J.-R Chazottes

P Collet

S Méléard

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

Keywords: 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

We consider a general class of birth-and-death processes with state space {0, 1, 2, 3, . . .} 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 → +∞. We assume that the di erential equation dx/dt = 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 (x * )d (x * )), n ≥ 0. The other one is the spectrum of a continuous-time binary branching process conditioned on nonextinction, and is given by n(d (0)b (0)), n ≥ 1. A major di culty is that di erent scales and function spaces are involved. We work at the level of the eigenfunctions that we split over di erent regions, and study their asymptotic dependence on K in each region. In particular, we prove that the spectral gap goes to min b (0)d (0), d (x * )b (x * ) . 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.

1 Introduction, heuristics, and main result

The context

We consider a class of birth-and-death processes (X K t ) t≥0 with state space Z ≥0 1 which describes how the size of a single population evolves according to birth and death rates of the form

λ (K) n = K b n K and µ (K) n = K d n K (1.1)
where n ≥ 1, and K ∈ Z >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 ν (K) on the positive integers is a quasi-stationary distribution if, for all t > 0 and for all subsets A ⊂ Z >0 , one has

P ν (K) (X K t ∈ A | T (K) 0 > t) = ν (K) (A)
, where T (K) 0 is the extinction time, that is, the smallest t > 0 such that X (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 [START_REF] Collet | Quasi-Stationary Distributions[END_REF][START_REF] Méléard | Quasi-stationary distributions and population processes[END_REF] for more informations about quasi-stationary distributions.

When K → ∞, the trajectories of the rescaled process (K -1 X (K) t ) t≥0 converge in probability, in any fixed time-window, to the solutions of the differential equation

dx dt = b(x) -d(x) (1.2)
if the initial condition state is for instance of the form Kx 0 for a given x 0 > 0. We assume that the functions b and d only vanish at 0, and that d (0)b (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 * ) and b (x * )d (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) = λx, d(x) = x(µ + x), where λ and µ are positive real numbers. We assume that λ > µ and we have x * = λµ.

In [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] we obtained the precise asymptotic behaviour of the first eigenvalue of the generator L 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 → ∞. 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.

Notations for basic function spaces

We denote by • D the space of C ∞ C-valued functions with compact support on R;

• c 00 the space of C-valued sequences with finitely many nonzero values;

• 2 the space of square-summable C-valued sequences equipped with the standard scalar product.

• L 2 the space of square-integrable C-valued functions with respect to Lebesgue measure on R.

We will define several operators on c 00 and will consider their closure on 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.

Heuristics

The fundamental object in this paper is the spectrum of the following operator that we momentarily define on c 00 :

L K v (n) = λ (K) n v(n + 1) -v(n) + µ (K) n v(n -1)1 {n>1} -v(n) (1.3)
for n ∈ Z >0 . The idea is to 'localize' this operator either around n = Kx * 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 di erentiate these two dynamics. However the main di culty is that the two di erent pieces involve di erent scales and di erent 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 di erent regions, and study their asymptotic dependence on K in each region.

To have an idea of the di erent 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

λ (K) n = K b x * + (n -Kx * ) b x * + O (n -Kx * ) 2 K and µ (K) n = K d x * + (n -Kx * ) d x * + O (n -Kx * ) 2 K .
Taking v (K) (n) = u (n -Kx * )/ √ K with u ∈ D, we get

L K v (K) (n) = OU * u n -Kx * √ K + O 1 √ K where OU * f (x) = b(x * )f (x) + b (x * ) -d (x * ) xf (x) (1.4)
is the generator of an Ornstein-Uhlenbeck process on R which satisfies the stochastic di erential equation

dX t = b (x * ) -d (x * ) X t dt + 2b(x * ) dB t ,
where (B t ) t≥0 is a one-dimensional Brownian motion. It is well known (see Remark 3.1) that the spectrum of OU * is -S 1 where

S 1 = d (x * ) -b (x * ) Z ≥0 (1.5) in the space L 2 d (x * )-b (x * ) 2πb(x * ) e - (d (x * )-b (x * ) 2b(x * )
x 2

dx . Now we look at n near 1. By Taylor expansion, we have

λ (K) n = n b (0) + O n K and µ (K) n = n b (0) + O n K .
If v ∈ c 00 , then we get

L K v -Q 0 v 2 ≤ O(1) K where Q 0 v (n) = b (0) n v(n + 1) -v(n) + d (0) n v(n -1) 1 {n>1} -v(n)
which is the generator of a (continuous-time) binary branching process killed at 0. We shall prove later on that, in a weighted 2 space defined below, the spectrum of Q 0 is -S 2 where

S 2 = b (0) -d (0) Z >0 .
(1.6)

The previous observations suggest that the limit of the spectrum of the generator of the birth-and-death process (X (K) t ) t≥0 , in an appropriate space, is

d (0) -b (0) Z >0 b (x * ) -d (x * ) Z ≥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 (0) -b (0) = b (x * ) -d (x * ) = µ -λ
, so we will have asymptotic double eigenvalues in this case. We will prove that the limit of the spectrum of L K is obtained from the explicit spectra of the above two operators. Notice that one is di erential operator and the other one is a finite-di erence operator. This is a reverse situation with respect to numerical analysis where the spectrum of limiting di erential operators are obtained from the knowledge of the spectrum of finite-di erence operators. See for instance [START_REF] Chatelin | Spectral Approximation of Linear Operators[END_REF].

Main result

For each K ∈ Z >0 , the sequence of numbers π (K) n :=

λ (K) 1 • • • λ (K) n-1 µ (K) 1 • • • µ (K) n , n ≥ 2 and π (K) 1 := 1 µ (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 di erential 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 2 (π (K) ) be the space of C-valued sequences

(v n ) n≥1 such that n≥1 |v n | 2 π (K) n < ∞ .
This is a Hilbert space when endowed with the scalar product v, w π (K) := n≥1 vn w n π (K) n . We know from [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] that the operator L K is closable in 2 (π (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 K ψ (K) j = -ρ (K) j ψ (K) j (1.7)
where we order the eigenvalues -ρ (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 -ρ, with ρ > 0.

As shown in [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF], the quasi-stationary distribution exists, is unique, and given by

ν (K) ({n}) = π (K) n ψ (K) 0 (n) ψ (K) 0 , 1 π (K) , n ∈ Z >0 (1.8)
where 1 = (1, 1, . . .). Note that it also follows from Theorem 3.2 and Lemma 9.3 in [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] that there exists D > 1 such that for all K ∈ Z >0 , we have

D -1 ≤ ψ (K) 0 ≤ D. Therefore, the Hilbert spaces 2 (ν (K) ) and 2 (π (K) ) are isomorphic. Let S (K) t f (n) := E n f (X (K) t )1 {T (K) 0
>t} (t ≥ 0) be the semigroup of the killed process, where n ∈ Z >0 and f ∈ ∞ . The following result justifies that we look for the spectrum of L K in 2 (π (K) ) .

P

1.1. The semigroup (S (K) t ) t≥0 , de ned on ∞ , extends to a C 0contraction semigroup on 2 (ν (K) ).

We refer to [START_REF] Yosida | Functional analysis[END_REF] for definitions and properties of C 0 -contraction semigroups.

PROOF. We follow the argument of Proposition 8.1.8 p. 162 in [START_REF] Bertoldi | Analytical Methods for Markov Semigroups[END_REF]. Since ν (K) is a quasi-stationary distribution, for f ∈ c 00 and t ≥ 0, we have

|S (K) t f | 2 dν (K) ≤ dν (K) (n) E n f (X (K) t ) 2 1 T (K) 0 >t = e -ρ (K) 0 t |f | 2 dν (K) .
Therefore

S (K) t f 2 (ν (K) ) ≤ e - ρ (K) 0 t 2 f 2 (ν (K) ) , t ≥ 0.
Since c 00 is dense in 2 (ν (K) ), we get

S (K) t 2 (ν (K) ) ≤ e - ρ (K) 0 t 2 , t ≥ 0.
This implies that (S (K) t ) t≥0 extends to a contraction semigroup in 2 (ν (K) ).

Now, since P n (X (K) t = m, T (K) 0 > t) → δ n,m , as t → 0, then for any f ∈ c 00 , S (K)
t f → f pointwise, hence by dominated convergence we obtain S (K) t f → f in 2 (ν (K) ) as t → 0. The proposition follows from the contraction property obtained above and the fact that c 00 is dense in 2 (ν (K) ). Observe that the same result holds in fact for any 1 ≤ p < ∞ (with a similar proof).

