rho^{{\scriptscriptstyle(K)}}_{j}$ are the same as
for $L_{{\scriptscriptstyle K}}$ (cf. (1.7)), and
$\phi^{{\scriptscriptstyle(K)}}=\big{(}\Pi^{{\scriptscriptstyle(K)}}\big{)}^{\frac{1}{2}}\psi^{{\scriptscriptstyle(K)}}$.
To capture the behavior of the eigenvectors of
$\mathcal{L}_{{\scriptscriptstyle K}}$ near $\lfloor Kx_{*}\rfloor$ at scale
$\sqrt{K}$, we are going to embed $\ell^{2}$ into $L^{2}$. For this purpose we
define for each $K\in\mathds{Z}_{{\scriptscriptstyle>0}}$ the functions

$e_{n}^{{\scriptscriptstyle(K)}}(x)=K^{\frac{1}{4}}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{n}}(x),\;x\in\real,\;n\in\mathds{Z}_{{\scriptscriptstyle>0}}$

where

$I^{\scriptscriptstyle(K)}_{n}=\left[\frac{n-0.5}{\sqrt{K}}-x_{*}\sqrt{K},\frac{n+0.5}{\sqrt{K}}-x_{*}\sqrt{K}\right[.$

The functions $e_{n}^{{\scriptscriptstyle(K)}}$ are orthogonal and of norm one
in $L^{2}$. They form a basis of a sub-Hilbert space
$\mathscr{H}_{{\scriptscriptstyle K}}$ of piecewise constant functions in
$L^{2}$. We define two maps denoted by $Q_{{\scriptscriptstyle K}}$ and
$P_{{\scriptscriptstyle K}}$ as follows:

$Q_{{\scriptscriptstyle K}}:\ell^{2}\to L^{2},\quad Q_{{\scriptscriptstyle
K}}u(x)=\sum_{n\geq 1}u(n)\,e_{n}^{{\scriptscriptstyle(K)}}(x)\ $

and

$P_{{\scriptscriptstyle K}}:L^{2}\to\ell^{2},\quad P_{{\scriptscriptstyle
K}}f(n)=\int
f(x)\,e_{n}^{{\scriptscriptstyle(K)}}(x)\,\mathrm{d}x\;,n\in\mathds{Z}_{{\scriptscriptstyle>0}}.$
(3.1)

We will use the following properties of $P_{{\scriptscriptstyle K}}$ and
$Q_{{\scriptscriptstyle K}}$ stated as two lemmas.

######  Lemma 3.1.

For each $K\in\mathds{Z}_{{\scriptscriptstyle>0}}$, the map
$Q_{{\scriptscriptstyle K}}$ is an isometry between $\ell^{2}$ and
$\mathscr{H}_{{\scriptscriptstyle K}}$.

The proof of this lemma is left to the reader.

######  Lemma 3.2.

Let $f\in C^{1}(\real)$ and assume that there exists $a>0$ and $A>0$ such that

$\big{|}f(x)\big{|}+\big{|}f^{\prime}(x)\big{|}\leq
A\,\mathrm{e}^{-a\,|x|},\,x\in\real.$

Then

  * (i)

$\lim_{K\to\infty}\big{\|}f-Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle
K}}f\big{\|}_{L^{2}}=0$ .

  * (ii)

$\lim_{K\to\infty}\big{\|}P_{{\scriptscriptstyle
K}}f\big{\|}_{\ell^{2}}=\big{\|}f\big{\|}_{L^{2}}$ .

PROOF. Let us prove (i). We have $Q_{{\scriptscriptstyle
K}}P_{{\scriptscriptstyle K}}f(x)=e^{{\scriptscriptstyle(K)}}_{n(x)}(x)\;\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)\,f(y)\,\mathrm{d}y$ for
$(n(x)-0.5)/\sqrt{K}-x_{*}\leq x\leq(n(x)+0.5)/\sqrt{K}-x_{*}$. We get from
our hypothesis

$Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle
K}}f(x)=f(x)+\mathcal{O}\big{(}K^{-\frac{1}{2}}\big{)}\,\mathrm{e}^{-a|x|}$

and the result follows.
We now prove (ii). From the isometric property of $Q_{K}$ on the space of
piecewise functions $\mathscr{H}_{{\scriptscriptstyle K}}$, we get

$\big{\|}P_{{\scriptscriptstyle
K}}f\big{\|}_{\ell^{2}}=\big{\|}Q_{{\scriptscriptstyle
K}}P_{{\scriptscriptstyle K}}f\big{\|}_{L^{2}}$

and the result follows from (i). $\;\blacksquare$

We now introduce the operator

$\mathscr{L}_{{\scriptscriptstyle K}}=Q_{{\scriptscriptstyle
K}}\mathcal{L}_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle K}}$

and we keep the same notation for its closure in $L^{2}$. Since
$\mathscr{L}_{{\scriptscriptstyle K}}$ when acting on
$\mathscr{H}_{{\scriptscriptstyle K}}$ is conjugated to
$\mathcal{L}_{{\scriptscriptstyle K}}$, we have

$\mathscr{L}_{{\scriptscriptstyle
K}}\varphi^{{\scriptscriptstyle(K)}}_{j}=-\rho^{{\scriptscriptstyle(K)}}_{j}\varphi^{{\scriptscriptstyle(K)}}_{j}.$

We will prove in the next proposition that the operator
$\mathscr{L}_{{\scriptscriptstyle K}}$ converges weakly, when $K\to+\infty$,
to the operator

$\displaystyle\mathcal{H}_{*}f(x)=$ (3.2) $\displaystyle
b(x_{*})\,\frac{\mathrm{d}^{2}f(x)}{\mathrm{d}x^{2}}-\frac{\big{(}d^{\prime}(x_{*})-b^{\prime}(x_{*})\big{)}^{2}}{4b(x_{*})}\,x^{2}f(x)+\frac{d^{\prime}(x_{*})-b^{\prime}(x_{*})}{2}f(x).$

######  Proposition 3.3.

Let $f\in C^{3}(\real)$ and assume that there exist $a>0$ and $A>0$ such that

$\sum_{j=0}^{3}\big{|}f^{(j)}(x)\big{|}\leq
A\,\mathrm{e}^{-a\,|x|},\,x\in\real.$

Then

$\lim_{K\to\infty}\big{\|}\mathscr{L}_{{\scriptscriptstyle
K}}f-\mathcal{H}_{*}f\big{\|}_{L^{2}}=0\,.$

PROOF. By the assumption made on $f$, it follows easily that

$\lim_{K\to\infty}\big{\|}{\mathds{1}}_{\\{|\,\cdot\,|>(\log
K)^{2}\\}}\;\mathcal{H}_{*}f\big{\|}_{L^{2}}=0.$

We have

$\displaystyle\mathscr{L}_{{\scriptscriptstyle K}}f(x)=\sum_{n\geq
1}e^{{\scriptscriptstyle(K)}}_{n}(x)\,\left(\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\,\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,\int
e^{{\scriptscriptstyle(K)}}_{n+1}(y)\,f(y)\,\mathrm{d}y\right.$ (3.3)
$\displaystyle\left.+\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,{\mathds{1}}_{\\{n>1\\}}\,\int
e^{{\scriptscriptstyle(K)}}_{n-1}(y)\,f(y)\,\mathrm{d}y-\big{(}\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}\big{)}\,\int
e^{{\scriptscriptstyle(K)}}_{n}(y)\,f(y)\,\mathrm{d}y\right).$

It follows easily from the assumption made on $f$ and assumption (2.1) that

$\lim_{K\to\infty}\big{\|}{\mathds{1}}_{\\{|\,\cdot\,|>(\log
K)^{2}\\}}\mathscr{L}_{{\scriptscriptstyle K}}f\big{\|}_{L^{2}}=0\,.$

Therefore we only have to consider $|x|\leq(\log K)^{2}$. Note also that for
such an $x$, the sum in (3.3) reduces to one element for $K$ large enough,
namely $n=n(x)=\big{\lfloor}Kx_{*}+\sqrt{K}x+\frac{1}{2}\big{\rfloor}$. For
$x\in\real$, we have

$\displaystyle\mathscr{L}_{{\scriptscriptstyle K}}f(x)$
$\displaystyle=\mathrm{e}^{{\scriptscriptstyle(K)}}_{n(x)}(x)\left(\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n(x)}\mu^{{\scriptscriptstyle(K)}}_{n(x)+1}}\,\int
e^{{\scriptscriptstyle(K)}}_{n(x)+1}(y)f(y)\,\mathrm{d}y\right.$
$\displaystyle\left.\hskip
59.75095pt+\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n(x)-1}\,\mu^{{\scriptscriptstyle(K)}}_{n(x)}}\int
e^{{\scriptscriptstyle(K)}}_{n(x)-1}(y)f(y)\,\mathrm{d}y\right.$
$\displaystyle\left.\hskip
59.75095pt-\,\big{(}\lambda^{{\scriptscriptstyle(K)}}_{n(x)}+\mu^{{\scriptscriptstyle(K)}}_{n(x)}\big{)}\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)f(y)\,\mathrm{d}y\right)$
$\displaystyle=e^{{\scriptscriptstyle(K)}}_{n(x)}(x)\left(\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n(x)}\,\mu^{{\scriptscriptstyle(K)}}_{n(x)+1}}\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)f\left(y-\frac{1}{\sqrt{K}}\right)\mathrm{d}y\right.$
$\displaystyle\left.\hskip
56.9055pt+\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n(x)-1}\,\mu^{{\scriptscriptstyle(K)}}_{n(x)}}\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)f\left(y+\frac{1}{\sqrt{K}}\right)\mathrm{d}y\right.$
$\displaystyle\left.\hskip
56.9055pt-\big{(}\lambda^{{\scriptscriptstyle(K)}}_{n(x)}+\mu^{{\scriptscriptstyle(K)}}_{n(x)}\big{)}\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)f(y)\,\mathrm{d}y\right).$

Now we have

$\displaystyle\int e^{{\scriptscriptstyle(K)}}_{n(x)}(y)\,f(y)\,\mathrm{d}y$
$\displaystyle=K^{\frac{1}{4}}\int_{-\frac{1}{2\sqrt{K}}}^{\frac{1}{2\sqrt{K}}}f\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}-s\right)\,\mathrm{d}s$
$\displaystyle=K^{-\frac{1}{4}}\,f\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}\right)+\frac{K^{-\frac{5}{4}}}{24}f^{\prime\prime}\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}\right)+\mathcal{O}\Big{(}K^{-\frac{7}{4}}\Big{)}$

where the error term is unifom in $x$. Similarly

$\displaystyle\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)\,f\left(y\pm\frac{1}{\sqrt{K}}\right)\mathrm{d}y$
$\displaystyle=\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)\,f(y)\,\mathrm{d}y\pm\frac{1}{\sqrt{K}}\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)\,f^{\prime}(y)\,\mathrm{d}y$
$\displaystyle\quad+\frac{1}{2K}\int
e^{{\scriptscriptstyle(K)}}_{n(x)}(y)\,f^{\prime\prime}(y)\,\mathrm{d}y+\mathcal{O}\Big{(}K^{-\frac{7}{4}}\Big{)}$
$\displaystyle=K^{-\frac{1}{4}}\,f\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}\right)\pm
K^{-\frac{3}{4}}\,f^{\prime}\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}\right)$
$\displaystyle\quad+\frac{13\,K^{-\frac{5}{4}}}{24}f^{\prime\prime}\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}\right)+\mathcal{O}\big{(}K^{-\frac{7}{4}}\big{)}.$

Recall that

$\lambda^{{\scriptscriptstyle(K)}}_{n}=Kb\left(\frac{n}{K}\right)\quad\text{and}\quad\mu^{{\scriptscriptstyle(K)}}_{n}=Kd\left(\frac{n}{K}\right)$

hence

$\lambda^{{\scriptscriptstyle(K)}}_{n}=Kb\big{(}x_{*}\big{)}+(n-Kx_{*})\,b^{\prime}\big{(}x_{*}\big{)}+\mathcal{O}\left(\frac{(n-Kx_{*})^{2}}{K}\right)$

and

$\mu^{{\scriptscriptstyle(K)}}_{n}=Kd\big{(}x_{*}\big{)}+(n-Kx_{*})\,d^{\prime}\big{(}x_{*}\big{)}+\mathcal{O}\left(\frac{(n-Kx_{*})^{2}}{K}\right).$

After a tedious but straightforward computation, we obtain that

$\displaystyle\mathscr{L}_{{\scriptscriptstyle K}}f(x)$
$\displaystyle=\mathcal{H}_{*}f\left(\frac{n(x)-Kx_{*}}{\sqrt{K}}\right)+\mathcal{O}\big{(}K^{-\frac{1}{4}}\big{)}$
$\displaystyle=\mathcal{H}_{*}f(x)+\mathcal{O}\big{(}K^{-\frac{1}{4}}\;(\log
K)^{4}\big{)}$

and the error term is uniform in $|x|\leq(\log K)^{2}$. We get

$\big{\|}{\mathds{1}}_{\\{|\,\cdot\,|\leq(\log
K)^{2}\\}}\;\big{(}\mathscr{L}_{{\scriptscriptstyle
K}}f-\mathcal{H}_{*}f\big{)}\big{\|}_{L^{2}}=\mathcal{O}\big{(}K^{-\frac{1}{4}}\;(\log
K)^{6}\big{)}$

and the result follows. $\;\blacksquare$

######  Remark 3.1.

Let us recall the relationship between the generator of the Ornstein-Uhlenbeck
process (1.4) and (3.2) which is, up to a minus sign and a shift, the
Schrödinger operator for the quantum harmonic oscillator. We refer to e.g. [2,
Chapter 3] or [10, Sections 4.4 and 4.9]. In $L^{2}$, the eigenvalues of
$\mathcal{H}_{*}$ are $-\big{(}d^{\prime}(x_{*})-b^{\prime}(x_{*}))n$,
$n\in\mathds{Z}_{{\scriptscriptstyle\geq 0}}$, and the corresponding
eigenfunctions are

$\displaystyle\psi_{n}(x)=$ (3.4)
$\displaystyle\frac{1}{\sqrt{2^{n}n!}}\left(\frac{\big{(}d^{\prime}(x_{*})-b^{\prime}(x_{*})\big{)}^{2}}{2\pi
b(x_{*})}\right)^{\frac{1}{4}}\mathrm{e}^{-\frac{d^{\prime}(x_{*})-b^{\prime}(x_{*})}{4b(x_{*})}x^{2}}H_{n}\left(\sqrt{\frac{d^{\prime}(x_{*})-b^{\prime}(x_{*})}{2b(x_{*})}}x\right)$

where $(H_{n})_{n}$ is the family of the physicists’ Hermite polynomials
defined by

$H_{n}(x)=(-1)^{n}\mathrm{e}^{x^{2}}\frac{\mathrm{d}^{n}}{\mathrm{d}x^{n}}\mathrm{e}^{-x^{2}}.$

One can check that $\mathcal{H}_{*}$ is conjugated to the generator of the
Ornstein-Uhlenbeck process (1.4) acting on
$L^{2}\Big{(}\sqrt{\frac{d^{\prime}(x_{*})-b^{\prime}(x_{*})}{2\pi
b(x_{*})}}\,\mathrm{e}^{-\frac{(d^{\prime}(x_{*})-b^{\prime}(x_{*})}{2b(x_{*})}x^{2}}\mathrm{d}x\Big{)}$
in the following way:
$\frac{1}{\psi_{0}}\mathcal{H}_{*}(\psi_{0}f)=\mathcal{OU}_{*}f$.