Recall from [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] that we also have

P ν (K) (T (K) 0 > t) = e -ρ (K) 0 t , t > 0
and the mean-time to extinction starting from ν (K) is

E ν (K) T (K) 0 = 1 ρ (K) 0 = √ 2π exp K x * 0 log b(x) d(x) dx b(x * ) b(1/K) d(1/K) -d(1/K) b(1/K) KH (x * ) 1+O (log K) 3 √ K .
(See the next section for the definition of H.) In the logistic model this gives (recall that x * = λµ)

E ν (K) T (K) 0 = √ 2π µ exp K λ -µ + µ log µ λ (λ -µ) 2 √ K 1+O (log K) 3 √ K .
Thus ρ (K) 0 is exponentially small in K, and we also proved in [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] that the 'spectral gap' satisfies (see Theorem 3.3 in [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF])

ρ (K) 1 -ρ (K) 0 ≥ O(1) log K , K ∈ Z >1 . (1.9)
This lower bound goes to 0 as K → +∞. 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 ∪ 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 (η n ) n≥0 . Let η 0 = 0. We construct this sequence recursively as follows.

If

η n ∈ S 1 ∆S 2 , then η n+1 = min{η : η ∈ S 1 ∪ S 2 : η > η n }. If η n ∈ S 1 ∩ S 2 , then • If η n-1 = η n , then η n+1 = min{η : η ∈ S 1 ∪ S 2 : η > η n }. • If η n-1 < η n , then η n+1 = η n .
The main result of this paper, whose assumptions will be stated in Section 2, is the following. 1.2 (Convergence of the spectrum). The spectrum of L K in 2 (π (K) ) converges pointwise to (-η n ) n≥0 when K tends to in nity. In other words

lim K→+∞ ρ (K) j = η j , ∀j ∈ Z ≥0 . C 1.3. The spectral gap ρ (K) 1 -ρ (K) 0 converges to min b (0) -d (0), d (x * ) -b (x * ) .
Let us give some examples. In the logistic model, we have d (0)b (0) = b (x * )d (x * ) = µλ (asymptotic double eigenvalues), and the spectral gap is equal to λµ. Another example is the Ayala-Gilpin-Ehrenfeld model [START_REF] Ayala | Competition between species: Theoretical models and experimental tests[END_REF] defined by b(x) = λx, d(x) = x(µ + x θ ) where θ ∈ (0, 1) is a parameter, and λ > µ. In this case, d (0)-b (0) = µ-λ, x * = (λ-µ) 1/θ , and b (x * )-d (x * ) = θ(µ-λ), so the spectral gap is θ(λ-µ). Yet another example is Smith's model [START_REF] Smith | Population dynamics in Daphnia magna and a new model for population growth[END_REF]

defined by b(x) = λx/(1 + x), d(x) = (x(µ + x))/(1 + x), where λ > µ. One easily finds d (0) -b (0) = µ -λ, x * = λ -µ, and b (x * ) -d (x * ) = (µ -λ)/(1 + λ -µ), so the spectral gap is (λ -µ)/(1 + λ -µ).

Consequences on relaxation times

Recall that the spectral gap is the inverse of the relaxation time to the quasistationary distribution, namely for ψ ∈ 2 (π (K) ), t > 0 and K ∈ Z >0 we have

e ρ (K) 0 t S (K) t ψ -ψ (K) 0 ψ (K) 0 , 1 π (K) ν (K) (ψ) 2 (π (K) ) ≤ ψ 2 (π (K) ) e -ρ (K) 1 -ρ (K) 0 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 Qprocess, namely the birth-and-death process conditioned on survival. Recall that the Q-process is the irreducible Markov process with state space Z >0 , defined by the semigroup

R (K) t g = e ρ (K) 0 t 1 ψ (K) 0 S (K) t gψ (K) 0 .
It satisfies R (K) t 1 {n≥1} = 1 {n≥1} , and its unique invariant distribution m (K) , defined by

R (K) t † m (K) = m (K) , is related to ν (K) by m (K) (g) = ν (K) (ψ (K)
0 g) where ν (K) has been defined in (1.8). Indeed we have g ∈ 2 (m (K) ) if and only if ψ (K) 0 g ∈ 2 (π (K) ), and

g 2 2 (m (K) ) = ψ (K) 0 g 2 2 (π (K) ) ψ (K) 0 , 1 π (K)
.

Hence, we get the following result.

P 1.4. Let g ∈ 2 (m (K) ). Then for all t > 0 R (K) t g -ψ (K) 0 , 1 π (K) m (K) (g) 2 (m (K) ) ≤ g 2 (m (K) ) e -(ρ (K) 1 -ρ (K) 0 ) t .
Furthermore, for g 1 , g 2 ∈ 2 (m (K) ) and for all t > 0, we have

R (K) t g 1 • g 2 dm (K) -g 1 dm (K) g 2 dm (K) ≤ g 1 2 (m (K) ) g 2 2 (m (K) ) e -ρ (K) 1 -ρ (K) 0 t .
As before, the rate of decay of correlations converges when K goes to infinity.

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 ∪ S 2 is contained in the set of accumulation points of the eigenvalues of 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 ∪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 L K into two dominant parts, one localised near the origin, the other one near Kx * . 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 di culties of the proof is that the two pieces of the spectrum correspond to limiting operators which are obtained at di erent scales and leave in di erent function spaces.

Standing assumptions

We work under the assumptions of [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] which we recall for convenience.

The functions b, d : R + → R + defining the di erential equation (1.2) are supposed to be such that b(0) = d(0) = 0 and the functions x → b(x)/x and x → d(x)/x are defined on R + and assumed to be positive, twice di erentiable and increasing (in particular the sequences (λ (K) n ) n and (µ (K) n ) n defined in (1.1) are increasing for each K). We start by the biologically relevant assumptions:

lim x→+∞ b(x) d(x) = 0 (deaths prevail over births for very large densities). b (0) > d (0) > 0 (

at low density births prevail).

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 (x * ) = d (x * ) (genericity condition). The remaining (technical) assumptions are the following:

+∞ x * 2 dx d(x) < +∞ and sup x∈R + d (x) d(x) -1 x < +∞. The function x → log d(x) b(x) is increasing on R + .
The function H : R + → R defined by H(x) =

x x * log d(s) b(s) ds is three times di erentiable, and

sup x∈R + (1 + x 2 )|H (x)| < +∞.
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 (x * ) < d (x * ). It also follows that H (x * ) > 0. These assumptions are satisfied for many classical examples.

As explained in [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF], the above conditions imply the following properties:

n≥1 (λ (K) n π (K) n ) -1
= +∞, which implies that the process reaches 0 in finite time with probability one. n≥1 π (K) n < +∞, which implies finiteness of the mean time to extinction.

n≥1 (λ (K) n π (K) n ) -1 i≥n+1 π (K) i
< +∞, 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→∞ log b(x) x = lim x→∞ 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

Some useful operators

Instead of working with L K on the weighted Hilbert space 2 (π (K) ), we find more convenient to work on the 'flat' Hilbert space 2 . We introduce the conjugated operator

L K = (Π (K) ) 1 2 L K (Π (K) ) -1 2
where Π (K) denotes the mutiplication operator

Π (K) v(n) = π (K) n v(n)
for v ∈ c 00 and n ∈ Z >0 . One can check that

L K v (n) = λ (K) n µ (K) n+1 v(n + 1) + λ (K) n-1 µ (K) n v(n -1) 1 {n>1} -λ (K) n + µ (K) n v(n)
for n ∈ Z >0 . We denote also by L K its closure in 2 and by Dom(L K ) its domain, and we have

L K φ (K) j = -ρ (K) j φ (K) j
where the eigenvalues -ρ (K) j are the same as for L K (cf. (1.7)), and

φ (K) = Π (K) 1 2 ψ (K) .
To capture the behavior of the eigenvectors of L K near Kx * at scale √ K, we are going to embed 2 into L 2 . For this purpose we define for each K ∈ Z >0 the functions

e (K) n (x) = K 1 4 1 I (K) n (x), x ∈ R, n ∈ Z >0
where

I (K) n = n -0.5 √ K -x * √ K, n + 0.5 √ K -x * √ K .
The functions e (K) n are orthogonal and of norm one in L 2 . They form a basis of a sub-Hilbert space H K of piecewise constant functions in L 2 . We define two maps denoted by Q K and P K as follows:

Q K : 2 → L 2 , Q K u(x) = n≥1 u(n) e (K)
n (x)

and

P K : L 2 → 2 , P K f (n) = f (x) e (K) n (x) dx , n ∈ Z >0 . (3.1)
We will use the following properties of P K and Q K stated as two lemmas.

L 3.1. For each K ∈ Z >0 , the map Q K is an isometry between 2 and H K .
The proof of this lemma is left to the reader.

L 3.2. Let f ∈ C 1 (R)
and assume that there exists a > 0 and

A > 0 such that f (x) + f (x) ≤ A e -a |x| , x ∈ R. Then (i) lim K→∞ f -Q K P K f L 2 = 0 . (ii) lim K→∞ P K f 2 = f L 2 .
PROOF. Let us prove (i). We have

Q K P K f (x) = e (K) n(x) (x) e (K) n(x) (y) f (y) dy for (n(x) -0.5)/ √ K -x * ≤ x ≤ (n(x) + 0.5)/ √ K -x * .
We get from our hypothesis

Q K P K f (x) = f (x) + O K -1 2 e -a|x|
and the result follows.

We now prove (ii). From the isometric property of Q K on the space of piecewise functions H K , we get

P K f 2 = Q K P K f L 2
and the result follows from (i).

We now introduce the operator

L K = Q K L K P K
and we keep the same notation for its closure in L 2 . Since L K when acting on H K is conjugated to L K , we have

L K ϕ (K) j = -ρ (K) j ϕ (K) j .
We will prove in the next proposition that the operator L K converges weakly, when K → +∞, to the operator

H * f (x) = (3.2) b(x * ) d 2 f (x) dx 2 - d (x * ) -b (x * ) 2 4b(x * ) x 2 f (x) + d (x * ) -b (x * ) 2 f (x). P 3.3. Let f ∈ C 3 (R)
and assume that there exist a > 0 and A > 0 such that

3 j=0 f (j) (x) ≤ A e -a |x| , x ∈ R. Then lim K→∞ L K f -H * f L 2 = 0 .
PROOF. By the assumption made on f , it follows easily that

lim K→∞ 1 {| • |>(log K) 2 } H * f L 2 = 0.
We have

L K f (x) = n≥1 e (K) n (x) λ (K) n µ (K) n+1 e (K) n+1 (y) f (y) dy (3.3)
+ λ (K) n-1 µ (K) n 1 {n>1} e (K) n-1 (y) f (y) dy -λ (K) n + µ (K) n e (K)
n (y) f (y) dy .

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

lim K→∞ 1 {| • |>(log K) 2 } L K f L 2 = 0 .
Therefore we only have to consider |x| ≤ (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) = Kx * + √ Kx + 1 2 . For x ∈ R, we have L K f (x) = e (K) n(x) (x) λ (K) n(x) µ (K) n(x)+1 e (K) n(x)+1 (y)f (y) dy + λ (K) n(x)-1 µ (K) n(x) e (K) n(x)-1 (y)f (y) dy -λ (K) n(x) + µ (K) n(x) e (K) n(x) (y)f (y) dy = e (K) n(x) (x) λ (K) n(x) µ (K) n(x)+1 e (K) n(x) (y)f y - 1 √ K dy + λ (K) n(x)-1 µ (K) n(x) e (K) n(x) (y)f y + 1 √ K dy -λ (K) n(x) + µ (K) n(x)
e (K) n(x) (y)f (y) dy .

Now we have

e (K) n(x) (y) f (y) dy = K 1 4 1 2 √ K -1 2 √ K f n(x) -Kx * √ K -s ds = K -1 4 f n(x) -Kx * √ K + K -5 4 24 f n(x) -Kx * √ K + O K -7 4
where the error term is unifom in x. Similarly

e (K) n(x) (y) f y ± 1 √ K dy = e (K) n(x) (y) f (y) dy ± 1 √ K e (K) n(x) (y) f (y) dy + 1 2K e (K) n(x) (y) f (y) dy + O K -7 4 = K -1 4 f n(x) -Kx * √ K ± K -3 4 f n(x) -Kx * √ K + 13 K -5 4 24 f n(x) -Kx * √ K + O K -7 4 .
Recall that

λ (K) n = Kb n K and µ (K) n = Kd n K hence λ (K) n = Kb x * + (n -Kx * ) b x * + O (n -Kx * ) 2 K and µ (K) n = Kd x * + (n -Kx * ) d x * + O (n -Kx * ) 2 K .
After a tedious but straightforward computation, we obtain that

L K f (x) = H * f n(x) -Kx * √ K + O K -1 4 = H * f (x) + O K -1 4 (log K) 4
and the error term is uniform in |x| ≤ (log K) 2 . We get

1 {| • |≤(log K) 2 } L K f -H * f L 2 = O K -1 4 (log K) 6
and the result follows.

ψ n (x) = (3.4) 1 √ 2 n n! d (x * ) -b (x * ) 2 2πb(x * ) 1 4 e - d (x * )-b (x * ) 4b(x * )
x 2

H n d (x * ) -b (x * ) 2b(x * ) x
where (H n ) n is the family of the physicists' Hermite polynomials de ned by

H n (x) = (-1) n e x 2 d n dx n e -x 2 .
One can check that H * is conjugated to the generator of the Ornstein-Uhlenbeck pro-

cess (1.4) acting on L 2 d (x * )-b (x * ) 2πb(x * ) e - (d (x * )-b (x * ) 2b(x * ) x 2
dx in the following way:

1 ψ 0 H * (ψ 0 f ) = OU * f .
In Proposition 3.4 we prove that the operator L K converges weakly, when K tends to infinity, to the operator M 0 defined for v ∈ c 00 by

M 0 v (n) = (3.5) b (0) d (0) n (n + 1) v(n + 1) + b (0) d (0) n (n -1) v(n -1) 1 {n>1} -n (b (0) + d (0))v(n)).
Here again we denote the operator on c 00 and its closure by the same letter.

P 3.4. Let u ∈ c 00 . Then lim K→∞ L K u = M 0 u
where M 0 is de ned in (3.5).

PROOF. Follows from the fact that for each fixed n

lim K→∞ λ (K) n = b (0) n and lim K→∞ µ (K) n = d (0) n .

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 Sp(L K ) is discrete, and let

G = ∞ j=0 ρ (K) j acc
where ρ (K) j acc is the set of accumulation points of ρ (K) j when K → +∞.

T 3.5. We have

S 1 ∪ S 2 ⊂ G
where S 1 and S 2 are de ned in (1.5) and (1.6).

This theorem is proved in Section 5.1.

C 3.6. For every xed j we have

lim sup K→+∞ ρ (K) j < +∞.
PROOF. We proceed by contradiction. Assume that there exists j 0 such that

lim sup K→+∞ ρ (K) j 0 = +∞.
Let j c = min{0 < ≤ j 0 : lim sup K→+∞ ρ (K) = +∞}. Hence there exists α < +∞ such that lim sup K→+∞ ρ (K) jc-1 = α. By definition of j c , there exists a diverging sequence (K p ) p such that lim p→+∞ ρ j+1ρ * , ρ

(Kp) j-1 -ρ * > 0 . (3.8)
Note that (3.6) means that if ρ * ∈ S 1 ∆S 2 then -ρ * is a simple asymptotic eigenvalue, and either -ρ * is an eigenvalue of

H * if ρ * ∈ S 1 , or of M 0 if ρ * ∈ S 2 .
In addition, (3.7) and (3.8) mean that if ρ * ∈ S 1 ∩ S 2 , then -ρ * is a double asymptotic eigenvalue which is an eigenvalue of both H * and M 0 . Proof of Theorem 1.2. The proof is recursive. For j = 0 it follows from [START_REF] Chazottes | Sharp asymptotics for the quasistationary distribution of birth-and-death processes[END_REF] that lim K→+∞ ρ (K) 0 = 0. Let j ≥ 0 and assume that for ≤ j (if any) lim K→+∞ ρ (K) = η . We now prove that lim K→+∞ ρ (K) j+1 = η j+1 . There are several cases to consider.

• If η j ∈ S 1 ∆S 2 , we claim that lim inf K→+∞ ρ (K) j+1 ≥ η j+1 . Otherwise, by Theorem 3.7 and the recursive hypothesis, there would exist K p → +∞ such that ρ (Kp) j+1 → η * < η j+1 . Since by the recursive hypothesis lim K→+∞ ρ (Kp) j = η j , we have η * ≥ η j . From the first statement of Theorem 3.7 it follows that η * = η j . This contradicts 1) of Theorem 3.7. We now claim that lim sup K→+∞ ρ (K) j+1 ≤ η j+1 . Otherwise, by Theorem 3.7 and the recursive hypothesis, there would exist K p → +∞ such that ρ (Kp) j+1 → η * > η j+1 . This implies that η j+1 ∈ G, contradicting Theorem 3.5. Hence, in this case, lim K→+∞ ρ (K) j+1 = η j+1 .