In Proposition 3.4 we prove that the operator
$\mathcal{L}_{{\scriptscriptstyle K}}$ converges weakly, when $K$ tends to
infinity, to the operator $\mathcal{M}_{0}$ defined for $v\in c_{00}$ by

$\displaystyle\big{(}\mathcal{M}_{0}v\big{)}(n)=$ (3.5)
$\displaystyle\sqrt{b^{\prime}(0)\,d^{\prime}(0)\,n\,(n+1)}\,v(n+1)+\sqrt{b^{\prime}(0)\,d^{\prime}(0)\,n\,(n-1)}\,v(n-1)\,{\mathds{1}}_{\\{n>1\\}}$
$\displaystyle-n\,(b^{\prime}(0)+d^{\prime}(0))v(n)).$

Here again we denote the operator on $c_{00}$ and its closure by the same
letter.

######  Proposition 3.4.

Let $u\in c_{00}$. Then

$\lim_{K\to\infty}\mathcal{L}_{{\scriptscriptstyle K}}u=\mathcal{M}_{0}u$

where $\mathcal{M}_{0}$ is defined in (3.5).

PROOF. Follows from the fact that for each fixed $n$

$\lim_{K\to\infty}\lambda^{{\scriptscriptstyle(K)}}_{n}=b^{\prime}(0)\,n\qquad\mathrm{and}\qquad\lim_{K\to\infty}\mu^{{\scriptscriptstyle(K)}}_{n}=d^{\prime}(0)\,n\;.$

$\;\blacksquare$

###  3.2 Steps of the proof of Theorem 1.2

The proof of Theorem 1.2 relies on the following two theorems whose proofs are
postponed to Section 5. Recall that for any fixed $K$, the spectrum
$\mathrm{Sp}(\mathcal{L}_{{\scriptscriptstyle K}})$ is discrete, and let

$G=\bigcup_{j=0}^{\infty}\big{(}\rho_{j}^{{\scriptscriptstyle(K)}}\big{)}^{\mathrm{{\scriptscriptstyle
acc}}}$

where
$\big{(}\rho_{j}^{{\scriptscriptstyle(K)}}\big{)}^{\mathrm{{\scriptscriptstyle
acc}}}$ is the set of accumulation points of
$\big{(}\rho_{j}^{{\scriptscriptstyle(K)}}\big{)}$ when $K\to+\infty$.

######  Theorem 3.5.

We have

$S_{1}\cup S_{2}\subset G$

where $S_{1}$ and $S_{2}$ are defined in (1.5) and (1.6).

This theorem is proved in Section 5.1.

######  Corollary 3.6.

For every fixed $j$ we have

$\limsup_{{\scriptscriptstyle
K}\to+\infty}\rho_{j}^{{\scriptscriptstyle(K)}}<+\infty.$

PROOF. We proceed by contradiction. Assume that there exists $j_{0}$ such that

$\limsup_{{\scriptscriptstyle
K}\to+\infty}\rho_{j_{0}}^{{\scriptscriptstyle(K)}}=+\infty.$

Let $j_{c}=\min\\{0<\ell\leq
j_{0}:\limsup_{K\to+\infty}\rho_{\ell}^{{\scriptscriptstyle(K)}}=+\infty\\}$.
Hence there exists $\alpha<+\infty$ such that
$\limsup_{K\to+\infty}\rho_{j_{c}-1}^{{\scriptscriptstyle(K)}}=\alpha$. By
definition of $j_{c}$, there exists a diverging sequence $(K_{p})_{p}$ such
that $\lim_{p\to+\infty}\rho_{j_{c}}^{{\scriptscriptstyle(K_{p})}}=+\infty$.
Let $\rho\in S_{1}\cup S_{2}$ such that $\rho>\alpha$.

If $j\leq j_{c}-1$, we have
$\limsup_{p\to+\infty}\rho_{j}^{{\scriptscriptstyle(K_{p})}}\leq\limsup_{p\to+\infty}\rho_{j_{c}-1}^{{\scriptscriptstyle(K_{p})}}\leq\alpha<\rho$.
For all $j\geq j_{c}$, we have
$\liminf_{p\to+\infty}\rho_{j}^{{\scriptscriptstyle(K_{p})}}\geq\liminf_{p\to+\infty}\rho_{j_{c}}^{{\scriptscriptstyle(K_{p})}}=+\infty$.
This implies $\rho\notin G$, contradicting Theorem 3.5. $\;\blacksquare$

######  Theorem 3.7.

We have

$S_{1}\cup S_{2}\supset G.$

Moreover, for each $j\in\mathds{Z}_{{\scriptscriptstyle\geq 0}}$, let
$(K_{p})_{p}$ be a diverging sequence such that

$\lim_{p\to\infty}\rho_{j}^{{\scriptscriptstyle(K_{p})}}=\rho_{*}$

where $\rho_{*}$ is finite by Corollary 3.6. Then

  1. 1)

If $\rho_{*}\in S_{1}\Delta S_{2}$ then

$\liminf_{p\to\infty}\;\min\left\\{\bigg{|}\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|},\;\bigg{|}\rho_{j-1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|}\right\\}>0\;.$
(3.6)

Moreover there are only two cases:

    1. (a)

If $\rho_{*}\in S_{1}$ then there exists a diverging sequence of integers
$(p_{\ell})$ such that $Q_{{\scriptscriptstyle
K_{p_{\ell}}}}\phi^{{\scriptscriptstyle(K_{p_{\ell}}})}_{j}\xrightarrow[]{L^{2}}\varphi_{*}$,
where $\rho_{*}$ and $\varphi_{*}$ are such that
$\mathcal{H}_{*}\varphi_{*}=-\rho_{*}\varphi_{*}$.

    2. (b)

If $\rho_{*}\in S_{2}$ then
$\phi^{{\scriptscriptstyle(K_{p})}}_{j}\xrightarrow[]{\ell^{2}}\phi_{*}$,
where $\rho_{*}$ and $\phi_{*}$ are such that
$\mathcal{M}_{0}\phi_{*}=-\rho_{*}\phi_{*}$.

  2. 2)

If $\rho_{*}\in S_{1}\cap S_{2}$ then we have the following two assertions:

    1. (a)

There exists a diverging sequence of integers $(p_{\ell})$ such that:

$\text{either}\quad\lim_{\ell\to\infty}\rho_{j+1}^{{\scriptscriptstyle(K_{p_{\ell}})}}=\rho_{*}\quad\text{or}\quad\lim_{\ell\to\infty}\rho_{j-1}^{{\scriptscriptstyle(K_{p_{\ell}})}}=\rho_{*}.$

    2. (b)

We have

$\liminf_{p\to\infty}\;\min\left\\{\bigg{|}\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|},\;\bigg{|}\rho_{j-1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|}\right\\}=0$
(3.7)

and

$\liminf_{p\to\infty}\;\max\left\\{\bigg{|}\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|},\;\bigg{|}\rho_{j-1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|}\right\\}>0\;.$
(3.8)

Note that (3.6) means that if $\rho_{*}\in S_{1}\Delta S_{2}$ then $-\rho_{*}$
is a simple asymptotic eigenvalue, and either $-\rho_{*}$ is an eigenvalue of
$\mathcal{H}_{*}$ if $\rho_{*}\in S_{1}$, or of $\mathcal{M}_{0}$ if
$\rho_{*}\in S_{2}$. In addition, (3.7) and (3.8) mean that if $\rho_{*}\in
S_{1}\cap S_{2}$, then $-\rho_{*}$ is a double asymptotic eigenvalue which is
an eigenvalue of both $\mathcal{H}_{*}$ and $\mathcal{M}_{0}$.

Proof of Theorem 1.2. The proof is recursive. For $j=0$ it follows from [5]
that $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{0}^{{\scriptscriptstyle(K)}}=0$. Let $j\geq 0$ and assume
that for $\ell\leq j$ (if any) $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{\ell}^{{\scriptscriptstyle(K)}}=\eta_{\ell}$. We now prove
that $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}=\eta_{j+1}$. There are
several cases to consider.

  * •

If $\eta_{j}\in S_{1}\Delta S_{2}$, we claim that
$\liminf_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}\geq\eta_{j+1}$. Otherwise,
by Theorem 3.7 and the recursive hypothesis, there would exist
$K_{p}\to+\infty$ such that
$\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}\to\eta_{*}<\eta_{j+1}$. Since by the
recursive hypothesis $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{j}^{{\scriptscriptstyle(K_{p})}}=\eta_{j}$, we have
$\eta_{*}\geq\eta_{j}$. From the first statement of Theorem 3.7 it follows
that $\eta_{*}=\eta_{j}$. This contradicts 1) of Theorem 3.7.
We now claim that $\limsup_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}\leq\eta_{j+1}$. Otherwise,
by Theorem 3.7 and the recursive hypothesis, there would exist
$K_{p}\to+\infty$ such that
$\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}\to\eta_{*}>\eta_{j+1}$. This implies
that $\eta_{j+1}\not\in G$, contradicting Theorem 3.5.
Hence, in this case, $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}=\eta_{j+1}$.

  * •

If $\eta_{j}\in S_{1}\cap S_{2}$ (which implies $j>0$), we have two cases:

    * –

If $\eta_{j-1}=\eta_{j}$, then we claim that $\liminf_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}\geq\eta_{j+1}$. Otherwise,
by the same argument as before, there would exist $K_{p}\to+\infty$ such that
$\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}\to\eta_{j}$, contradicting (3.8) in
Theorem 3.7.
We now claim that $\limsup_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}\leq\eta_{j+1}$. Otherwise,
as before, this would contradict that $\eta_{j+1}\in G$.
Hence, in this case, $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}=\eta_{j+1}$.

    * –

If $\eta_{j-1}<\eta_{j}$, then we obviously have $\liminf_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}\geq\eta_{j}$. If
$\limsup_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}>\eta_{j}$, then there exists
$K_{p}\to+\infty$ such that
$\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}\to\eta_{*}>\eta_{j}$, contradicting
(3.7) in Theorem 3.7.
Hence, in this case, $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}=\eta_{j+1}$.

Therefore we $\lim_{{\scriptscriptstyle
K}\to+\infty}\rho_{j+1}^{{\scriptscriptstyle(K)}}=\eta_{j+1}$. As announced,
the proof of Theorem 1.2 follows recursively.

##  4 Properties of the eigenvectors

Our aim in this part is to prove that for $K$ large enough, the eigenvectors
$\phi^{{\scriptscriptstyle(K)}}_{j}$ of $\mathcal{L}_{{\scriptscriptstyle K}}$
(see (1.7)) are functions whose representation is sketched in the figure. An
eigenvector is ‘negligible’ outside the union of a neighborhood of $1$, and a
neighborhood of $Kx_{*}$. It is ‘non-negligible’ in at least one of these
neighborhoods.

Figure: Schematic representation of one of the three possible ‘shapes’ of the
eigenvectors $\phi^{{\scriptscriptstyle(K)}}$ (with distortion).

To separate the different behaviors, we introduce a ‘potential’ defined by

$V_{n}(K)=\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\mu^{{\scriptscriptstyle(K)}}_{n}}\,{\mathds{1}}_{\\{n>1\\}}.$
(4.1)

For $\eta>0$, let $n_{\mathrm{g}}({\scriptstyle K},\eta)$ and
$n_{\mathrm{d}}({\scriptstyle K},\,\eta)$ be integers such that $\llbracket
n_{\mathrm{g}}({\scriptstyle K},\eta),n_{\mathrm{d}}({\scriptstyle
K},\,\eta)\rrbracket$ is the maximal interval containing $Kx_{*}/2$ such that

$\inf_{n\,\in\,\llbracket n_{\mathrm{g}}({\scriptstyle
K},\eta),\,n_{\mathrm{d}}({\scriptstyle
K},\,\eta)\rrbracket}\big{(}V_{n}(K)-\eta\big{)}>0\;.$

Let $n_{\ell}{\scriptstyle(K)}=\big{\lfloor}(\log K)^{2}\big{\rfloor}$ and
$n_{r}{\scriptstyle(K)}=\big{\lfloor}Kx_{*}-K^{\frac{2}{3}}\,\log
K\big{\rfloor}$. It follows from our assumptions that for $K$ large enough

$1<n_{\mathrm{g}}({\scriptstyle
K},\eta)<n_{\ell}{\scriptstyle(K)}\ll\frac{Kx_{*}}{2}<n_{r}{\scriptstyle(K)}<n_{\mathrm{d}}({\scriptstyle
K},\,\eta)\;.$

######  Proposition 4.1.

For any $\eta>0$ there exists $a_{\eta}>0$, and $K_{\eta}>0$ such that, if
$K>K_{\eta}$ and $\phi$ of norm one in $\ell^{2}$ satisfies

$\mathcal{L}_{{\scriptscriptstyle
K}}\phi=-\rho\,\phi\quad\text{where}\quad\rho<\eta$

then

$\sup_{n_{\ell}{\scriptstyle(K)}\leq n\leq
n_{r}{\scriptstyle(K)}}\big{|}\phi(n)\big{|}\leq\mathrm{e}^{-a_{\eta}(\log
K)^{2}}\;.$

PROOF. Let us consider an eigenvector $\phi$ of norm one in $\ell^{2}$
satisfying $\ \mathcal{L}_{{\scriptscriptstyle K}}\,\phi=-\rho\,\phi$.

If $\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})\neq 0$, define
$\tilde{n}_{\mathrm{g}}({\scriptstyle K},\eta)=n_{\mathrm{g}}({\scriptstyle
K},\eta)$ and if needed, change the sign of $\phi$ such that
$\phi({\tilde{n}_{\mathrm{g}}({\scriptstyle K},\eta)})>0$. If
$\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})=0$, define
$\tilde{n}_{\mathrm{g}}({\scriptstyle K},\eta)=n_{\mathrm{g}}({\scriptstyle
K},\eta)+1$ and if needed, change the sign of $\phi$ such that
$\phi({\tilde{n}_{\mathrm{g}}({\scriptstyle K},\eta)})>0$ (note that
$\phi({n_{\mathrm{g}}({\scriptstyle
K},\eta)})=\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)+1})=0$ contradicts the
normalisation since $\phi$ solves a second-order recurrence relation). Then,
changing the definition of $n_{\mathrm{g}}({\scriptstyle K},\eta)$ if
necessary, we can assume that $\phi({n_{\mathrm{g}}({\scriptstyle
K},\eta)})>0$.

Thanks to the local maximum-minimum principle (see Proposition 8.2) we only
have four cases.

  1. 1)

$\phi({n_{\mathrm{g}}({\scriptstyle
K},\eta)+1})\geq\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})$ and $\phi({n})$
is increasing on $\llbracket n_{\mathrm{g}}({\scriptstyle
K},\eta),n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$.

  2. 2)

$\phi({n_{\mathrm{g}}({\scriptstyle
K},\eta)+1})<\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})$ and $\phi({n})$ is
decreasing and stays nonnegative on $\llbracket n_{\mathrm{g}}({\scriptstyle
K},\eta),\,n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$.

  3. 3)

$\phi({n_{\mathrm{g}}({\scriptstyle
K},\eta)+1})<\phi({n_{\mathrm{g}}({\scriptstyle K},\eta)})$ and $\phi({n})$
has a minimum in the interval $n_{\mathrm{min}}{\scriptstyle(K)}$ in
$\llbracket n_{\mathrm{g}}({\scriptstyle
K},\eta)-1,\,n_{\mathrm{d}}({\scriptstyle K},\,\eta)-1\rrbracket$ and
$\phi({n_{\mathrm{min}}{\scriptstyle(K)}})\geq 0$. Note that $\phi({n})$ is
decreasing on $\llbracket n_{\mathrm{g}}({\scriptstyle
K},\eta),\,n_{\mathrm{min}}{\scriptstyle(K)}\rrbracket$ and increasing on
$\llbracket n_{\mathrm{min}}{\scriptstyle(K)},\,n_{\mathrm{d}}({\scriptstyle
K},\,\eta)\rrbracket$.

  4. 4)

$\phi({n})$ is decreasing on $\llbracket n_{\mathrm{g}}({\scriptstyle
K},\eta),\,n_{\mathrm{d}}({\scriptstyle K},\,\eta)\rrbracket$ and
$\phi({n_{\mathrm{d}}({\scriptstyle K},\,\eta)})<0$.

We first observe that since $b^{\prime}(0)>d^{\prime}(0)$ we have

$\displaystyle\lim_{K\to\infty}\sup_{n\,\in\big{\llbracket}\frac{n_{\ell}{\scriptstyle(K)}}{2},\,n_{\ell}{\scriptstyle(K)}\big{\rrbracket}}\frac{\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}}{\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}-\eta-\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}}$
$\displaystyle=\frac{\sqrt{b^{\prime}(0)\,d^{\prime}(0)}}{b^{\prime}(0)+d^{\prime}(0)-\sqrt{b^{\prime}(0)\,d^{\prime}(0)}}<1.$

We also observe that there exists $c>0$ such that for $K$ large enough, and
any $n\in\llbracket n_{r}{\scriptstyle(K)},\,n_{\mathrm{d}}({\scriptstyle
K},\,\eta)\rrbracket$ we have

$\frac{\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}}{\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}-\eta-\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}}\leq\frac{1}{1+\frac{c\,(n-Kx_{*})^{2}}{K^{2}}}\;.$

The result follows by inspecting the monotonicity in the different cases and
using the last part of Proposition (8.1). $\;\blacksquare$

######  Theorem 4.2.

Let $\phi\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})\subset\ell^{2}$
of norm $1$, satisfying

$\mathcal{L}_{{\scriptscriptstyle K}}\phi=-\rho\,\phi$

for some real $\rho$. Then there exist $C(\phi)>0$ and an integer $r(\phi)$
such that for all $n\geq r(\phi)$ we have

$|\phi(n)|\leq C(\phi)\,2^{-n}.$

Moreover, $(\phi(n))_{n}$ does not vanish and is of constant sign.

PROOF. It follows from our hypothesis that for any $K>1$ there exists an
integer $r_{0}{\scriptstyle(K)}$ such that, for all $n\geq
r_{0}{\scriptstyle(K)}$, we have

$0<\frac{\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}}{\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n+1}+\rho-\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}}\leq\frac{1}{2}.$

We can assume that $(\phi(n))_{n}$ is a sequence of real numbers and
$\phi(r_{0}{\scriptstyle(K)})>0$. We start by proving that $(\phi(n))_{n}$ is
positive and decreasing for $n\geq r_{0}{\scriptstyle(K)}$. There are only the
following four possibilities.

  1. 1.

$\phi(r_{0}{\scriptstyle(K)}+1)\geq\phi(r_{0}{\scriptstyle(K)})$. It follows
from Proposition 8.3 that $\phi$ is increasing for $n\geq
r_{0}{\scriptstyle(K)}$, contradicting that $\phi$ has norm $1$.

  2. 2.

$\phi(r_{0}{\scriptstyle(K)}+1)<\phi(r_{0}{\scriptstyle(K)})$, and there
exists $r^{\prime}>r_{0}{\scriptstyle(K)}$ such that $\phi(r^{\prime})<0$ and
$\phi$ decreases on $\llbracket r_{0}{\scriptstyle(K)},r^{\prime}\rrbracket$,
and $\phi\geq 0$ on $\llbracket
r_{0}{\scriptstyle(K)},r^{\prime}-1\rrbracket$. Then by Proposition 8.3,
$\phi$ is decreasing for $n\geq r^{\prime}$, contradicting that $\phi$ is
normalized.

  3. 3.

$\phi(r_{0}{\scriptstyle(K)}+1)<\phi(r_{0}{\scriptstyle(K)})$, and there
exists $r^{\prime}>r_{0}{\scriptstyle(K)}$ such that $\phi(r^{\prime})<0$ and
$\phi$ is not monotonous on $\llbracket
r_{0}{\scriptstyle(K)},r^{\prime}\rrbracket$, and $\phi\geq 0$ on $\llbracket
r_{0}{\scriptstyle(K)},r^{\prime}-1\rrbracket$. Then there exists
$r^{\prime\prime}<r^{\prime}$ such that $\phi$ is decreasing on $\llbracket
r_{0}{\scriptstyle(K)},r^{\prime\prime}\rrbracket$, and such that
$\phi(r^{\prime\prime}+1)\geq\phi(r^{\prime\prime})$. If
$\phi(r^{\prime\prime})>0$, then we are in case 1. If
$\phi(r^{\prime\prime}+1)>0$, it follows from Proposition 8.3 that $\phi$ is
increasing for $n\geq r^{\prime\prime}+1$, contradicting that $\phi$ has norm
$1$. If $\phi(r^{\prime\prime\prime}+1)=\phi(r^{\prime\prime\prime})=0$ then
$\phi$ is the null sequence as solution of a second order equation, which
leads to a contradiction.

  4. 4.

$\phi(r_{0}{\scriptstyle(K)}+1)<\phi(r_{0}{\scriptstyle(K)})$, $\phi\geq 0$.
Suppose that there exists a local minimum at $r^{\prime\prime\prime}$
(finite). Then if
$\phi(r^{\prime\prime\prime}+1)\geq\phi(r^{\prime\prime\prime})>0$, we are in
case 1. If $\phi(r^{\prime\prime\prime}+1)>\phi(r^{\prime\prime\prime})=0$,
then it follows from Proposition 8.3 that $\phi$ is increasing for $n\geq
r^{\prime\prime\prime}+1$, contradicting that $\phi$ has norm $1$. If
$\phi(r^{\prime\prime\prime}+1)=\phi(r^{\prime\prime\prime})=0$ then $\phi$ is
the null sequence, which leads to a contradiction.

Therefore $\phi$ is striclty positive and monotone decreasing. The result then
follows by using the last part of Proposition 8.1.

$\;\blacksquare$

Let us now prove two key lemmas.

######  Lemma 4.3.

Let $\phi_{{\scriptscriptstyle
K}}\in\mathrm{Dom}\big{(}\mathcal{L}_{K}\big{)}$ be a normalized sequence such
that

$\mathcal{L}_{{\scriptscriptstyle
K}}\phi^{{\scriptscriptstyle(K)}}=-\rho^{{\scriptscriptstyle(K)}}\phi^{{\scriptscriptstyle(K)}}.$

Assume that there exists a diverging sequence $(K_{p})$ such that

$0\leq\lim_{p\to\infty}\rho^{{\scriptscriptstyle(K_{p})}}=\rho_{*}<+\infty$

and

$\limsup_{p\to\infty}\big{\|}\phi^{{\scriptscriptstyle(K_{p})}}{\mathds{1}}_{\\{\cdot\,\leq
n_{l}(K_{p})\\}}\big{\|}_{\ell^{2}}>0.$

Then $\rho_{*}\in S_{2}$ and there exists a diverging subsequence
$\big{(}K_{p_{\ell}}\big{)}$ such that the limit

$\lim_{\ell\to\infty}\phi^{(K_{p_{\ell}})}=\phi_{*}$

exists in $\ell^{2}$, $\|\phi_{*}\|_{\ell^{2}}>0$ and $\phi_{*}$ is an
eigenvector of $\mathcal{M}_{0}$ with eigenvalue $-\rho_{*}$.

PROOF. Let $(p_{\ell})$ be a diverging sequence of integers such that

$\lim_{\ell\to\infty}\big{\|}\phi^{(K_{p_{\ell}})}\;{\mathds{1}}_{\\{\cdot\,\leq
n_{l}(K)\\}}\big{\|}_{\ell^{2}}=\limsup_{p\to\infty}\big{\|}\phi^{(K_{p})}\;{\mathds{1}}_{\\{\cdot\,\leq
n_{l}(K_{p})\\}}\big{\|}_{\ell^{2}}\;.$

We define for each $\ell$ a normalized sequences in $\ell^{2}$ by

$\psi_{\ell}(n)=\frac{\phi^{(K_{p_{\ell}})}(n)\;{\mathds{1}}_{\\{n\,\leq
n_{l}(K_{p_{\ell}})\\}}}{\big{\|}\phi^{(K_{p_{\ell}})}\;{\mathds{1}}_{\\{\cdot\,\leq
n_{l}(K_{p_{\ell}})\\}}\big{\|}_{\ell^{2}}}\,.$

It is easy to verify using Proposition 4.1 that
$\psi_{\ell}\in\mathrm{Dom}(\mathcal{M}_{0})$ and

$\big{\|}\mathcal{M}_{0}\psi_{\ell}+\rho^{(K_{p_{\ell}})}\psi_{\ell}\|_{\ell^{2}}\leq\mathcal{O}(1)\;\frac{(\log
K_{p_{\ell}})^{2}\,\mathrm{e}^{-a\,(\log
K_{p_{\ell}})^{2}}}{\big{\|}\phi^{(K_{p_{\ell}})}\;{\mathds{1}}_{\\{\cdot\,\leq
n_{l}(K_{p_{\ell}})\\}}\big{\|}_{\ell^{2}}}\;.$

The first result follows from Proposition A.1 since the r.h.s. tends to zero.

The second result follows from Proposition A.2 since the spectrum of
$\mathcal{M}_{0}$ is discrete and simple by Theorem 7.1. $\;\blacksquare$

######  Lemma 4.4.

Let $\phi^{(K)}\in\mathrm{Dom}\big{(}\mathcal{L}_{K}\big{)}$ be a normalized
sequence such that

$\mathcal{L}_{{\scriptscriptstyle
K}}\phi^{{\scriptscriptstyle(K)}}=-\rho^{{\scriptscriptstyle(K)}}\;\phi^{{\scriptscriptstyle(K)}}.$

Let us assume that there exists a diverging subsequence $(K_{p})$ such that

$0\leq\lim_{p\to\infty}\rho^{(K_{p})}=\rho_{*}<+\infty$

and

$\lim_{p\to\infty}\big{\|}\phi^{(K_{p})}\;{\mathds{1}}_{\\{\cdot\,\geq
n_{r}(K_{p})\\}}\big{\|}_{\ell^{2}}>0.$

Then $\rho_{*}\in S_{1}$ and there exists a diverging subsequence of integers
$(p_{\ell})$ such that

$\lim_{\ell\to\infty}\frac{Q_{K_{p_{\ell}}}\phi^{(K_{p_{\ell}})}}{\big{\|}Q_{K_{p_{\ell}}}\phi^{(K_{p_{\ell}})}\big{\|}_{L^{2}}}=\psi_{*}$

exists in $L^{2}$, $\|\psi_{*}\|_{L^{2}}=1$, and $\psi_{*}$ is an eigenvector
of $\mathcal{H}_{*}$ with eigenvalue $-\rho_{*}$.