• If η j ∈ S 1 ∩ S 2 (which implies j > 0), we have two cases:

-If η j-1 = η j , then we claim that lim inf K→+∞ ρ (K) j+1 ≥ η j+1 . Otherwise, by the same argument as before, there would exist K p → +∞ such that ρ (Kp) j+1 → η j , contradicting (3.8) in Theorem 3.7.

We now claim that lim sup K→+∞ ρ (K) j+1 ≤ η j+1 . Otherwise, as before, this would contradict that η j+1 ∈ G. Hence, in this case, lim K→+∞ ρ (K) j+1 = η j+1 . -If η j-1 < η j , then we obviously have lim inf K→+∞ ρ (K) j+1 ≥ η j . If lim sup K→+∞ ρ (K) j+1 > η j , then there exists K p → +∞ such that ρ (Kp) j+1 → η * > η j , contradicting (3.7) in Theorem 3.7. Hence, in this case, lim K→+∞ ρ (K) j+1 = η j+1 .

Therefore we lim K→+∞ ρ (K) j+1 = η j+1 . As announced, the proof of Theorem 1.2 follows recursively.

Properties of the eigenvectors

Our aim in this part is to prove that for K large enough, the eigenvectors φ (K) j of L 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. To separate the di erent behaviors, we introduce a 'potential' defined by

n g n l n r n d Kx * O (1)
V n (K) = λ (K) n + µ (K) n -λ (K) n µ (K) n+1 -λ (K) n-1 µ (K) n 1 {n>1} . (4.1)
For η > 0, let n g (K, η) and n d (K, η) be integers such that n g (K, η), n d (K, η) is the maximal interval containing Kx * /2 such that

inf n ∈ ng(K,η), n d (K, η) V n (K) -η > 0 .
Let n (K) = (log K) 2 and n r (K) = Kx * -K 2 3 log K . It follows from our assumptions that for K large enough

1 < n g (K, η) < n (K) Kx * 2 < n r (K) < n d (K, η) . P 4.1.
For any η > 0 there exists a η > 0, and K η > 0 such that, if K > K η and φ of norm one in 2 satis es

L K φ = -ρ φ where ρ < η then sup n (K)≤n≤nr(K) φ(n) ≤ e -aη(log K) 2 .
PROOF. Let us consider an eigenvector φ of norm one in 2 satisfying L K φ = -ρ φ.

If φ(n g (K, η)) = 0, define ñg (K, η) = n g (K, η) and if needed, change the sign of φ such that φ(ñ g (K, η)) > 0. If φ(n g (K, η)) = 0, define ñg (K, η) = n g (K, η)+1 and if needed, change the sign of φ such that φ(ñ g (K, η)) > 0 (note that φ(n g (K, η)) = φ(n g (K, η) + 1) = 0 contradicts the normalisation since φ solves a second-order recurrence relation). Then, changing the definition of n g (K, η) if necessary, we can assume that φ(n g (K, η)) > 0.

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

1) φ(n g (K, η) + 1) ≥ φ(n g (K, η)) and φ(n) is increasing on n g (K, η), n d (K, η) .
2) φ(n g (K, η) + 1) < φ(n g (K, η)) and φ(n) is decreasing and stays nonnegative on n g (K, η), n d (K, η) .

3) φ(n g (K, η) + 1) < φ(n g (K, η)) and φ(n) has a minimum in the interval n min (K) in n g (K, η) -1, n d (K, η) -1 and φ(n min (K)) ≥ 0. Note that φ(n) is decreasing on n g (K, η), n min (K) and increasing on n min (K), n d (K, η) .

4) φ(n) is decreasing on n g (K, η), n d (K, η) and φ(n d (K, η)) < 0.

We first observe that since b (0) > d (0) we have

lim K→∞ sup n ∈ n (K) 2 , n (K) λ (K) n-1 µ (K) n λ (K) n + µ (K) n -η -λ (K) n µ (K) n+1 = b (0) d (0) b (0) + d (0) -b (0) d (0) < 1.
We also observe that there exists c > 0 such that for K large enough, and any n ∈ n r (K), n d (K, η) we have

λ (K) n µ (K) n+1 λ (K) n + µ (K) n -η -λ (K) n-1 µ (K) n ≤ 1 1 + c (n-Kx * ) 2 K 2 .
The result follows by inspecting the monotonicity in the di erent cases and using the last part of Proposition (8.1).

T 4.2. Let φ ∈ Dom(L K ) ⊂ 2 of norm 1, satisfying L K φ = -ρ φ
for some real ρ. Then there exist C(φ) > 0 and an integer r(φ) such that for all n ≥ r(φ) we have

|φ(n)| ≤ C(φ) 2 -n .
Moreover, (φ(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 (K) such that, for all n ≥ r 0 (K), we have

0 < λ (K) n µ (K) n+1 λ (K) n + µ (K) n+1 + ρ -λ (K) n-1 µ (K) n ≤ 1 2 .
3. φ(r 0 (K) + 1) < φ(r 0 (K)), and there exists r > r 0 (K) such that φ(r ) < 0 and φ is not monotonous on r 0 (K), r , and φ ≥ 0 on r 0 (K), r -1 .

Then there exists r < r such that φ is decreasing on r 0 (K), r , and such that φ(r + 1) ≥ φ(r ). If φ(r ) > 0, then we are in case 1. If φ(r + 1) > 0, it follows from Proposition 8.3 that φ is increasing for n ≥ r + 1, contradicting that φ has norm 1. If φ(r + 1) = φ(r ) = 0 then φ is the null sequence as solution of a second order equation, which leads to a contradiction.

4. φ(r 0 (K) + 1) < φ(r 0 (K)), φ ≥ 0. Suppose that there exists a local minimum at r (finite). Then if φ(r + 1) ≥ φ(r ) > 0, we are in case 1. If φ(r + 1) > φ(r ) = 0, then it follows from Proposition 8.3 that φ is increasing for n ≥ r + 1, contradicting that φ has norm 1. If φ(r + 1) = φ(r ) = 0 then φ is the null sequence, which leads to a contradiction.

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

Let us now prove two key lemmas.

L K φ (K) = -ρ (K) φ (K) .
Assume that there exists a diverging sequence (K p ) such that

0 ≤ lim p→∞ ρ (Kp) = ρ * < +∞ and lim sup p→∞ φ (Kp) 1 {• ≤n l (Kp)} 2 > 0.
Then ρ * ∈ S 2 and there exists a diverging subsequence K p such that the limit

lim →∞ φ (Kp ) = φ *
exists in 2 , φ * 2 > 0 and φ * is an eigenvector of M 0 with eigenvalue -ρ * .

PROOF. Let (p ) be a diverging sequence of integers such that

lim →∞ φ (Kp ) 1 {• ≤n l (K)} 2 = lim sup p→∞ φ (Kp) 1 {• ≤n l (Kp)} 2 .
We define for each a normalized sequences in 2 by

ψ (n) = φ (Kp ) (n) 1 {n ≤n l (Kp )} φ (Kp ) 1 {• ≤n l (Kp )} 2 .
It is easy to verify using Proposition 4.1 that ψ ∈ Dom(M 0 ) and

M 0 ψ + ρ (Kp ) ψ 2 ≤ O(1) (log K p ) 2 e -a (log Kp ) 2 φ (Kp ) 1 {• ≤n l (Kp )} 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 M 0 is discrete and simple by Theorem 7.1. L 4.4. Let φ (K) ∈ Dom L K be a normalized sequence such that

L K φ (K) = -ρ (K) φ (K) .
Let us assume that there exists a diverging subsequence (K p ) such that

0 ≤ lim p→∞ ρ (Kp) = ρ * < +∞ and lim p→∞ φ (Kp) 1 {• ≥nr(Kp)} 2 > 0.
Then ρ * ∈ S 1 and there exists a diverging subsequence of integers (p ) such that

lim →∞ Q Kp φ (Kp ) Q Kp φ (Kp ) L 2 = ψ * exists in L 2 , ψ * L 2 = 1
, and ψ * is an eigenvector of H * with eigenvalue -ρ * .

PROOF. We define for each p a normalized sequence in 2 by

ψ (Kp) (n) = φ (Kp) (n) 1 {n ≥ nr(Kp)} φ (Kp) 1 {• ≥ nr(Kp)} 2 .
It is easy to verify using Proposition 4.1 that ψ (Kp) ∈ Dom L Kp ) and

L Kp ψ (Kp) + ρ (Kp) ψ (Kp) 2 ≤ O(1) K p e -a (log Kp) 2 φ (Kp) 1 {• ≥ nr(Kp)} 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 ) such that the sequence of functions Q Kp ψ (Kp ) converges in L 2 to a normalised fonction ψ * . Let u ∈ D. We have from (4.2)

lim →∞ P Kp u , L Kp ψ (Kp ) + ρ (Kp ) ψ (Kp ) 2 = 0 hence (since P Kp u ∈ Dom L Kp ) lim →∞ L Kp P Kp u , ψ (Kp ) 2 = -ρ * lim →∞ P Kp u , ψ (Kp ) 2 .
From the isometric property of Q (see Lemma 3.1) we get

lim →∞ Q Kp L Kp P Kp u , Q Kp ψ (Kp ) L 2 = -ρ * lim →∞ Q Kp P Kp u , Q Kp ψ (Kp ) L 2 .
In other words

lim →∞ L Kp u, Q Kp ψ (Kp ) L 2 = -ρ * lim →∞ Q Kp P Kp u, Q Kp ψ (Kp ) L 2 .
Using the convergence in L 2 of Q Kp ψ (Kp ) to ψ * , Proposition 3.3, and Lemma 3.2, we get for all u ∈ D,

H * u, ψ * L 2 = -ρ * u , ψ * L 2 .
Since D is dense in the domain of the self-adjoint operator H * , we conclude that ψ * is an eigenvector of H * .

We now state three key propositions. . Then there exists an in nite sequence of integers (p ) such that

• φ (Kp ) j 1 {• < n l (Kp )} 2 -→ φ * ,
where φ * is either the null sequence or an eigenvector of M 0 with eigenvalue -ρ * , namely ρ * ∈ S 2 .

• Q Kp φ (Kp ) j 1 {• ≥ nr(Kp )} L 2 -→ ϕ * , where ϕ * is either the null function or an eigenvector of H * with eigenvalue -ρ * , namely ρ * ∈ S 1 . Moreover, φ * 2 2 + ϕ * 2 L 2 = 1.
PROOF. We have either

lim p→∞ φ (Kp) j 1 {• <n l (Kp)} 2 = 0
or there exists an infinite sequence of integers (p ) such that

lim →∞ φ (Kp ) j 1 {• <n l (Kp )} 2 > 0 .
The first statement of the proposition follows from Lemma 4.3 applied to the diverging sequence (K p ). Similarly, either

lim p→∞ φ (Kp) j 1 {• ≥ nr(K)} 2 = 0
or there exists an infinite sequence of integers (p ) such that

lim →∞ φ (Kp ) j 1 {• ≥ nr(Kp )} 2 > 0.
The second statement of the proposition follows from Lemma 4.4 applied to the diverging sequence (K p ). The last statement follows from the normalisation φ (Kp) j and Proposition 4.1. P 4.6. Let ρ ∈ S 2 . Then there exists a normalized vector v ρ ∈ Dom(L K ) with eigenvalue -ρ such that

lim K→∞ L K v ρ + ρ v ρ 2 = 0. Moreover lim K→+∞ v ρ 1 {• > n l (K)} 2 = 0.
PROOF. Let v ρ be a normalized eigenvector corresponding to the eigenvalue -ρ for the operator M 0 in the space 2 . From the assumptions we have for

n ≤ log K λ (K) n = b (0) + O log K K and µ (K) n = d (0) + O log K K .
Since M 0 v ρ + ρ v ρ = 0, the reader can easily check that

L K v ρ + ρ v ρ 2 (Z >0 ∩{1,..., log K }) ≤ O log K K .
Now using Theorem 7.1 we have

v ρ (n) = √ n d (0) b (0) n 2 P (n)
where P is certain polynomial. Hence there exists c v > 0 such that for any

n ∈ Z >0 , |v ρ (n)| ≤ c v d (0) b (0)
L K v ρ + ρ v ρ 2 (Z >0 ∩{ log K +1,...,∞}) ≤ O(1) d (0) b (0) log K 4 .
It follows that v ρ ∈ Dom(L K ), and (remember that b (0

) > d (0)) lim K→∞ L K v ρ + ρ v ρ 2 = 0.
The other statement follows at once from the exponential decay of v. P 4.7. Let ρ ∈ S 1 . Then there exists a sequence of normalized vectors (ψ

(K) ρ ) K ⊂ 2 such that ψ (K)
ρ ∈ Dom(L K ), and

lim K→∞ L K ψ (K) ρ + ρ ψ (K) ρ 2 = 0. Moreover lim K→∞ ψ (K) ρ 1 {• <nr(K)} 2 = 0.
PROOF. Let ϕ ρ be a real normalized eigenvector of H * corresponding to the eigenvalue -ρ (in L 2 ). Since ϕ ρ is a (rescaled) Hermite function (see [START_REF] Babusci | Mathematical Methods for Physicists[END_REF] or Remark 3.1), it satisfies the hypothesis of Proposition 3.3, hence

lim K→∞ L K ϕ ρ + ρ ϕ ρ L 2 = 0.
Using Lemma 3.2 we get

lim K→∞ L K ϕ ρ + ρ Q K P K ϕ ρ L 2 = 0
and then lim K→∞

L K P K ϕ ρ + ρ P K ϕ ρ 2 = 0 and lim K→∞ P K ϕ ρ 2 = 1.
The first statement follows by letting ψ (K) ρ (n) = (P K ϕ ρ )(n) where P K is defined in (3.1). The other statement follows from an exponential bound on the decay of ϕ ρ .

5 Proof of Theorems 3.5 and 3.7 The proof of Theorem 3.5 is an immediate consequence of the following two propositions. P 5.1. We have S 2 ⊂ G.

PROOF. The proof is by contradiction. Let ρ ∈ S 2 and assume ρ / ∈ G. Then there exists η > 0 be such that for K large enough, [ ρη, ρ + η ] ∩ G = ∅. It follows from Proposition 4.6 that there exists a normalized vector v in 2 , such that v ∈ Dom(L K ), and

lim K→∞ L K v + ρ v 2 = 0.
Therefore, using Proposition A.1, we obtain a contradiction, hence the proof is finished.

P 5.2. We have S 1 ⊂ G.
PROOF. The proof is by contradiction. Let ρ ∈ S 1 and assume ρ / ∈ G. Then there exists η > 0 be such that for K large enough, [ρη, ρ + η] ∩ G = ∅. It follows from Proposition 4.7 that there exists a sequence of normalized vectors

(ψ (K) ) K ⊂ 2 such that ψ (K) ∈ Dom(L K ) and lim K→∞ L K ψ (K) + ρ ψ (K) 2 = 0.
Therefore, using Proposition A.1, we obtain a contradiction, hence the proof is finished.

Proof of Theorem 3.7

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

lim inf p→∞ min ρ (Kp) j+1 -ρ * , ρ (Kp)
j-1ρ * = 0.

Assume ρ * ∈ S 1 \S 2 and let (p ) be a diverging sequence of positive integers such that ρ

(Kp ) i → ρ * and ρ (Kp ) i+1 → ρ * (where i = j or i = j -1). Let φ (Kp ) i
be a normalized eigenvector of L Kp corresponding to the eigenvalue -ρ

We define φ (Kp ) i+1 similarly. We claim that

lim sup →+∞ φ (Kp ) i 1 {• < n l (Kp )} 2 = 0.
Otherwise Proposition 4.5 would imply that ρ * ∈ S 2 , a contradiction. By Proposition 4.1 we have

lim →+∞ φ (Kp ) i 1 {• > nr(Kp )} 2 = 1.
This implies 1-a). By a similar argument, we have

lim →+∞ φ (Kp ) i+1 1 {• > nr(Kp )} 2 = 1.
By Proposition 4.5, there exists a diverging sequence of integers ( r ) such that

Q Kp r φ (Kp r ) i 1 {• > nr(Kp r )} -ϕ * L 2 → 0 where ϕ * ∈ Dom(H * )
is a normalized eigenfunction of H * corresponding to the eigenvalue -ρ * . By Proposition 4.5 again, there exists a diverging sequence of integers (r s ) such that

Q Kp rs φ (Kp rs ) i+1 1 {• > nr(Kp rs )} -ϕ * L 2 → 0 where ϕ * ∈ Dom(H * ) is a normalized eigenfunction of H * corresponding to the eigenvalue -ρ * . Since φ (Kp rs ) i and φ (Kp rs ) i+1
are orthogonal in2 , it follows from the previous estimates that

lim s→+∞ φ (Kp rs ) i 1 {• > nr(Kp rs )} , φ (Kp rs ) i+1 1 {• > nr(Kp rs )} 2 = 0. By Lemma 3.1 we have Q Kp rs φ (Kp rs ) i 1 {• > nr(Kp rs )} , Q Kp rs φ (Kp rs ) i+1 1 {• > nr(Kp rs )} L 2 ---→ s→∞ 0.
In other words, ϕ * , ϕ * L 2 = 0. This is a contradiction since ϕ * and ϕ * are normalized eigenfunctions of H * corresponding to the same eigenvalue -ρ * which is simple. The case ρ * ∈ S 2 \S 1 is similar (using again Proposition 4.5), so it is left to the reader.