PROOF. We define for each $p$ a normalized sequence in $\ell^{2}$ by

$\psi^{(K_{p})}(n)=\frac{\phi^{(K_{p})}(n)\;{\mathds{1}}_{\\{n\,\geq\,n_{r}(K_{p})\\}}}{\big{\|}\phi^{(K_{p})}\;{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}(K_{p})\\}}\big{\|}_{\ell^{2}}}\;.$

It is easy to verify using Proposition 4.1 that
$\psi^{(K_{p})}\in\mathrm{Dom}\big{(}\mathcal{L}_{K_{p}})$ and

$\big{\|}\mathcal{L}_{K_{p}}\psi^{(K_{p})}+\rho^{(K_{p})}\psi^{(K_{p})}\|_{\ell^{2}}\leq\mathcal{O}(1)\;\frac{K_{p}\,\mathrm{e}^{-a\,(\log
K_{p})^{2}}}{\big{\|}\phi^{(K_{p})}\;{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}(K_{p})\\}}\big{\|}_{\ell^{2}}}\;.$
(4.2)

We apply Proposition 6.6, Lemma 6.2, Lemma 6.3, and Theorem 6.1 to conclude
that there exists a diverging sequence of integers $(p_{\ell})$ such that the
sequence of functions $Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}$ converges in
$L^{2}$ to a normalised fonction $\psi_{*}$.

Let $u\in\mathscr{D}$. We have from (4.2)

$\lim_{\ell\to\infty}\big{\langle}P_{K_{p_{\ell}}}u\,,\,\mathcal{L}_{K_{p_{\ell}}}\,\psi^{(K_{p_{\ell}})}+\rho^{(K_{p_{\ell}})}\,\psi^{(K_{p_{\ell}})}\big{\rangle}_{\ell^{2}}=0$

hence (since
$P_{K_{p_{\ell}}}\,u\in\mathrm{Dom}\big{(}\mathcal{L}_{K_{p_{\ell}}}\big{)}$)

$\lim_{\ell\to\infty}\big{\langle}\mathcal{L}_{K_{p_{\ell}}}\,P_{K_{p_{\ell}}}u\,,\,\psi^{(K_{p_{\ell}})}\big{\rangle}_{\ell^{2}}=-\rho_{*}\lim_{\ell\to\infty}\big{\langle}P_{K_{p_{\ell}}}u\,,\,\psi^{(K_{p_{\ell}})}\big{\rangle}_{\ell^{2}}\;.$

From the isometric property of $Q$ (see Lemma 3.1) we get

$\displaystyle\lim_{\ell\to\infty}\big{\langle}Q_{K_{p_{\ell}}}\mathcal{L}_{K_{p_{\ell}}}\,P_{K_{p_{\ell}}}u\,,\,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big{\rangle}_{L^{2}}$
$\displaystyle=-\rho_{*}\lim_{\ell\to\infty}\big{\langle}Q_{K_{p_{\ell}}}P_{K_{p_{\ell}}}u\,,\,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big{\rangle}_{L^{2}}\;.$

In other words

$\lim_{\ell\to\infty}\big{\langle}\mathscr{L}_{K_{p_{\ell}}}u,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big{\rangle}_{L^{2}}=-\rho_{*}\lim_{\ell\to\infty}\big{\langle}Q_{K_{p_{\ell}}}P_{K_{p_{\ell}}}u,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big{\rangle}_{L^{2}}.$

Using the convergence in $L^{2}$ of
$\big{(}\,Q_{K_{p_{\ell}}}\psi^{(K_{p_{\ell}})}\big{)}_{\ell}$ to $\psi_{*}$,
Proposition 3.3, and Lemma 3.2, we get for all $u\in\mathscr{D}$,

$\big{\langle}\mathcal{H}_{*}u,\psi_{*}\big{\rangle}_{L^{2}}=-\rho_{*}\big{\langle}u\,,\,\psi_{*}\big{\rangle}_{L^{2}}\;.$

Since $\mathscr{D}$ is dense in the domain of the self-adjoint operator
$\mathcal{H}_{*}$, we conclude that $\psi_{*}$ is an eigenvector of
$\mathcal{H}_{*}$. $\;\blacksquare$

We now state three key propositions.

######  Proposition 4.5.

Let $j$ be fixed and let $(K_{p})$ be a diverging sequence such that

$\lim_{p\to\infty}\rho_{j}^{{\scriptscriptstyle(K_{p})}}=\rho_{*}\;.$

Let $\phi_{j}^{(K_{p})}$ be a normalized eigenvector of
$\mathcal{L}_{{\scriptscriptstyle K}_{p}}$ with eigenvalue
$-\rho_{j}^{{\scriptscriptstyle(K_{p})}}$. Then there exists an infinite
sequence of integers $(p_{\ell})$ such that

  * •

$\phi_{j}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,<\,n_{l}({\scriptscriptstyle
K}_{p_{\ell}})\\}}\xrightarrow[]{\ell^{2}}\phi_{*}$, where $\phi_{*}$ is
either the null sequence or an eigenvector of $\mathcal{M}_{0}$ with
eigenvalue $-\rho_{*}$, namely $\rho_{*}\in S_{2}$.

  * •

$Q_{K_{p_{\ell}}}\phi_{j}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}({\scriptscriptstyle
K}_{p_{\ell}})\\}}\xrightarrow[]{L^{2}}\varphi_{*}$, where $\varphi_{*}$ is
either the null function or an eigenvector of $\mathcal{H}_{*}$ with
eigenvalue $-\rho_{*}$, namely $\rho_{*}\in S_{1}$.

Moreover, $\|\phi_{*}\|_{\ell^{2}}^{2}+\|\varphi_{*}\|_{L^{2}}^{2}=1$.

PROOF. We have either

$\lim_{p\to\infty}\Big{\|}\phi_{j}^{(K_{p})}\,{\mathds{1}}_{\\{\cdot\,<n_{l}(K_{p})\\}}\Big{\|}_{\ell^{2}}=0$

or there exists an infinite sequence of integers $(p_{\ell})$ such that

$\lim_{\ell\to\infty}\Big{\|}\phi_{j}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,<n_{l}(K_{p_{\ell}})\\}}\Big{\|}_{\ell^{2}}>0\,.$

The first statement of the proposition follows from Lemma 4.3 applied to the
diverging sequence $(K_{p_{\ell}})$. Similarly, either

$\lim_{p\to\infty}\Big{\|}\phi_{j}^{(K_{p})}\,{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}{\scriptstyle(K)}\\}}\Big{\|}_{\ell^{2}}=0$

or there exists an infinite sequence of integers $(p_{\ell})$ such that

$\lim_{\ell\to\infty}\Big{\|}\phi_{j}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}(K_{p_{\ell}})\\}}\Big{\|}_{\ell^{2}}>0.$

The second statement of the proposition follows from Lemma 4.4 applied to the
diverging sequence $(K_{p_{\ell}})$. The last statement follows from the
normalisation $\phi_{j}^{(K_{p})}$ and Proposition 4.1. $\;\blacksquare$

######  Proposition 4.6.

Let $\rho\in S_{2}$. Then there exists a normalized vector
$v_{\rho}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ with
eigenvalue $-\rho$ such that

$\lim_{K\to\infty}\|\mathcal{L}_{{\scriptscriptstyle
K}}v_{\rho}+\rho\,v_{\rho}\|_{\ell^{2}}=0.$

Moreover

$\lim_{K\to+\infty}\big{\|}v_{\rho}{\mathds{1}}_{\\{\cdot\,>\,n_{l}({\scriptscriptstyle
K})\\}}\big{\|}_{\ell^{2}}=0.$

PROOF. Let $v_{\rho}$ be a normalized eigenvector corresponding to the
eigenvalue $-\rho$ for the operator $\mathcal{M}_{0}$ in the space $\ell^{2}$.
From the assumptions we have for $n\leq\lfloor\log K\rfloor$

$\lambda^{{\scriptscriptstyle(K)}}_{n}=b^{\prime}(0)+\mathcal{O}\left(\frac{\log
K}{K}\right)\quad\text{and}\quad\mu^{{\scriptscriptstyle(K)}}_{n}=d^{\prime}(0)+\mathcal{O}\left(\frac{\log
K}{K}\right).$

Since $\mathcal{M}_{0}v_{\rho}+\rho\,v_{\rho}=0$, the reader can easily check
that

$\|\mathcal{L}_{{\scriptscriptstyle
K}}v_{\rho}+\rho\,v_{\rho}\|_{\ell^{2}(\mathds{Z}_{{\scriptscriptstyle>0}}\cap\\{1,\ldots,\lfloor\log
K\rfloor\\})}\leq\mathcal{O}\left(\frac{\log K}{K}\right).$

Now using Theorem 7.1 we have

$v_{\rho}(n)=\sqrt{n}\,\left(\frac{d^{\prime}(0)}{b^{\prime}(0)}\right)^{\frac{n}{2}}P(n)$

where $P$ is certain polynomial. Hence there exists $c_{v}>0$ such that for
any $n\in\mathds{Z}_{{\scriptscriptstyle>0}}$, $|v_{\rho}(n)|\leq
c_{v}\left(\frac{d^{\prime}(0)}{b^{\prime}(0)}\right)^{\frac{n}{4}}$. By (2.1)
there exists $c>0$ such $b(x)+d(x)\leq c\,\mathrm{e}^{x}$ for all $x\geq 0$.
This implies that $\lambda_{n}^{{\scriptscriptstyle(K_{p})}}\leq
cK_{p}\,\mathrm{e}^{n/K_{p}}$ and $\mu_{n}^{{\scriptscriptstyle(K_{p})}}\leq
cK_{p}\,\mathrm{e}^{n/K_{p}}$. The reader can easily check that

$\|\mathcal{L}_{{\scriptscriptstyle
K}}v_{\rho}+\rho\,v_{\rho}\|_{\ell^{2}(\mathds{Z}_{{\scriptscriptstyle>0}}\cap\\{\lfloor\log
K\rfloor+1,\ldots,\infty\\})}\leq\mathcal{O}(1)\left(\frac{d^{\prime}(0)}{b^{\prime}(0)}\right)^{\frac{\log
K}{4}}.$

It follows that $v_{\rho}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle
K}})$, and (remember that $b^{\prime}(0)>d^{\prime}(0)$)

$\lim_{K\to\infty}\|\mathcal{L}_{{\scriptscriptstyle
K}}v_{\rho}+\rho\,v_{\rho}\|_{\ell^{2}}=0.$

The other statement follows at once from the exponential decay of $v$.
$\;\blacksquare$

######  Proposition 4.7.

Let $\rho\in S_{1}$. Then there exists a sequence of normalized vectors
$(\psi_{\rho}^{(K)})_{K}\subset\ell^{2}$ such that
$\psi_{\rho}^{{\scriptscriptstyle(K)}}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle
K}})$, and

$\lim_{K\to\infty}\big{\|}\mathcal{L}_{{\scriptscriptstyle
K}}\psi_{\rho}^{{\scriptscriptstyle(K)}}+\rho\,\psi_{\rho}^{{\scriptscriptstyle(K)}}\big{\|}_{\ell^{2}}=0.$

Moreover

$\lim_{K\to\infty}\big{\|}\psi_{\rho}^{{\scriptscriptstyle(K)}}{\mathds{1}}_{\\{\cdot\,<n_{r}{\scriptscriptstyle(K)}\\}}\big{\|}_{\ell^{2}}=0.$

PROOF. Let $\varphi_{\rho}$ be a real normalized eigenvector of
$\mathcal{H}_{*}$ corresponding to the eigenvalue $-\rho$ (in $L^{2}$). Since
$\varphi_{\rho}$ is a (rescaled) Hermite function (see [2] or Remark 3.1), it
satisfies the hypothesis of Proposition 3.3, hence

$\lim_{K\to\infty}\big{\|}\mathscr{L}_{{\scriptscriptstyle
K}}\varphi_{\rho}+\rho\,\varphi_{\rho}\big{\|}_{L^{2}}=0.$

Using Lemma 3.2 we get

$\lim_{K\to\infty}\big{\|}\mathscr{L}_{{\scriptscriptstyle
K}}\varphi_{\rho}+\rho\;Q_{{\scriptscriptstyle K}}P_{{\scriptscriptstyle
K}}\varphi_{\rho}\big{\|}_{L^{2}}=0$

and then

$\lim_{K\to\infty}\big{\|}\mathcal{L}_{{\scriptscriptstyle
K}}\,P_{{\scriptscriptstyle K}}\varphi_{\rho}+\rho\,P_{{\scriptscriptstyle
K}}\varphi_{\rho}\big{\|}_{\ell^{2}}=0\;$

and

$\lim_{K\to\infty}\big{\|}P_{{\scriptscriptstyle
K}}\varphi_{\rho}\big{\|}_{\ell^{2}}=1.$

The first statement follows by letting
$\psi_{\rho}^{{\scriptscriptstyle(K)}}(n)=(P_{{\scriptscriptstyle
K}}\varphi_{\rho})(n)$ where $P_{{\scriptscriptstyle K}}$ is defined in (3.1).
The other statement follows from an exponential bound on the decay of
$\varphi_{\rho}$. $\;\blacksquare$

##  5 Proof of Theorems 3.5 and 3.7

###  5.1 Proof of Theorem 3.5

The proof of Theorem 3.5 is an immediate consequence of the following two
propositions.

######  Proposition 5.1.

We have $\ S_{2}\subset G$.

PROOF. The proof is by contradiction. Let $\rho\in S_{2}$ and assume
$\rho\notin G$. Then there exists $\eta>0$ be such that for $K$ large enough,
$\left[\,\rho-\eta,\rho+\eta\,\right]\cap G=\emptyset$. It follows from
Proposition 4.6 that there exists a normalized vector $v$ in $\ell^{2}$, such
that $v\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$, and

$\lim_{K\to\infty}\|\mathcal{L}_{{\scriptscriptstyle
K}}v+\rho\,v\|_{\ell^{2}}=0.$

Therefore, using Proposition A.1, we obtain a contradiction, hence the proof
is finished. $\;\blacksquare$

######  Proposition 5.2.

We have $\ S_{1}\subset G$.