Let us now assume that ρ * ∈ S 1 ∩S 2 . We now prove (3.7) by contradiction. So we assume that there exists δ > 0 such that

lim inf p→∞ min ρ (Kp) j+1 -ρ * , ρ (Kp) j-1 -ρ * > δ.
Since ρ * ∈ S 1 and from Proposition 4.7, there exists a sequence of normalized vectors (ψ

(Kp) ρ * ) p ⊂ 2 such that ψ (Kp) ρ *
∈ Dom(L Kp ) for all p, and

lim p→∞ L Kp ψ (Kp) ρ * -ρ * ψ (Kp) ρ * 2 = 0.
We also have (since ρ

(Kp) j → ρ * ) that lim p→∞ L Kp ψ (Kp) ρ * -ρ (Kp) j ψ (Kp) ρ *
For each p, let φ We now apply Proposition 4.5 with j and K p . Hence there exists a diverging sequence of integers ( s ) such that

φ (Kp s ) j 1 {• < n l (Kp s )} 2 -→ φ * ,1 and Q Kp s φ (Kp s ) j 1 {• ≥ nr(Kp s )} L 2 -→ ϕ * ,1
and we have φ * ,1

2 2 + ϕ * ,1 2 L 2 = 1.
We now apply Proposition 4.5 with j -1 and K p s . Hence there exists a diverging sequence of integers (s r ) such that

φ (Kp sr ) j-1 1 {• < n l (Kp sr )} 2 -→ φ * ,2 and Q Kp sr φ (Kp sr ) j-1 1 {• ≥ nr(Kp sr )} L 2 -→ ϕ * ,2
and we have φ * ,2

2 2 + ϕ * ,2 2 L 2 = 1.
We now apply Proposition 4.5 with j + 1 and K p sr . Hence there exists a diverging sequence of integers (r q ) such that

φ (Kp sr q ) j+1 1 {• < n l (Kp sr q )} 2 -→ φ * ,3
and

Q Kp sr q φ (Kp sr q ) j+1 1 {• ≥ nr(Kp sr q )} L 2 -→ ϕ * ,3
and we have φ * ,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.

2 2 + ϕ * ,3 2 L 2 = 1. Moreover, since φ (Kp sr q ) j-1 , φ (Kp
6 Fréchet-Kolmogorov-Riesz compactness criterion and Dirichlet form

Fréchet-Kolmogorov-Riesz compactness criterion

We recall the Fréchet-Kolmogorov-Riesz compactness criterion in L 2 . T 6.1. Let (f p ) p be a normalized sequence in L 2 such that the following two conditions are satis ed.

(i) There exists 0 > 0 such that there exists a function R( ) > 0 on (0, 0 ) such that sup

p {|x|>R( )} f p (x) 2 dx ≤ .
(ii) There exits a positive function α on ]0, 1] satisfying lim y 0 α(y) = 0 and such that for any p and y ∈ ]-1, 1]

f p (x + y) -f p (x) 2 dx ≤ α(|y|).
Then one can extract from (f p ) p a convergent subsequence in L 2 .

We refer to Remark 5 on page 387 in [START_REF] Hanche-Olsen | The Kolmogorov-Riesz compactness theorem[END_REF]. The following Lemmas provide expressions for R( ) and α(y) in our case. Recall that the potential V n (K) and n r (K) have been defined in Section 4 (see (4.1)). L 6.2. Let C > 0. Let F C, K be the set of normalized sequences (φ(n)) n in 2 such that φ(n) = 0 for any n < n r (K) and

∞ n=nr(K) 1 ∨ V n (K) φ 2 (n) ≤ C .
Then, for any φ ∈ F C,K , the function Q K φ(x) satis es condition (i) in Theorem 6.1 with R( ) = C for any 0 < < 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 φ ∈ F C,K {|x|>R}

Q K φ(x) 2 dx ≤ C R .
We observe that {|x|>R}

Q K φ(x) 2 dx ≤ n:|n-Kx * |>R √ K-1 φ 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 λ (K) n and µ (K) n that there exist constants K 0 > 4, 1 > C 2 > 0, Γ > 0 and ζ > 0, such that for any K > K 0 there exists an integer Γ K > m * (K) > 2 Kx * (hence of order K) such that µ (K) n > ζ n for any n ≥ m * (K) and

m * (K) n=nr(k) (1 ∨ V n (K)) φ 2 (n) ≥ C 2 K -1 m * (K) n=nr(K) (n -Kx * ) 2 φ 2 (n) ∞ n=m * (K)+1 (1 ∨ V n (K)) φ 2 (n) ≥ C 2 ∞ n=m * (K)+1 n φ 2 (n) .
These estimates imply the following bounds for any integer L > 0

{nr(K)≤n≤m * (K)}∩{|n-Kx * |>L} φ 2 (n) ≤ KC C 2 L 2 1 {L<m * (K)} (6.1) {n>m * (K)}∩{|n-Kx * |>L} φ 2 (n) ≤ C C 2 (L ∨ m * (K)) . (6.2)
We now replace L with R √ K -1 in the above estimates. Let K > 4 be fixed. We distinguish two cases according to the value of R.

1) 1 ≤ R < m * (K)/ √ K. Then √ K -1 ≤ L < m * (K) ≤ ΓK. Since L = R √ K -1 > R √ K/2 (because K > 4), we have {|n-Kx * |>R √ K-1} φ 2 (n) ≤ 4 C R 2 + C C 2 m * (K) ≤ 4C C 2 R 2 + C Γ C 2 R 2 ≤ C R if R > C -1 2 4 + Γ). 2) R ≥ m * (K)/ √ K. Then L ≥ m * (K). We get {|n-Kx * |>R √ K-1} φ 2 (n) ≤ C C 2 L ≤ 2C R C 2 √ K ≤ C R if C 2 2 K > 4. We define K 0 = 5 + 4 C -2 2 and R 0 = 1 + C -1 2
4 + Γ). The result follows. L 6.3. Let (φ (K) ) K be a sequence of normalized elements of 2 such that φ (K) (n) = 0 for n ≤ n r (K). Assume also that there exists C > 0 such that

sup K>1 ∞ n=nr(K) λ (K) n µ (K) n+1 φ (K) (n + 1) -φ (K) (n) 2 + ∞ n=nr(K) (1 ∨ V n (K)) φ (K) (n) 2 ≤ C.
Then there exits a positive constant C such that for any |h| ≤ 1 and any K > 1

Q K φ (K) (x + h) -Q K φ (K) (x) 2 dx ≤ α(h) = C |h| .
Hence, for any (φ (K) ) K satisfying the above assumptions, the sequence of functions (Q K φ (K) ) K satis es condition ii) in Theorem 6.1 with α(y) = Cy .