PROOF. The proof is by contradiction. Let $\rho\in S_{1}$ and assume
$\rho\notin G$. Then there exists $\eta>0$ be such that for $K$ large enough,
$[\rho-\eta,\rho+\eta]\cap G=\emptyset$. It follows from Proposition 4.7 that
there exists a sequence of normalized vectors
$(\psi^{(K)})_{K}\subset\ell^{2}$ such that
$\psi^{(K)}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ and

$\lim_{K\to\infty}\big{\|}\mathcal{L}_{{\scriptscriptstyle
K}}\psi^{{\scriptscriptstyle(K)}}+\rho\,\psi^{{\scriptscriptstyle(K)}}\big{\|}_{\ell^{2}}=0.$

Therefore, using Proposition A.1, we obtain a contradiction, hence the proof
is finished. $\;\blacksquare$

###  5.2 Proof of Theorem 3.7

The first statement follows at once from Proposition 4.5.
The proof of (3.6) is by contradiction. Assume

$\liminf_{p\to\infty}\;\min\left\\{\bigg{|}\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|},\bigg{|}\rho_{j-1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|}\right\\}=0.$

Assume $\rho_{*}\in S_{1}\backslash S_{2}$ and let $(p_{\ell})$ be a diverging
sequence of positive integers such that $\rho_{i}^{({\scriptscriptstyle
K}_{p_{\ell}})}\to\rho_{*}$ and $\rho_{i+1}^{({\scriptscriptstyle
K}_{p_{\ell}})}\to\rho_{*}$ (where $i=j$ or $i=j-1$). Let
$\phi_{i}^{({\scriptscriptstyle K}_{p_{\ell}})}$ be a normalized eigenvector
of $\mathcal{L}_{K_{p_{\ell}}}$ corresponding to the eigenvalue
$-\rho_{i}^{({\scriptscriptstyle K}_{p_{\ell}})}$. We define
$\phi^{({\scriptscriptstyle K}_{p_{\ell}})}_{i+1}$ similarly. We claim that

$\limsup_{\ell\to+\infty}\Big{\|}\phi_{i}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,<\,n_{l}({\scriptscriptstyle
K}_{p_{\ell}})\\}}\Big{\|}_{\ell^{2}}=0.$

Otherwise Proposition 4.5 would imply that $\rho_{*}\in S_{2}$, a
contradiction. By Proposition 4.1 we have

$\lim_{\ell\to+\infty}\Big{\|}\phi_{i}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell}})\\}}\Big{\|}_{\ell^{2}}=1.$

This implies 1-a).
By a similar argument, we have

$\lim_{\ell\to+\infty}\Big{\|}\phi_{i+1}^{(K_{p_{\ell}})}\,{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell}})\\}}\Big{\|}_{\ell^{2}}=1.$

By Proposition 4.5, there exists a diverging sequence of integers $(\ell_{r})$
such that

$\Big{\|}Q_{K_{p_{\ell_{r}}}}\Big{(}\phi_{i}^{(K_{p_{\ell_{r}}})}\,{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{r}}})\\}}\Big{)}-\varphi_{*}\Big{\|}_{L^{2}}\to 0$

where $\varphi_{*}\in\mathrm{Dom}(\mathcal{H}_{*})$ is a normalized
eigenfunction of $\mathcal{H}_{*}$ corresponding to the eigenvalue
$-\rho_{*}$. By Proposition 4.5 again, there exists a diverging sequence of
integers $(r_{s})$ such that

$\Big{\|}Q_{K_{p_{\ell_{r_{s}}}}}\Big{(}\phi_{i+1}^{(K_{p_{\ell_{r_{s}}}})}\,{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{r_{s}}}})\\}}\Big{)}-\varphi^{\prime}_{*}\Big{\|}_{L^{2}}\to 0$

where $\varphi^{\prime}_{*}\in\mathrm{Dom}(\mathcal{H}_{*})$ is a normalized
eigenfunction of $\mathcal{H}_{*}$ corresponding to the eigenvalue
$-\rho_{*}$. Since $\phi_{i}^{(K_{p_{\ell_{r_{s}}}})}$ and
$\phi_{i+1}^{(K_{p_{\ell_{r_{s}}}})}$ are orthogonal in $\ell^{2}$, it follows
from the previous estimates that

$\lim_{s\to+\infty}\left\langle\phi_{i}^{(K_{p_{\ell_{r_{s}}}})}\,{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{r_{s}}}})\\}},\phi_{i+1}^{(K_{p_{\ell_{r_{s}}}})}\,{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{r_{s}}}})\\}}\right\rangle_{\ell^{2}}=0.$

By Lemma 3.1 we have

$\left\langle\\!Q_{K_{p_{\ell_{r_{s}}}}}\\!\\!\Big{(}\phi_{i}^{(K_{p_{\ell_{r_{s}}}})}{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{r_{s}}}})\\}}\Big{)},Q_{K_{p_{\ell_{r_{s}}}}}\\!\\!\Big{(}\phi_{i+1}^{(K_{p_{\ell_{r_{s}}}})}{\mathds{1}}_{\\{\cdot\,>\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{r_{s}}}})\\}}\Big{)}\\!\right\rangle_{\\!\\!L^{2}}\xrightarrow[s\to\infty]{}0.$

In other words, $\langle\varphi_{*},\varphi^{\prime}_{*}\rangle_{L^{2}}=0$.
This is a contradiction since $\varphi_{*}$ and $\varphi^{\prime}_{*}$ are
normalized eigenfunctions of $\mathcal{H}_{*}$ corresponding to the same
eigenvalue $-\rho_{*}$ which is simple.

The case $\rho_{*}\in S_{2}\backslash S_{1}$ is similar (using again
Proposition 4.5), so it is left to the reader.

Let us now assume that $\rho_{*}\in S_{1}\cap S_{2}$. We now prove (3.7) by
contradiction. So we assume that there exists $\delta>0$ such that

$\liminf_{p\to\infty}\;\min\left\\{\bigg{|}\rho_{j+1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|},\bigg{|}\rho_{j-1}^{{\scriptscriptstyle(K_{p})}}-\rho_{*}\bigg{|}\right\\}>\delta.$

Since $\rho_{*}\in S_{1}$ and from Proposition 4.7, there exists a sequence of
normalized vectors $(\psi_{\rho_{*}}^{(K_{p})})_{p}\subset\ell^{2}$ such that
$\psi_{\rho_{*}}^{(K_{p})}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle
K}_{p}})$ for all $p$, and

$\lim_{p\to\infty}\Big{\|}\mathcal{L}_{{\scriptscriptstyle
K}_{p}}\psi_{\rho_{*}}^{(K_{p})}-\rho_{*}\psi_{\rho_{*}}^{(K_{p})}\Big{\|}_{\ell^{2}}=0.$

We also have (since $\rho^{({\scriptscriptstyle K}_{p})}_{j}\to\rho_{*}$) that

$\lim_{p\to\infty}\Big{\|}\mathcal{L}_{{\scriptscriptstyle
K}_{p}}\psi_{\rho_{*}}^{(K_{p})}-\rho^{({\scriptscriptstyle
K}_{p})}_{j}\psi_{\rho_{*}}^{(K_{p})}\Big{\|}_{\ell^{2}}=0.$

For each $p$, let $\phi^{({\scriptscriptstyle K}_{p})}_{j}$ be a normalized
eigenvector of $\mathcal{L}_{{\scriptscriptstyle K}_{p}}$ corresponding to the
eigenvalue $-\rho^{({\scriptscriptstyle K}_{p})}_{j}$. By using Lemma A.2 with
$A=\mathcal{L}_{{\scriptscriptstyle K}_{p}}$, we deduce that there exists a
sequence of real numbers $(\theta_{p})_{p}$ such that

$\lim_{p\to\infty}\Big{\|}\psi_{\rho_{*}}^{(K_{p})}-\mathrm{e}^{\mathrm{i}\theta_{p}}\phi^{({\scriptscriptstyle
K}_{p})}_{j}\Big{\|}_{\ell^{2}}=0.$ (5.1)

Since $\rho_{*}\in S_{2}$ and from Proposition 4.6, there exists a normalized
vector $v_{\rho_{*}}\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}_{p}})$
for all $p$ such that

$\lim_{p\to\infty}\|\mathcal{L}_{{\scriptscriptstyle
K}_{p}}v_{\rho_{*}}+\rho_{*}v_{\rho_{*}}\|_{\ell^{2}}=0.$

We also have

$\lim_{p\to\infty}\Big{\|}\mathcal{L}_{{\scriptscriptstyle
K}_{p}}v_{\rho_{*}}+\rho^{({\scriptscriptstyle
K}_{p})}_{j}v_{\rho_{*}}\Big{\|}_{\ell^{2}}=0.$

By using Lemma A.2 with $A=\mathcal{L}_{{\scriptscriptstyle K}_{p}}$, we
deduce that there exists a sequence of real numbers
$(\theta^{\prime}_{p})_{p}$ such that

$\lim_{p\to\infty}\Big{\|}v_{\rho_{*}}-\mathrm{e}^{\mathrm{i}\theta^{\prime}_{p}}\phi^{({\scriptscriptstyle
K}_{p})}_{j}\Big{\|}_{\ell^{2}}=0.$ (5.2)

Moreover, from Propositions 4.7 and 4.6, it follows that

$\lim_{p\to\infty}\Big{\|}\psi_{\rho_{*}}^{(K_{p})}{\mathds{1}}_{\\{\cdot\,<\,n_{r}({\scriptscriptstyle
K}_{p})\\}}\Big{\|}_{\ell^{2}}=0\quad\text{and}\quad\lim_{K\to+\infty}\big{\|}v_{\rho_{*}}{\mathds{1}}_{\\{\cdot\,>\,n_{l}({\scriptscriptstyle
K})\\}}\big{\|}_{\ell^{2}}=0$

which implies (using Proposition 4.1) that

$\lim_{p\to\infty}\langle\psi_{\rho_{*}}^{(K_{p})},v_{\rho_{*}}\rangle=0.$

This is a contradiction with (5.1) and (5.2). Finally, we prove (3.8) by
contradiction. So we assume that there exists a diverging sequence of integers
$(p_{\ell})$ such that

$\lim_{\ell\to+\infty}\Big{|}\,\rho_{j+1}^{{\scriptscriptstyle(K_{p_{\ell}})}}-\rho_{*}\Big{|}+\Big{|}\,\rho_{j-1}^{{\scriptscriptstyle(K_{p_{\ell}})}}-\rho_{*}\Big{|}=0.$

We now apply Proposition 4.5 with $j$ and $K_{p_{\ell}}$. Hence there exists a
diverging sequence of integers $(\ell_{s})$ such that

$\phi_{j}^{(K_{p_{\ell_{s}}})}\,{\mathds{1}}_{\\{\cdot\,<\,n_{l}({\scriptscriptstyle
K}_{p_{\ell_{s}}})\\}}\xrightarrow[]{\ell^{2}}\phi_{*,1}$

and

$Q_{K_{p_{\ell_{s}}}}\phi_{j}^{(K_{p_{\ell_{s}}})}\,{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{s}}})\\}}\xrightarrow[]{L^{2}}\varphi_{*,1}$

and we have $\|\phi_{*,1}\|_{\ell^{2}}^{2}+\|\varphi_{*,1}\|_{L^{2}}^{2}=1$.

We now apply Proposition 4.5 with $j-1$ and $K_{p_{\ell_{s}}}$. Hence there
exists a diverging sequence of integers $(s_{r})$ such that

$\phi_{j-1}^{(K_{p_{\ell_{s_{r}}}})}\,{\mathds{1}}_{\\{\cdot\,<\,n_{l}({\scriptscriptstyle
K}_{p_{\ell_{s_{r}}}})\\}}\xrightarrow[]{\ell^{2}}\phi_{*,2}$

and

$Q_{K_{p_{\ell_{s_{r}}}}}\phi_{j-1}^{(K_{p_{\ell_{s_{r}}}})}\,{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{s_{r}}}})\\}}\xrightarrow[]{L^{2}}\varphi_{*,2}$

and we have $\|\phi_{*,2}\|_{\ell^{2}}^{2}+\|\varphi_{*,2}\|_{L^{2}}^{2}=1$.

We now apply Proposition 4.5 with $j+1$ and $K_{p_{\ell_{s_{r}}}}$. Hence
there exists a diverging sequence of integers $(r_{q})$ such that

$\phi_{j+1}^{(K_{p_{\ell_{s_{r_{q}}}}})}\,{\mathds{1}}_{\\{\cdot\,<\,n_{l}({\scriptscriptstyle
K}_{p_{\ell_{s_{r_{q}}}}})\\}}\xrightarrow[]{\ell^{2}}\phi_{*,3}$

and

$Q_{K_{p_{\ell_{s_{r_{q}}}}}}\phi_{j+1}^{(K_{p_{\ell_{s_{r_{q}}}}})}\,{\mathds{1}}_{\\{\cdot\,\geq\,n_{r}({\scriptscriptstyle
K}_{p_{\ell_{s_{r_{q}}}}})\\}}\xrightarrow[]{L^{2}}\varphi_{*,3}$

and we have $\|\phi_{*,3}\|_{\ell^{2}}^{2}+\|\varphi_{*,3}\|_{L^{2}}^{2}=1$.

Moreover, since
$\phi_{j-1}^{(K_{p_{\ell_{s_{r_{q}}}}})},\phi_{j}^{(K_{p_{\ell_{s_{r_{q}}}}})},\phi_{j+1}^{(K_{p_{\ell_{s_{r_{q}}}}})}$
are pairwise orthogonal, we have

$\left\langle\phi_{*,m},\phi_{*,m^{\prime}}\right\rangle_{\ell^{2}}+\left\langle\varphi_{*,m},\varphi_{*,m^{\prime}}\right\rangle_{L^{2}}=0,\quad\forall
m\neq m^{\prime}.$ (5.3)

The linear subspace of $\ell^{2}$ spanned by
$\phi_{*,1},\phi_{*,2},\phi_{*,3}$ is of dimension at most one because they
are eigenvectors of $\mathcal{M}_{0}$ for the same simple eigenvalue
$-\rho_{*}$. The linear subspace of $L^{2}$ spanned by
$\varphi_{*,1},\varphi_{*,2},\varphi_{*,3}$ is of dimension at most one
because they are eigenfunctions of $\mathcal{H}_{*}$ for the same simple
eigenvalue $-\rho_{*}$. Therefore the subspace of $\ell^{2}\oplus L^{2}$
spanned by the three vectors $(\phi_{*,1},\varphi_{*,1})$,
$(\phi_{*,2},\varphi_{*,2})$, $(\phi_{*,3},\varphi_{*,3})$ is of dimension at
most two. However, these three vectors are normalized and pairwise orthogonal
by (5.3). We thus arrive at a contradiction.

##  6 Fréchet-Kolmogorov-Riesz compactness criterion and Dirichlet form

###  6.1 Fréchet-Kolmogorov-Riesz compactness criterion

We recall the Fréchet-Kolmogorov-Riesz compactness criterion in $L^{2}$.

######  Theorem 6.1.

Let $(f_{p})_{p}$ be a normalized sequence in $L^{2}$ such that the following
two conditions are satisfied.

  1. (i)

There exists $\epsilon_{0}>0$ such that there exists a function
$R(\epsilon)>0$ on $(0,\epsilon_{0})$ such that

$\sup_{p}\int_{\\{|x|>R(\epsilon)\\}}\big{|}f_{p}(x)\big{|}^{2}\mathrm{d}x\leq\epsilon.$

  2. (ii)

There exits a positive function $\alpha$ on $]0,1]$ satisfying

$\lim_{y\searrow\,0}\alpha(y)=0$

and such that for any $p$ and $y\in\left]-1,1\right]$

$\int\big{|}f_{p}(x+y)-f_{p}(x)\big{|}^{2}\mathrm{d}x\leq\alpha(|y|).$

Then one can extract from $(f_{p})_{p}$ a convergent subsequence in $L^{2}$.

We refer to Remark 5 on page 387 in [8]. The following Lemmas provide
expressions for $R(\epsilon)$ and $\alpha(y)$ in our case. Recall that the
potential $V_{n}(K)$ and $n_{r}(K)$ have been defined in Section 4 (see
(4.1)).

######  Lemma 6.2.

Let $C>0$. Let $\mathscr{F}_{C,\,K}$ be the set of normalized sequences
$(\phi(n))_{n}$ in $\ell^{2}$ such that $\phi(n)=0$ for any $n<n_{r}(K)$ and

$\sum_{n=n_{r}(K)}^{\infty}\big{(}1\vee
V_{n}(K)\big{)}\big{)}\,\phi^{2}(n)\leq C\;.$

Then, for any $\phi\in\mathscr{F}_{C,K}$, the function $Q_{{\scriptscriptstyle
K}}\phi(x)$ satisfies condition (i) in Theorem 6.1 with

$R(\epsilon)=\frac{C}{\epsilon}$

for any $0<\epsilon<1$.

PROOF. We are going to prove that there exist $R_{0}>1$ and $K_{0}>4$ such
that for any $K>K_{0}$, any $R>R_{0}$ and any $\phi\in\mathscr{F}_{C,K}$

$\int_{\\{|x|>R\\}}\big{(}Q_{{\scriptscriptstyle
K}}\phi(x)\big{)}^{2}\mathrm{d}x\leq\frac{C}{R}.$

We observe that

$\int_{\\{|x|>R\\}}\big{(}Q_{{\scriptscriptstyle
K}}\phi(x)\big{)}^{2}\mathrm{d}x\leq\sum_{n:|n-Kx_{*}|>R\,\sqrt{K}-1}\phi^{2}(n).$

Our aim is to prove that the right-hand side of the above inequality is
bounded above by $C/R$.

It follows from the hypotheses on $\lambda^{{\scriptscriptstyle(K)}}_{n}$ and
$\mu^{{\scriptscriptstyle(K)}}_{n}$ that there exist constants $K_{0}>4$,
$1>C_{2}>0$, $\Gamma>0$ and $\zeta>0$, such that for any $K>K_{0}$ there
exists an integer $\Gamma\,K>m_{*}{\scriptstyle(K)}>2\,Kx_{*}$ (hence of order
$K$) such that $\mu^{{\scriptscriptstyle(K)}}_{n}>\zeta\,n$ for any $n\geq
m_{*}{\scriptstyle(K)}$ and

$\displaystyle\sum_{n=n_{r}(k)}^{m_{*}{\scriptstyle(K)}}(1\vee
V_{n}(K))\;\phi^{2}(n)$ $\displaystyle\geq
C_{2}\,K^{-1}\sum_{n=n_{r}(K)}^{m_{*}{\scriptstyle(K)}}(n-Kx_{*})^{2}\phi^{2}(n)$
$\displaystyle\sum_{n=m_{*}{\scriptstyle(K)}+1}^{\infty}(1\vee
V_{n}(K))\;\phi^{2}(n)$ $\displaystyle\geq
C_{2}\,\sum_{n=m_{*}{\scriptstyle(K)}+1}^{\infty}n\,\phi^{2}(n)\;.$

These estimates imply the following bounds for any integer $L>0$

$\displaystyle\sum_{\\{n_{r}(K)\leq n\leq
m_{*}{\scriptstyle(K)}\\}\cap\\{|n-Kx_{*}|>L\\}}\phi^{2}(n)$
$\displaystyle\leq\frac{KC}{C_{2}\,L^{2}}\,{\mathds{1}}_{\\{L<m_{*}{\scriptstyle(K)}\\}}$
(6.1)
$\displaystyle\sum_{\\{n>m_{*}{\scriptstyle(K)}\\}\cap\\{|n-Kx_{*}|>L\\}}\phi^{2}(n)$
$\displaystyle\leq\frac{C}{C_{2}(L\vee m_{*}{\scriptstyle(K)})}\;.$ (6.2)

We now replace $L$ with $R\,\sqrt{K}-1$ in the above estimates.

Let $K>4$ be fixed. We distinguish two cases according to the value of $R$.

  1. 1)

$1\leq R<m_{*}{\scriptstyle(K)}/\sqrt{K}$. Then $\sqrt{K}-1\leq
L<m_{*}{\scriptstyle(K)}\leq\Gamma K$. Since $L=R\sqrt{K}-1>R\sqrt{K}/2$
(because $K>4$), we have

$\sum_{\\{|n-Kx_{*}|>R\,\sqrt{K}-1\\}}\phi^{2}(n)\leq\frac{4\,\,C}{R^{2}}+\frac{C}{C_{2}m_{*}{\scriptstyle(K)}}\leq\frac{4C}{C_{2}R^{2}}+\frac{C\,\Gamma}{C_{2}R^{2}}\leq\frac{C}{R}$

if $R>C_{2}^{-1}\big{(}4+\,\Gamma)$.

  2. 2)

$R\geq m_{*}{\scriptstyle(K)}/\sqrt{K}$. Then $L\geq m_{*}{\scriptstyle(K)}$.
We get

$\sum_{\\{|n-Kx_{*}|>R\,\sqrt{K}-1\\}}\phi^{2}(n)\leq\frac{C}{C_{2}L}\leq\frac{2C}{R\,C_{2}\sqrt{K}}\leq\frac{C}{R}$

if $C_{2}^{2}K>4$.

We define $K_{0}=5+4\,C_{2}^{-2}$ and $R_{0}=1+C_{2}^{-1}\big{(}4+\,\Gamma)$.
The result follows. $\;\blacksquare$

######  Lemma 6.3.

Let $(\phi^{{\scriptscriptstyle(K)}})_{K}$ be a sequence of normalized
elements of $\ell^{2}$ such that ${\phi^{{\scriptscriptstyle(K)}}(n)}=0$ for
$n\leq n_{r}(K)$. Assume also that there exists $C>0$ such that

$\displaystyle\sup_{K>1}\left\\{\sum_{n=n_{r}{\scriptscriptstyle(K)}}^{\infty}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\big{(}{\phi^{{\scriptscriptstyle(K)}}(n+1)}-{\phi^{{\scriptscriptstyle(K)}}(n)}\big{)}^{2}\right.$
$\displaystyle\left.+\sum_{n=n_{r}{\scriptscriptstyle(K)}}^{\infty}(1\vee
V_{n}(K))\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(n)}\big{)}^{2}\right\\}\leq
C.$

Then there exits a positive constant $\tilde{C}$ such that for any $|h|\leq 1$
and any $K>1$

$\int\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x\leq\alpha(h)=\tilde{C}\,|h|\,.$

Hence, for any $({\phi^{{\scriptscriptstyle(K)}}})_{K}$ satisfying the above
assumptions, the sequence of functions $(Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}})_{K}$ satisfies condition ii) in Theorem
6.1 with

$\alpha(y)=\tilde{C}y\;.$

PROOF. It is enough to consider the case $0<h<1$. We first consider the case
$0<h\leq 1/\sqrt{K}$. We have

$\int\\!\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x=\sum_{q\geq
1}\int_{I^{(K)}_{q}}\\!\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x.$

Since

$Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)=K^{\frac{1}{4}}\,\sum_{q\geq
1}{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q}}(x)$

and since the intervals $I^{{\scriptscriptstyle(K)}}_{q}$ are disjoint, we get

$\displaystyle\int\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x$
$\displaystyle=K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I^{{\scriptscriptstyle(K)}}_{q}}\Big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\mathds{1}}_{I^{(K)}_{q}}(x+h)$
$\displaystyle\hskip
79.6678pt+{\phi^{{\scriptscriptstyle(K)}}(q+1)}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q+1}}(x+h)-{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle(K)}}}(x)\Big{)}^{2}\mathrm{d}x$
$\displaystyle=K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I^{{\scriptscriptstyle(K)}}_{q}}\Big{(}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q}}(x+h)+\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}\big{)}^{2}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q+1}}(x+h)$
$\displaystyle\hskip
79.6678pt+\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q}}(x)-2\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\,{\mathds{1}}_{I_{q}}(x){\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q}}(x+h)$
$\displaystyle\hskip
79.6678pt-2\,{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\phi^{{\scriptscriptstyle(K)}}(q+1)}\,{\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q}}(x){\mathds{1}}_{I^{{\scriptscriptstyle(K)}}_{q+1}}(x+h)\Big{)}\mathrm{d}x$

Let us consider each term separately. Since
$\int_{I^{(K)}_{q}}\mathrm{d}x=K^{-\frac{1}{2}}$, we have

$K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I_{q}^{{\scriptscriptstyle(K)}}}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\,\mathrm{d}x=\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}.$

Then we have

$\displaystyle K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I_{q}^{{\scriptscriptstyle(K)}}}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle(K)}}}(x+h)\,\mathrm{d}x$
$\displaystyle=K^{\frac{1}{2}}\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\int_{\frac{q}{\sqrt{K}}-\frac{1}{2\sqrt{K}}-x_{*}}^{\frac{q}{\sqrt{K}}+\frac{1}{2\sqrt{K}}-x_{*}-h}\,\mathrm{d}x$
$\displaystyle=\big{(}1-hK^{\frac{1}{2}}\big{)}\,\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}.$

We also have

$\displaystyle K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I_{q}^{{\scriptscriptstyle(K)}}}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}\big{)}^{2}\,{\mathds{1}}_{I_{q+1}^{{\scriptscriptstyle(K)}}}(x+h)\,\mathrm{d}x$
$\displaystyle=K^{\frac{1}{2}}\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}\big{)}^{2}\int_{\frac{q}{\sqrt{K}}+\frac{1}{2\sqrt{K}}-x_{*}-h}^{\frac{q}{\sqrt{K}}+\frac{1}{2\sqrt{K}}-x_{*}}\mathrm{d}x$
$\displaystyle=K^{\frac{1}{2}}\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}\big{)}^{2}\,h$
$\displaystyle=K^{\frac{1}{2}}\,h\,\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}.$

Similarly

$\displaystyle-2\,K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I_{q}^{{\scriptscriptstyle(K)}}}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle(K)}}}(x){\mathds{1}}_{I_{q}}(x+h)\,\mathrm{d}x$
$\displaystyle=-2\,\big{(}1-hK^{\frac{1}{2}}\big{)}\,\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}$

and

$\displaystyle-2K^{\frac{1}{2}}\sum_{q\geq
1}\int_{I_{q}^{{\scriptscriptstyle(K)}}}{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\phi^{{\scriptscriptstyle(K)}}(q+1)}\,{\mathds{1}}_{I_{q}^{{\scriptscriptstyle(K)}}}(x){\mathds{1}}_{I_{q+1}^{{\scriptscriptstyle(K)}}}(x+h)\,\mathrm{d}x$
$\displaystyle=-2hK^{\frac{1}{2}}\sum_{q\geq
1}\,{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\phi^{{\scriptscriptstyle(K)}}(q+1)}\,.$

We rewrite the last term:

$\displaystyle-2\,h\,\sqrt{K}\,\sum_{q\geq
1}\,{\phi^{{\scriptscriptstyle(K)}}(q)}\,{\phi^{{\scriptscriptstyle(K)}}(q+1)}$
$\displaystyle=-2\,h\,\sqrt{K}\,\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}+2\,h\,\sqrt{K}\,\sum_{q\geq
1}\,{\phi^{{\scriptscriptstyle(K)}}(q)}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}-{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}.$

Summing up, we get

$\displaystyle\int\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x$
$\displaystyle=\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}+\big{(}1-h\,\sqrt{K}\big{)}\,\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}+K^{\frac{1}{2}}\,h\,\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}$
$\displaystyle\quad-2\,\big{(}1-h\,\sqrt{K}\big{)}\,\sum_{q\geq
1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}-2\,h\,\sqrt{K}\,\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}$
$\displaystyle\quad+2\,h\,\sqrt{K}\,\sum_{q\geq
1}\,{\phi^{{\scriptscriptstyle(K)}}(q)}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}-{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}$
$\displaystyle=2\,h\,\sqrt{K}\,\sum_{q\geq
1}\,{\phi^{{\scriptscriptstyle(K)}}(q)}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}-{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}.$