PROOF. It is enough to consider the case 0 < h < 1. We first consider the case 0 < h ≤ 1/ √ K. We have

Q K φ (K) (x+h)-Q K φ (K) (x) 2 dx = q≥1 I (K) q Q K φ (K) (x+h)-Q K φ (K) (x) 2 dx. Since Q K φ (K) (x) = K 1 4 q≥1 φ (K) (q) 1 I (K) q (x)
and since the intervals I (K) q are disjoint, we get

Q K φ (K) (x + h) -Q K φ (K) (x) 2 dx = K 1 2 q≥1 I (K) q φ (K) (q) 1 I (K) q (x + h) + φ (K) (q + 1) 1 I (K) q+1 (x + h) -φ (K) (q) 1 I (K) q (x) 2 dx = K 1 2 q≥1 I (K) q φ (K) (q) 2 1 I (K) q (x + h) + φ (K) (q + 1) 2 1 I (K) q+1 (x + h) + φ (K) (q) 2 1 I (K) q (x) -2 φ (K) (q) 2 1 Iq (x)1 I (K) q (x + h) -2 φ (K) (q) φ (K) (q + 1) 1 I (K) q (x)1 I (K) q+1 (x + h) dx
Let us consider each term separately. Since I (K)

q dx = K -1 2 , we have K 1 2 q≥1 I (K) q φ (K) (q) 2 dx = q≥1 φ (K) (q) 2 .
Then we have

K 1 2 q≥1 I (K) q φ (K) (q) 2 1 I (K) q (x + h) dx = K 1 2 q≥1 φ (K) (q) 2 q √ K + 1 2 √ K -x * -h q √ K -1 2 √ K -x * dx = 1 -hK 1 2 q≥1 φ (K) (q) 2 .
We also have

K 1 2 q≥1 I (K) q φ (K) (q + 1) 2 1 I (K) q+1 (x + h) dx = K 1 2 q≥1 φ (K) (q + 1) 2 q √ K + 1 2 √ K -x * q √ K + 1 2 √ K -x * -h dx = K 1 2 q≥1 φ (K) (q + 1) 2 h = K 1 2 h q≥1 φ (K) (q) 2 .
Similarly

-2 K 1 2 q≥1 I (K) q φ (K) (q) 2 1 I (K) q (x)1 Iq (x + h) dx = -2 1 -hK 1 2 q≥1 φ (K) (q) 2 and -2K 1 2 q≥1 I (K) q φ (K) (q) φ (K) (q + 1) 1 I (K) q (x)1 I (K) q+1 (x + h) dx = -2hK 1 2
q≥1 φ (K) (q) φ (K) (q + 1) .

We rewrite the last term:

-2 h √ K q≥1 φ (K) (q) φ (K) (q + 1) = -2 h √ K q≥1 φ (K) (q) 2 + 2 h √ K q≥1 φ (K) (q) φ (K) (q + 1) -φ (K) (q) .
Summing up, we get

Q K φ (K) (x + h) -Q K φ (K) (x) 2 dx = q≥1 φ (K) (q) 2 + 1 -h √ K q≥1 φ (K) (q) 2 + K 1 2 h q≥1 φ (K) (q) 2 -2 1 -h √ K q≥1 φ (K) (q) 2 -2 h √ K q≥1 φ (K) (q) 2 + 2 h √ K q≥1 φ (K) (q) φ (K) (q + 1) -φ (K) (q) = 2 h √ K q≥1 φ (K) (q) φ (K) (q + 1) -φ (K) (q) .
By Cauchy-Schwarz inequality we get

Q K φ (K) (x + h) -Q K φ (K) (x) 2 dx ≤ 2h √ K   q≥1 φ (K) (q + 1) -φ (K) (q) 2   1 2 φ (K) 2 .
From the assumption of the lemma and using

inf n ≥ Kx * 3 -1 λ (K) n µ (K) n+1 > ζK for some ζ > 0 independent of K, we get K 1 2   q≥1 φ (K) (q + 1) -φ (K) (q) 2   1/2 ≤ C ζ .
Therefore, since the sequence φ (K) is normalized, we get

Q K φ (K) (x + h) -Q K φ (K) (x) 2 dx ≤ 2h C ζ .
We now consider the case

1 > h > 1/ √ K. Let r = h √ K and h = h -r/ √ K. Note that 0 ≤ h ≤ 1/ √ K. We have Q K φ (K) (x + h) -Q K φ (K) (x) 2 dx = r-1 j=1 Q K φ (K) x + j + 1 √ K -Q K φ (K) x + j √ K + Q K φ (K) x + r √ K + h -Q K φ (K) x + r √ K 2 dx ≤ 2 r-1 j=1 Q K φ (K) x + j + 1 √ K -Q K φ (K) x + j √ K 2 dx + 2 Q K φ (K) x + r √ K + h -Q K φ (K) x + r √ K 2 dx.
We have already estimated the last term. For the first term we now observe that e (K) n x + j √ K = e (K) n-j (x). Therefore we can write r-1 j=1

Q K φ (K) x + j + 1 √ K -Q K φ (K) x + j √ K = r-1 j=1 n φ (K) (n) e (K)
n-j-1 (x) - φ (K) (p + j + 1)φ (K) (p + j) .

This implies

r-1 j=1

Q K φ (K) x + j + 1 √ K -Q K φ (K) x + j √ K

Dirichlet form for the operator L 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. P 6.4. If φ is a normalized sequence in Dom(L K ) decaying exponentially fast at in nity, then

-φ, L K φ = ∞ n=1 λ (K) n µ (K) n+1 φ(n + 1) -φ(n) 2 + ∞ n=1 V n (K) φ(n) 2 - 1 2 λ (K) 1 µ (K) 2 φ(1) 2 .
The eigenvector v m corresponding to the eigenvalue (d (0)-b (0) m, where m ∈ Z >0 , is given (up to a multiplicative factor) by where P m is the monic orthogonal polynomial of degree m -1 associated with the measure q on Z >0 de ned by It is easy to verify that M 0 is closable and we denote by M 0 its closure. Since for any m ∈ Z >0 , the sequence v m (n) defined by (7.1) decays exponentially fast with n, it is easy to verify that v m ∈ Dom(M 0 ). Note also that if m = m , v m is orthogonal to v m in 2 .

v m (n) = √ n d (0) b (0)
q(n) = n d (
By a direct computation one checks that M 0 v 1 = (d (0)b (0))v 1 (recall that P 1 (n) = 1). It is left to the reader to check that

M 0 v m (n) = √ n b (0) d (0) n 2 Q m (n)
where Q m is a polynomial in n in which the coe cient of n m-1 is m(d (0)b (0)) .

To check that the v m are eigenvectors, we use a recursive argument. Assume that m ≥ 2 and for 1 ≤ k ≤ m -1

M 0 v k = k (d (0) -b (0)) v k .
We can write

M 0 v m = m (d (0) -b (0)) v m + r m with r m = √ n b (0) d (0) n 2 R m
where R m is a polynomial in n of degree at most m -2. Therefore r m ∈ Span v 1 , . . . , v m-1 .

From our recursive assumption, the symmetry of M 0 , and the orthogonality of the v k (following from the orthogonality of the P k ), we get that for any

1 ≤ k ≤ m -1 0 = v k , M 0 v m 2 = v k , r m 2 .
Therefore r m = 0. Hence M 0 v m = m (d (0)b (0)) v m , and we can proceed with the recursion. We now prove that the v m form a basis of 2 . Assume the contrary, namely there exists u ∈ 2 of norm one such that for any m

∞ n=1 u(n) v m (n) = 0.
We observe that the sequence

w(n) = 1 √ n b (0) d (0) n 2 u(n)
belongs to 2 (q), whith q defined in (7.2). Therefore our assumption on u implies that w is orthogonal to all the polynomials in 2 (q). Let us show that the set of polynomials is dense in 2 (q). It is su cient 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 γ m of q, satisfy the following property: there exists C > 0 such that, for any m ∈ Z >0 ,

γ m = ∞ n=1 n m+1 d (0) b (0) n ≤ C m m! .
The proof is left to the reader. Therefore the set of all polynomials is dense in 2 (q) implying w = 0 and we get a contradiction with the existence of a u nonzero orthogonal to all the v m in 2 . Therefore, the v m form a basis of 2 . We now observe that 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 2 , for any B > 0 we have ker(M † 0 -B) = {0}. Hence M 0 is self adjoint (see for instance [START_REF] Schmüdgen | Unbounded Self-adjoint Operators on Hilbert Space[END_REF]Prop. 3.9,p. 43]) and the spectrum is given by

Sp(M 0 ) = (d (0) -b (0))Z >0 .
This ends the proof.

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.

T

R 3 . 1 .

 31 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 H * ared (x * )-b (x * ))n, n ∈ Z ≥0 , and the corresponding eigenfunctions are

T 3 . 7 . 6 ) 2 - 2 -

 37622 Let ρ ∈ S 1 ∪ S 2 such that ρ > α.If j ≤ j c -1, we have lim sup p→+∞ ρ(Kp) j ≤ lim sup p→+∞ ρ (Kp)jc-1 ≤ α < ρ. For all j ≥ j c , we have lim inf p→+∞ ρ (Kp) j ≥ lim inf p→+∞ ρ (Kp) jc = +∞. This implies ρ / ∈ G, contradicting Theorem 3.5. We have S 1 ∪ S 2 ⊃ G. Moreover, for each j ∈ Z ≥0 , let (K p ) p be a diverging sequence such that lim where ρ * is nite by Corollary 3.6. Then 1) If ρ * ∈ S 1 ∆S 2 then Moreover there are only two cases: (a) If ρ * ∈ S 1 then there exists a diverging sequence of integers (p ) such that Q Kp φ (Kp ) j L → ϕ * , where ρ * and ϕ * are such that H * ϕ * = -ρ * ϕ * . (b) If ρ * ∈ S 2 then φ (Kp) j → φ * , where ρ * and φ * are such that M 0 φ * = -ρ * φ * . 2) If ρ * ∈ S 1 ∩ S 2 then we have the following two assertions: (a) There exists a diverging sequence of integers (p ) such that:

Figure :

 : Figure: Schematic representation of one of the three possible 'shapes' of the eigenvectors φ (K) (with distortion).