By Cauchy-Schwarz inequality we get

$\displaystyle\int\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x$
$\displaystyle\leq 2h\sqrt{K}\,\left(\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}-{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\right)^{\frac{1}{2}}\big{\|}{\phi^{{\scriptscriptstyle(K)}}}\big{\|}_{\ell^{2}}.$

From the assumption of the lemma and using

$\inf_{n\,\geq\big{\lfloor}\frac{Kx_{*}}{3}\big{\rfloor}-1}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}>\zeta
K$

for some $\zeta>0$ independent of $K$, we get

$K^{\frac{1}{2}}\left(\sum_{q\geq
1}\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(q+1)}-{\phi^{{\scriptscriptstyle(K)}}(q)}\big{)}^{2}\right)^{1/2}\leq\sqrt{{C\over\zeta}}.$

Therefore, since the sequence ${\phi^{{\scriptscriptstyle(K)}}}$ is
normalized, we get

$\int\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x\leq
2h\;\sqrt{{C\over\zeta}}.$

We now consider the case $1>h>1/\sqrt{K}$. Let $r=\lfloor h\sqrt{K}\rfloor$
and $h^{\prime}=h-r/\sqrt{K}$. Note that $0\leq h^{\prime}\leq 1/\sqrt{K}$. We
have

$\displaystyle\int\big{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x+h)-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}(x)\big{]}^{2}\mathrm{d}x$
$\displaystyle=\int\bigg{[}\sum_{j=1}^{r-1}\left(Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j+1}{\sqrt{K}}\Big{)}-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j}{\sqrt{K}}\Big{)}\right)$
$\displaystyle\hskip 34.14322pt+Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{r}{\sqrt{K}}+h^{\prime}\Big{)}-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{r}{\sqrt{K}}\Big{)}\bigg{]}^{2}\mathrm{d}x$
$\displaystyle\leq 2\int\bigg{[}\sum_{j=1}^{r-1}\left(Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j+1}{\sqrt{K}}\Big{)}-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j}{\sqrt{K}}\Big{)}\right)\bigg{]}^{2}\mathrm{d}x$
$\displaystyle\hskip 39.83368pt+2\int\bigg{[}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{r}{\sqrt{K}}+h^{\prime}\Big{)}-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{r}{\sqrt{K}}\Big{)}\bigg{]}^{2}\mathrm{d}x.$

We have already estimated the last term. For the first term we now observe
that
$e^{{\scriptscriptstyle(K)}}_{n}\Big{(}x+\frac{j}{\sqrt{K}}\Big{)}=e^{{\scriptscriptstyle(K)}}_{n-j}(x)$.
Therefore we can write

$\displaystyle\sum_{j=1}^{r-1}\left(Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j+1}{\sqrt{K}}\Big{)}-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j}{\sqrt{K}}\Big{)}\right)$
$\displaystyle=\sum_{j=1}^{r-1}\left(\sum_{n}{\phi^{{\scriptscriptstyle(K)}}(n)}\,e^{{\scriptscriptstyle(K)}}_{n-j-1}(x)-\sum_{n}{\phi^{{\scriptscriptstyle(K)}}(n)}\,e^{{\scriptscriptstyle(K)}}_{n-j}(x)\right)$
$\displaystyle=\sum_{j=1}^{r-1}\left(\sum_{p}e^{{\scriptscriptstyle(K)}}_{p}(x)\,\big{(}{\phi^{{\scriptscriptstyle(K)}}(p+j+1)}-{\phi^{{\scriptscriptstyle(K)}}(p+j)}\right)$
$\displaystyle=\sum_{p}e^{{\scriptscriptstyle(K)}}_{p}(x)\,\sum_{j=1}^{r-1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(p+j+1)}-{\phi^{{\scriptscriptstyle(K)}}(p+j)}\big{)}.$

This implies

$\displaystyle\int\bigg{[}\sum_{j=1}^{r-1}Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j+1}{\sqrt{K}}\Big{)}-Q_{{\scriptscriptstyle
K}}{\phi^{{\scriptscriptstyle(K)}}}\Big{(}x+\frac{j}{\sqrt{K}}\Big{)}\bigg{]}^{2}\mathrm{d}x$
$\displaystyle=\sum_{p}\left(\sum_{j=1}^{r-1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(p+j+1)}-{\phi^{{\scriptscriptstyle(K)}}(p+j)}\big{)}\right)^{2}$
$\displaystyle\leq
r\,\sum_{p}\sum_{j=1}^{r-1}\big{(}{\phi^{{\scriptscriptstyle(K)}}(p+j+1)}-{\phi^{{\scriptscriptstyle(K)}}(p+j)}\big{)}^{2}$
$\displaystyle=r^{2}\,\sum_{p}\big{(}{\phi^{{\scriptscriptstyle(K)}}(p+1)}-{\phi^{{\scriptscriptstyle(K)}}(p)}\big{)}^{2}\leq\frac{C}{\zeta}\;\frac{r^{2}}{K}$

as we have seen before. We observe that $r^{2}/K\leq h^{2}\leq h$ since $h\leq
1$, and the result follows by taking
$\tilde{C}=2\sqrt{\frac{C}{\zeta}}+\frac{C}{\zeta}$. $\;\blacksquare$

###  6.2 Dirichlet form for the operator $\mathcal{L}_{{\scriptscriptstyle
K}}$

We need an estimate on the decay at infinity of the eigenfunctions. Note that
since the eigenvalues are real, we can assume that the eigenfunctions are
real.

######  Proposition 6.4.

If $\phi$ is a normalized sequence in
$\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ decaying exponentially
fast at infinity, then

$\displaystyle-\langle\phi,\mathcal{L}_{{\scriptscriptstyle K}}\phi\rangle$
$\displaystyle=\sum_{n=1}^{\infty}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,\big{(}\phi(n+1)-\phi(n)\big{)}^{2}+\sum_{n=1}^{\infty}V_{n}(K)\,{\phi(n)}^{2}-\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{1}\mu^{{\scriptscriptstyle(K)}}_{2}}\,\phi(1)^{2}.$

PROOF. For any fixed positive integer $N$ we have

$\displaystyle\sum_{n=1}^{N}\phi(n)\,(\mathcal{L}_{{\scriptscriptstyle
K}}\phi)(n)$
$\displaystyle=\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,\phi(n)\,\phi(n+1)+\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,\phi(n)\,\phi(n-1)$
$\displaystyle\quad-\sum_{n=1}^{N}\big{(}\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}\big{)}\,\phi(n)^{2}$
$\displaystyle=-\frac{1}{2}\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,(\phi(n)-\phi(n+1))^{2}+\frac{1}{2}\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,\phi(n)^{2}$
$\displaystyle\quad+\frac{1}{2}\sum_{n=1}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,\phi(n+1)^{2}-\frac{1}{2}\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,(\phi(n)-\phi(n-1))^{2}$
$\displaystyle\quad+\frac{1}{2}\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,\phi(n)^{2}+\frac{1}{2}\sum_{n=2}^{N}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,\phi(n-1)^{2}$
$\displaystyle\quad-\sum_{n=1}^{N}\big{(}\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}\big{)}\,\phi(n)^{2}$
$\displaystyle=-\sum_{n=1}^{N-1}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,(\phi(n)-\phi(n+1))^{2}$
$\displaystyle\quad-\sum_{n=1}^{N}\left(\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,{\mathds{1}}_{\\{n>1\\}}\right)\,\phi(n)^{2}$
$\displaystyle\quad-\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{N}\mu^{{\scriptscriptstyle(K)}}_{N+1}}\,(\phi(N)-\phi(N+1))^{2}+\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{N}\mu^{{\scriptscriptstyle(K)}}_{N+1}}\,\phi(N+1)^{2}$
$\displaystyle\quad-\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{1}\mu^{{\scriptscriptstyle(K)}}_{2}}\,\phi_{1}^{2}-\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{N}\mu^{{\scriptscriptstyle(K)}}_{N+1}}\,\phi(N)^{2}.$

Since $\phi\in\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$, the
functions $\phi\,{\mathds{1}}_{\\{n\leq N\\}}$ and
$\mathcal{L}_{{\scriptscriptstyle K}}(\phi{\mathds{1}}_{\\{n\leq N\\}})$
converge to $\phi$, respectively $\mathcal{L}_{{\scriptscriptstyle K}}\phi$,
in $\ell^{2}$ when $N$ tends to infinity. The result follows by letting $N$
tend to infinity, since $V_{n}(K)$ is positive for $n$ large enough, and since
$\lambda_{N}(K)$ and $\mu_{N}(K)$ are exponential in $N$ and $\phi$ decays
exponentially fast by assumption. $\;\blacksquare$

######  Lemma 6.5.

There exists $\xi>0$ such that for all
$K\in\mathds{Z}_{{\scriptscriptstyle>0}}$, $\inf_{n\geq 1}V_{n}(K)\geq-\xi$.

PROOF. Since $(\lambda^{{\scriptscriptstyle(K)}}_{n})_{n}$ is an increasing
sequence we have

$\displaystyle V_{n}(K)$
$\displaystyle=\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n-1}\,\mu^{{\scriptscriptstyle(K)}}_{n}}\,{\mathds{1}}_{\\{n>1\\}}$
$\displaystyle\geq\frac{\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}}{2}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}.$

It follows from the general assumptions (see Section 2) that there exists
$\tilde{x}\geq 1$ such that all $K\in\mathds{Z}_{{\scriptscriptstyle>0}}$ and
for all $n\geq K\tilde{x}$ we have

$\frac{\lambda^{{\scriptscriptstyle(K)}}_{n}}{\mu^{{\scriptscriptstyle(K)}}_{n}}\leq\frac{1}{5}\quad\text{and}\quad\frac{\lambda^{{\scriptscriptstyle(K)}}_{n+1}}{\mu^{{\scriptscriptstyle(K)}}_{n}}\leq\frac{5}{4}.$

For all $n\geq K\tilde{x}$ we have $V_{n}(K)\geq 0$. When $n\leq K\tilde{x}$,
we write

$\displaystyle\frac{\lambda^{{\scriptscriptstyle(K)}}_{n}+\mu^{{\scriptscriptstyle(K)}}_{n}}{2}-\,\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}$
$\displaystyle=\frac{\big{(}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}}-\sqrt{\mu^{{\scriptscriptstyle(K)}}_{n}}\big{)}^{2}}{2}+\frac{\lambda^{{\scriptscriptstyle(K)}}_{n}}{\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}}+\sqrt{\mu^{{\scriptscriptstyle(K)}}_{n+1}}}\big{(}\mu^{{\scriptscriptstyle(K)}}_{n}-\mu^{{\scriptscriptstyle(K)}}_{n+1}\big{)}.$

Now observe that

$\mu^{{\scriptscriptstyle(K)}}_{n}-\mu^{{\scriptscriptstyle(K)}}_{n+1}=K\left(d\left(\frac{n}{K}\right)-d\left(\frac{n+1}{K}\right)\right)\geq-\sup_{0\,\leq\,x\,\leq\,\tilde{x}+1}d^{\prime}(x).$

The rest of the proof is obvious. $\;\blacksquare$

######  Proposition 6.6.

Let $\delta>0$ and $\phi$ be a real normalized sequence in
$\mathrm{Dom}(\mathcal{L}_{{\scriptscriptstyle K}})$ decaying exponentially
fast, such that $\big{\|}\mathcal{L}_{{\scriptscriptstyle
K}}\phi+\rho\,\phi\big{\|}_{\ell^{2}}\leq\delta.$ Then

$\displaystyle\sum_{n=1}^{\infty}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{n}\mu^{{\scriptscriptstyle(K)}}_{n+1}}\,\big{(}\phi(n+1)-\phi(n)\big{)}^{2}+\sum_{n=1}^{\infty}(1\vee
V_{n}(K))\,\big{(}\phi(n)\big{)}^{2}$ $\displaystyle\leq
1+\rho+\xi+\delta+\frac{1}{2}\sqrt{\lambda^{{\scriptscriptstyle(K)}}_{1}\mu^{{\scriptscriptstyle(K)}}_{2}}.$

The proof is left to the reader. It is a direct consequence of Proposition 6.4
and Lemma 6.5.

##  7 Spectral theory of $\mathcal{M}_{0}$

Recall that (cf. (3.5))

$\displaystyle\big{(}\mathcal{M}_{0}v\big{)}(n)=$
$\displaystyle\sqrt{b^{\prime}(0)\,d^{\prime}(0)\,n\,(n+1)}\,v(n+1)+\sqrt{b^{\prime}(0)\,d^{\prime}(0)\,n\,(n-1)}\,v(n-1)\,{\mathds{1}}_{\\{n>1\\}}$
$\displaystyle-n\,(b^{\prime}(0)+d^{\prime}(0))v(n)).$

######  Theorem 7.1.

The operator $\mathcal{M}_{0}$ defined on $c_{00}$ is symmetric for the scalar
product of $\ell^{2}$. We denote by $\mathcal{M}_{0}$ its closure which is
self-adjoint and bounded above. The spectrum of $\mathcal{M}_{0}$ is discrete,
all eigenvalues are simple, and we have

$\mathrm{Sp}(\mathcal{M}_{0})=\big{(}d^{\prime}(0)-b^{\prime}(0)\big{)}\mathds{Z}_{{\scriptscriptstyle>0}}=-S_{2}.$

The eigenvector $v_{m}$ corresponding to the eigenvalue
$(d^{\prime}(0)-b^{\prime}(0)\big{)}m$, where
$m\in\mathds{Z}_{{\scriptscriptstyle>0}}$, is given (up to a multiplicative
factor) by

$v_{m}(n)=\sqrt{n}\,\left(\frac{d^{\prime}(0)}{b^{\prime}(0)}\right)^{\frac{n}{2}}P_{m}(n)$
(7.1)

where $P_{m}$ is the monic orthogonal polynomial of degree $m-1$ associated
with the measure $q$ on $\mathds{Z}_{{\scriptscriptstyle>0}}$ defined by

$q(n)=n\left(\frac{d^{\prime}(0)}{b^{\prime}(0)}\right)^{n}.$ (7.2)

PROOF. It is easy to verify that $\mathcal{M}_{0}$ is a symmetric operator on
$c_{00}$, which is bounded above since from $b^{\prime}(0)>d^{\prime}(0)$, we
have

$\inf_{n}\big{(}n\,(b^{\prime}(0)+d^{\prime}(0))-\sqrt{b^{\prime}(0)\,d^{\prime}(0)\,n\,(n+1)}-\sqrt{b^{\prime}(0)\,d^{\prime}(0)\,n\,(n-1)}>-\infty.$

It is easy to verify that $\mathcal{M}_{0}$ is closable and we denote by
$\mathcal{M}_{0}$ its closure.

Since for any $m\in\mathds{Z}_{{\scriptscriptstyle>0}}$, the sequence
$v_{m}(n)$ defined by (7.1) decays exponentially fast with $n$, it is easy to
verify that $v_{m}\in\mathrm{Dom}(\mathcal{M}_{0})$. Note also that if $m\neq
m^{\prime}$, $v_{m}$ is orthogonal to $v_{m^{\prime}}$ in $\ell^{2}$.

By a direct computation one checks that
$\mathcal{M}_{0}v_{1}=(d^{\prime}(0)-b^{\prime}(0))v_{1}$ (recall that
$P_{1}(n)=1$). It is left to the reader to check that

$\mathcal{M}_{0}v_{m}(n)=\sqrt{n}\left(\frac{b^{\prime}(0)}{d^{\prime}(0)}\right)^{\frac{n}{2}}\,Q_{m}(n)$

where $Q_{m}$ is a polynomial in $n$ in which the coefficient of $n^{m-1}$ is

$m(d^{\prime}(0)-b^{\prime}(0))\;.$

To check that the $v_{m}$ are eigenvectors, we use a recursive argument.
Assume that $m\geq 2$ and for $1\leq k\leq m-1$

$\mathcal{M}_{0}v_{k}=k\,(d^{\prime}(0)-b^{\prime}(0))\,v_{k}.$

We can write

$\mathcal{M}_{0}v_{m}=m\,(d^{\prime}(0)-b^{\prime}(0))\,v_{m}+r_{m}$

with

$r_{m}=\sqrt{n}\left(\frac{b^{\prime}(0)}{d^{\prime}(0)}\right)^{\frac{n}{2}}R_{m}$

where $R_{m}$ is a polynomial in $n$ of degree at most $m-2$. Therefore

$r_{m}\in\text{Span}\big{\\{}v_{1},\ldots,v_{m-1}\big{\\}}\;.$

From our recursive assumption, the symmetry of $\mathcal{M}_{0}$, and the
orthogonality of the $v_{k}$ (following from the orthogonality of the
$P_{k}$), we get that for any $1\leq k\leq m-1$

$0=\langle v_{k},\mathcal{M}_{0}v_{m}\rangle_{\ell^{2}}=\langle
v_{k},r_{m}\rangle_{\ell^{2}}.$

Therefore $r_{m}=0$. Hence
$\mathcal{M}_{0}v_{m}=m\,(d^{\prime}(0)-b^{\prime}(0))\,v_{m}$, and we can
proceed with the recursion.

We now prove that the $v_{m}$ form a basis of $\ell^{2}$. Assume the contrary,
namely there exists $u\in\ell^{2}$ of norm one such that for any $m$

$\sum_{n=1}^{\infty}\overline{u}(n)\,v_{m}(n)=0.$

We observe that the sequence

$w(n)=\frac{1}{\sqrt{n}}\left(\frac{b^{\prime}(0)}{d^{\prime}(0)}\right)^{\frac{n}{2}}u(n)$

belongs to $\ell^{2}(q)$, whith $q$ defined in (7.2). Therefore our assumption
on $u$ implies that $w$ is orthogonal to all the polynomials in $\ell^{2}(q)$.
Let us show that the set of polynomials is dense in $\ell^{2}(q)$. It is
sufficient to prove that the measure $q$ is the solution of a determinate
moment problem, see [7, Corollary 2.50, p. 30]. Following [12, Proposition
1.5, p. 88], it is enough to prove that the moments of order $m$, denoted by
$\gamma_{m}$ of $q$, satisfy the following property: there exists $C>0$ such
that, for any $m\in\mathds{Z}_{{\scriptscriptstyle>0}}$,

$\gamma_{m}=\sum_{n=1}^{\infty}n^{m+1}\left(\frac{d^{\prime}(0)}{b^{\prime}(0)}\right)^{n}\leq
C^{m}\,m!\,.$

The proof is left to the reader. Therefore the set of all polynomials is dense
in $\ell^{2}(q)$ implying $w=0$ and we get a contradiction with the existence
of a $u$ nonzero orthogonal to all the $v_{m}$ in $\ell^{2}$. Therefore, the
$v_{m}$ form a basis of $\ell^{2}$.

We now observe that $\mathcal{M}_{0}$ is bounded above. The proof is similar
to that of Proposition 6.4 and left to the reader. Since the $v_{m}$’s form a
basis of $\ell^{2}$, for any $B>0$ we have
$\text{ker}(\mathcal{M}_{0}^{\dagger}-B)=\\{0\\}$. Hence $\mathcal{M}_{0}$ is
self adjoint (see for instance [11, Prop. 3.9, p. 43]) and the spectrum is
given by

$\mathrm{Sp}(\mathcal{M}_{0})=(d^{\prime}(0)-b^{\prime}(0))\mathds{Z}_{{\scriptscriptstyle>0}}.$

This ends the proof. $\;\blacksquare$

##  8 Local maximum principle and consequences thereof

We will state and prove a maximum/minimum principle in a form which is well
suited for our purposes. We start with a proposition giving elementary
inequalities following from the order on the real line.

######  Proposition 8.1.

Assume $a>0$, $c>0$ and $b>a+c$. Let $u,w\in\real$.
If $v>0$ is such that $a\,u+c\,w-b\,v\geq 0$, then $v<\max\\{u,w\\}$.
If $v<0$ is such that $a\,u+c\,w-b\,v\leq 0$, then $v>\min\\{u,w\\}$.
Moreover, if $u\geq v\geq w$ are such that $a\,u+c\,w-b\,v\geq 0$, then
$v\leq\frac{a}{b-c}\,u$.

PROOF. If $v>0$ we have $0<(b-a-c)\,v\leq a\,(u-v)+c\,(w-v)$ leads to a
contradiction if $v\geq\max\\{u,w\\}$. The case $v<0$ is similar. The last
statement is trivial since $b\,v\leq a\,u+c\,w\leq a\,u+c\,v$.
$\;\blacksquare$

######  Proposition 8.2.

Let $1<n_{1}<n_{2}$ be integers such $n_{2}>n_{1}+1$. Let $(\alpha_{n})$ be a
finite sequence of strictly positive real numbers defined for
$n_{1}-1,\ldots,n_{2}$. Let $(\beta_{n})$ be a finite sequence of strictly
positive real numbers defined for $n_{1},\ldots,n_{2}$. Let $(u_{n})$ be a
finite sequence of real numbers defined for $n_{1}-1,\ldots,n_{2}+1$. Assume
that, for all $n_{1}\leq n\leq n_{2}$, we have
$\beta_{n}>\alpha_{n}+\alpha_{n-1}$.
If $\alpha_{n}u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_{n}u_{n}\geq 0$, then the
sequence $(u_{n})$ has no positive local maxima for
$n\in\\{n_{1}+1,\ldots,n_{2}-1\\}$. Moreover, if there exists some $u_{n}>0$
then the maximum is attained only at the boundary, that is, on the set
$\\{n_{1},n_{2}\\}$.
If $\alpha_{n}u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_{n}u_{n}\leq 0$, then the
sequence $(u_{n})$ has no positive local minima for
$n\in\\{n_{1}+1,\ldots,n_{2}-1\\}$, and if there exists some $u_{n}<0$ then
the minimum is attained only at the boundary, that is, on the set
$\\{n_{1},n_{2}\\}$.

PROOF. It follows from Proposition 8.1. $\;\blacksquare$

######  Proposition 8.3.

Let $1<n_{1}<n_{2}$ be integers such $n_{2}>n_{1}+1$. Let $(\alpha_{n})$ be a
finite sequence of strictly positive real numbers defined for
$n_{1}-1,\ldots,n_{2}$. Let $(\beta_{n})$ be a finite sequence of strictly
positive real numbers defined for $n_{1},\ldots,n_{2}$. Let $(u_{n})$ be a
finite sequence of real numbers defined for $n_{1}-1,\ldots,n_{2}+1$. Assume
that, for all $n_{1}\leq n\leq n_{2}$, we have
$\beta_{n}>\alpha_{n}+\alpha_{n-1}$.
If $\alpha_{n}u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_{n}u_{n}\geq 0$,
$u_{n_{1}+1}>0$ and $u_{n_{1}+1}\geq u_{n_{1}}$, then the sequence $(u_{n})$
is increasing.
If $\alpha_{n}u_{n+1}+\alpha_{n-1}u_{n-1}-\beta_{n}u_{n}\leq 0$,
$u_{n_{1}+1}<0$ and $u_{n_{1}+1}\leq u_{n_{1}}$, then the sequence $(u_{n})$
is decreasing.
Finally, if $(u_{n})$ is a positive sequence then there cannot be two local
(positive) minima separated by a distance larger than one.

PROOF. It follows recursively from Proposition 8.1. $\;\blacksquare$

##  Appendix A Quasi-eigenvalues and quasi-eigenvectors of self-adjoint
operators

######  Proposition A.1.

Let $A$ be a self-adjoint operator in a Hilbert space $\mathscr{H}$ with
domain $\mathrm{Dom}(A)$. Assume there exists $u\in\mathrm{Dom}(A)$ of norm
$1$, $\omega\in\real$ and $\epsilon>0$ such that

$\big{\|}A\,u-\omega\,u\big{\|}\leq\epsilon.$

Then

$\mathrm{Sp}(A)\,\cap\left[\,\omega-\epsilon,\omega+\epsilon\,\right]\neq\emptyset\;.$

PROOF. We will assume that $\omega\notin\mathrm{Sp}(A)$, otherwise the result
is trivial. The proof is then by contradiction. If $R_{\omega}$ denotes the
resolvent of $A$ at $\omega$, we have

$u=-R_{\omega}\,(A\,u-\omega\,u)$

hence

$1\leq\epsilon\,\big{\|}R_{\omega}\big{\|}.$

The result follows from the estimate (a direct consequence of the spectral
decomposition)

$\big{\|}R_{\omega}\big{\|}\leq\frac{1}{\text{d}\big{(}\omega,\mathrm{Sp}(A)\big{)}}$

where $d$ denotes the Euclidean distance on the real line. If
$\mathrm{Sp}(A)\,\cap\,[\omega-\epsilon,\omega+\epsilon]=\emptyset$, since
$\mathrm{Sp}(A)$ is closed, then

$\delta=\text{d}\big{(}\omega,\mathrm{Sp}(A)\big{)}>\epsilon$

and we get

$1\leq\frac{\epsilon}{\delta}<1$

which is a contradiction. $\;\blacksquare$

######  Proposition A.2.

Let $A$ be a self-adjoint operator in a Hilbert space $\mathscr{H}$ with
domain $\mathrm{Dom}(A)$. Assume there exists $u\in\mathrm{Dom}(A)$ of norm
$1$, $\omega\in\real$ and $\epsilon>0$ such that

$\big{\|}A\,u-\omega\,u\big{\|}\leq\epsilon\;.$

Assume $A$ has discrete spectrum with eigenvalues of multiplicity one, and let
$\delta>0$ denote the minimum distance between two consecutive eigenvalues.
Then if $\epsilon<\delta$ there is a $\lambda\in\mathrm{Sp}(A)$ with a
normalized eigenvector $e$ such that $|\omega-\lambda|\leq\epsilon$ and

$\big{\|}u-e\big{\|}\leq\frac{2\,\epsilon}{\delta-\epsilon}.$

PROOF. We will denote by $P_{z}$ the one-dimensional spectral projector of $A$
corresponding to $z\in\mathrm{Sp}(A)$.

Since $\epsilon<\delta$, using Proposition A.1 we conclude that there is only
one eigenvalue of $A$ in $\left[\,\omega-\epsilon,\omega+\epsilon\,\right]$
and we denote by $\lambda$ this eigenvalue and by $\tilde{e}$ one of the
corresponding eigenvectors (they all differ only by a phase factor). Let

$v=\omega\,u-A\,u.$

Since $u=P_{\lambda}u+(\mathrm{Id}-P_{\lambda})u$, we get from the spectral
decomposition

$A(\mathrm{Id}-P_{\lambda})\,u-\omega(\mathrm{Id}-P_{\lambda})\,u=(\mathrm{Id}-P_{\lambda})\,v.$

This implies that

$\|(\mathrm{Id}-P_{\lambda})\,u\|\leq\frac{\epsilon}{\delta-\epsilon}.$

Since $P_{\lambda}\,u$ is proportional to $\tilde{e}$, we can write

$u=\beta\,\tilde{e}+(\mathrm{Id}-P_{\lambda})\,u$

with $\beta\in\mathds{C}$. Since $u$ and $\tilde{e}$ are of norm one, and
$\tilde{e}$ and $(\mathrm{Id}-P_{\lambda})\,u$ are orthogonal, we get

$1=|\beta|^{2}+\big{\|}(\mathrm{Id}-P_{\lambda})\,u\big{\|}^{2}$

which implies

$1\geq|\beta|\geq 1-\frac{\epsilon}{\delta-\epsilon}\;.$

Let $\beta=|\beta|\exp(\mathrm{i}\theta)$, we define
$e=\mathrm{e}^{\mathrm{i}\,\theta}\,\tilde{e}$, and the result follows.
$\;\blacksquare$

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