L 4 . 3 .

 43 Let φ K ∈ Dom L K be a normalized sequence such that

P 4 . 5 .

 45 Let j be xed and let (K p ) be a diverging sequence such that lim eigenvector of L Kp with eigenvalue -ρ (Kp) j

n 4 .

 4 By (2.1) there exists c > 0 such b(x) + d(x) ≤ c e x for all x ≥ 0. This implies that λ (Kp) n ≤ cK p e n/Kp and µ (Kp) n ≤ cK p e n/Kp . The reader can easily check that

5. 1

 1 Proof of Theorem 3.5

1 )Lρ * 1 {

 11 eigenvector of L Kp corresponding to the eigenvalue -ρ (Kp) j. By using Lemma A.2 with A = L Kp , we deduce that there exists a sequence of real numbers (θ p ) p such that lim Since ρ * ∈ S 2 and from Proposition 4.6, there exists a normalized vector v ρ * ∈ Dom(L Kp ) for all p such that limp→∞ Kp v ρ * + ρ * v ρ * 2 = 0.We also have lim p→∞L Kp v ρ * + ρ (Kp) j v ρ * 2 = 0.By using Lemma A.2 with A = L Kp , we deduce that there exists a sequence of real numbers (θ p ) p such thatlim p→∞ v ρ *e iθ p φ • < nr(Kp)} 2 = 0 and lim K→+∞ v ρ * 1 {• > n l (K)} 2 =0 which implies (using Proposition 4.1) that lim p→∞ ψ (Kp) ρ * , v ρ * = 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 ) such that lim →+∞ ρ (Kp ) j+1ρ * + ρ (Kp ) j-1ρ * = 0.

  orthogonal, we have φ * ,m , φ * ,m 2 + ϕ * ,m , ϕ * ,m L 2 = 0, ∀m = m . (5.3) The linear subspace of 2 spanned by φ * ,1 , φ * ,2 , φ * ,3 is of dimension at most one because they are eigenvectors of M 0 for the same simple eigenvalue -ρ * . The linear subspace of L 2 spanned by ϕ * ,1 , ϕ * ,2 , ϕ * ,3 is of dimension at most one because they are eigenfunctions of H * for the same simple eigenvalue -ρ * . Therefore the subspace of 2 ⊕ L 2 spanned by the three vectors (φ * ,1 , ϕ * ,1 ), (φ * ,2 , ϕ * ,2 ), (φ * ,3 , ϕ * ,3

  p (x) φ (K) (p + j + 1)φ (K) (p + j)

2 = r 2 pφ 2 K

 222 ) (p + j + 1)φ (K) (p + j) ) (p + j + 1)φ (K) (p + j) (K) (p + 1)φ (K) (p)as we have seen before. We observe that r2 /K ≤ h 2 ≤ h since h ≤ 1,and the result follows by taking C = 2 C ζ + C ζ .

2 )

 2 PROOF. It is easy to verify that M 0 is a symmetric operator on c 00 , which is bounded above since from b (0) > d (0), we haveinf n n (b (0) + d (0))b (0) d (0) n (n + 1)b (0) d (0) n (n -1) > -∞.

We denote by Z≥0 the set of non-negative integers, and by Z>0 the set of positive integers.

= 0.

We can assume that (φ(n)) n is a sequence of real numbers and φ(r 0 (K)) > 0. We start by proving that (φ(n)) n is positive and decreasing for n ≥ r 0 (K). There are only the following four possibilities.

1. φ(r 0 (K) + 1) ≥ φ(r 0 (K)). It follows from Proposition 8.3 that φ is increasing for n ≥ r 0 (K), contradicting that φ has norm 1.

2. φ(r 0 (K) + 1) < φ(r 0 (K)), and there exists r > r 0 (K) such that φ(r ) < 0 and φ decreases on r 0 (K), r , and φ ≥ 0 on r 0 (K), r -1 . Then by Proposition 8.3, φ is decreasing for n ≥ r , contradicting that φ is normalized.

PROOF. For any fixed positive integer N we have

Since φ ∈ Dom(L K ), the functions φ 1 {n≤N } and L K (φ1 {n≤N } ) converge to φ, respectively L K φ, in 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 λ N (K) and µ N (K) are exponential in N and φ decays exponentially fast by assumption. L 6.5. There exists ξ > 0 such that for all K ∈ Z >0 , inf n≥1 V n (K) ≥ -ξ.

PROOF. Since (λ (K) n ) n is an increasing sequence we have

It follows from the general assumptions (see Section 2) that there exists x ≥ 1 such that all K ∈ Z >0 and for all n ≥ K x we have

For all n ≥ K x we have V n (K) ≥ 0. When n ≤ K x, we write

Now observe that

The rest of the proof is obvious. P 6.6. Let δ > 0 and φ be a real normalized sequence in Dom(L K ) decaying exponentially fast, such that

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

Spectral theory of M 0

Recall that (cf. (3.5))

The operator M 0 de ned on c 00 is symmetric for the scalar product of 2 . We denote by M 0 its closure which is self-adjoint and bounded above. The spectrum of M 0 is discrete, all eigenvalues are simple, and we have

Let (α n ) be a nite sequence of strictly positive real numbers de ned for n 1 -1, . . . , n 2 . Let (β n ) be a nite sequence of strictly positive real numbers de ned for n 1 , . . . , n 2 . Let (u n ) be a nite sequence of real numbers de ned for n 1 -1, . . . , n 2 + 1. Assume that, for all n 1 ≤ n ≤ n 2 , we have

then the sequence (u n ) has no positive local maxima for n ∈ {n 1 + 1, . . . , 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 α n u n+1 + α n-1 u n-1β n u n ≤ 0, then the sequence (u n ) has no positive local minima for n ∈ {n 1 + 1, . . . , 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. P 8.3. Let 1 < n 1 < n 2 be integers such n 2 > n 1 + 1. Let (α n ) be a nite sequence of strictly positive real numbers de ned for n 1 -1, . . . , n 2 . Let (β n ) be a nite sequence of strictly positive real numbers de ned for n 1 , . . . , n 2 . Let (u n ) be a nite sequence of real numbers de ned for n 1 -1, . . . , n 2 + 1. Assume that, for all n 1 ≤ n ≤ n 2 , we have A Quasi-eigenvalues and quasi-eigenvectors of selfadjoint operators P A.1. Let A be a self-adjoint operator in a Hilbert space H with domain Dom(A). Assume there exists u ∈ Dom(A) of norm 1, ω ∈ R and > 0 such that

PROOF. We will assume that ω / ∈ Sp(A), otherwise the result is trivial. The proof is then by contradiction. If R ω denotes the resolvent of A at ω, we have

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

where d denotes the Euclidean distance on the real line. If

which is a contradiction. P A.2. Let A be a self-adjoint operator in a Hilbert space H with domain Dom(A). Assume there exists u ∈ Dom(A) of norm 1, ω ∈ R and > 0 such that A uω u ≤ .

Assume A has discrete spectrum with eigenvalues of multiplicity one, and let δ > 0 denote the minimum distance between two consecutive eigenvalues. Then if < δ there is a λ ∈ Sp(A) with a normalized eigenvector e such that |ω -λ| ≤ and ue ≤ 2 δ -.

PROOF. We will denote by P z the one-dimensional spectral projector of A corresponding to z ∈ Sp(A). Since < δ, using Proposition A.1 we conclude that there is only one eigenvalue of A in [ ω -, ω + ] and we denote by λ this eigenvalue and by ẽ one of the corresponding eigenvectors (they all di er only by a phase factor). Let v = ω u -A u.

Since u = P λ u + (Id -P λ )u, we get from the spectral decomposition A(Id -P λ ) uω(Id -P λ ) u = (Id -P λ ) v.

This implies that (Id -P λ ) u ≤ δ -.

Since P λ u is proportional to ẽ, we can write Let β = |β| exp(iθ), we define e = e i θ ẽ, and the result follows.