PRECISE ESTIMATES FOR BIORTHOGONAL FAMILIES UNDER ASYMPTOTIC GAP CONDITIONS

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PRECISE ESTIMATES FOR BIORTHOGONAL FAMILIES UNDER ASYMPTOTIC GAP

CONDITIONS

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble

To cite this version:

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. PRECISE ESTIMATES FOR

BIORTHOGONAL FAMILIES UNDER ASYMPTOTIC GAP CONDITIONS. Discrete and Con-

tinuous Dynamical Systems - Series S, American Institute of Mathematical Sciences, In press. �hal-

01834726�

UNDER ASYMPTOTIC GAP CONDITIONS

P. CANNARSA, P. MARTINEZ, AND J. VANCOSTENOBLE

Abstract. A classical and useful way to study controllability problems is the moment method developped by Fattorini-Russell [12, 13], and based on the construction of suitable biorthogonal families. Several recent problems exhibit the same behavior: the eigenvalues of the problem satisfy a uniform but rather ’bad’ gap condition, and a rather ’good’ but only asymptotic one.

The goal of this work is to obtain general and precise upper and lower bounds for biorthogonal families under these two gap conditions, and so to measure the influence of the ’bad’ gap condition and the good influence of the ’good’

asymptotic one. To achieve our goals, we extend some of the general results of Fattorini-Russell [12, 13] concerning biorthogonal families, using complex analysis techniques developped by Seidman [36], G¨uichal [20], Tenenbaum- Tucsnak [37] and Lissy [26, 27].

1. Introduction 1.1. Presentation of the subject.

Biorthogonal families are a classical tool in analysis. In particular, they play a crucial role in the so-called moment method, which was developed by Fattorini- Russell [12, 13] to study controllability for parabolic equations.

Given any sequence of nonnegative real numbers, (λ_n)_n≥1, we recall that a se- quence (σm)_m≥1 is biorthogonal to the sequence (e^λnt)_n≥1 inL²(0, T) if

∀m, n≥1, Z T

0

σm(t)e^λntdt=

(1 ifm=n 0 ifm6=n .

The goal of this paper is to provide explicit and precise upper and lower bounds for the biorthogonal family (σm)m≥1 under the following gap conditions:

• a ’global gap condition’:

(1. 1) ∀n≥1, 0< γmin≤p

λn+1−p

λn≤γmax,

• and an ’asymptotic gap condition’:

(1. 2) ∀n≥N^∗, γ_min^∗ ≤p

λn+1−p

λn≤γ_max^∗ , where γ_max^∗ −γ_min^∗ < γmax−γmin.

Before explaining why we are interested in such a question, let us describe some of the main results of the literature on this subject.

1991Mathematics Subject Classification. 11B05, 30B10, 30D15.

Key words and phrases. Biorthogonal families, gap conditions.

This research was partly supported by the Institut Mathematique de Toulouse and Istituto Nazionale di Alta Matematica through funds provided by the national group GNAMPA and the GDRE CONEDP.

1

1.2. The context.

Among the most important applications of biorthogonal families to control the- ory are those to the null controllability and sensitivity of control costs to parameters.

Major contributions in such directions are the following:

• Fattorini-Russell [12, 13], Hansen [21], and Ammar Khodja-Benabdallah- Gonz´alez Burgos-de Teresa [1] studied the existence of biorthogonal se- quences and their application to controllability for various equations;

• for nondegenerate parabolic equations and dispersive equations, Seidman [35], G¨uichal [20], Seidman-Avdonin-Ivanov [36], Miller [31], Tenenbaum- Tucsnak [37], and Lissy [26, 27] studied the dependence of the null control- lability cost C_T with respect to the time T (as T →0, the so-called ’fast control problem’) and with respect to the domain, obtaining extremely sharp estimates of the constantsc(Ω) and C(Ω) that appear in

e^c(Ω)/T ≤CT ≤e^C(Ω)/T;

• Coron-Guerrero [8], Glass [17], Lissy [27] investigated the vanishing viscos- ity problem:

(yt+M yx−εyxx= 0, x∈(0, L), y(0, t) =f(t),

obtaining sharp estimates of the null controllability cost with respect to the timeT, the transport coefficientM, the size of the domainL, and the diffusion coefficientε;

• in [5, 6], we studied the dependence of the controllability cost with respect to the degeneracy parameter αfor the degenerate parabolic equation

u_t−(x^αu_x)_x= 0, x∈(0, `);

• and recently Gueye-Lissy [19] studied a 1-D parabolic-hyperbolic degener- ate equation.

There is a common feature in these works: they depend on some parameter p, and this parameter forces the eigenvalues to satisfy (1. 1) (sometimes after normal- ization) with gap boundsγ_min(p) andγ_max(p) such that

γmin(p)→0 and/or γmax(p)→ ∞.

This fact makes it necessary to have general and precise estimates with respect to the main parameters that appear in the problem.

In [6], we proved the following general result: givenT >0 and a family (λn)_n≥1 of nonnegative real numbers that satisfy the ’global gap condition’ (1. 1), then:

• every family (σm)m≥1, biorthogonal to (e^λnt)n≥1 in L²(0, T), satisfies the lower estimate

(1. 3) kσmk²_L2(0,T)≥bme^−2λmTe

1 2T(γmax )2,

with an explicit value of bm=bm(T, γmax, m) (rational inT);

• there exists a family (σm)_m≥1, biorthogonal to (e^λnt)_n≥1 inL²(0, T), that satisfies the upper estimate

(1. 4) kσmk²_L2(0,T)≤Bme^−2λmTe

Cu T(γmin)2e^Cu

√

λm γmin,

with an explicit value of B_m=B_m(T, γ_min, m) (rational inT).

This generates several comments:

• concerning the results:

– the bounds (1. 3) and (1. 4) above describe quite precisely the behav- ior of the biorthogonal family, in particular in short time; (however, we were not able to make appear the exponential factore^Cu

√

λm

γmin in (1. 3)), and it would be interesting to know if (1. 3) could be improved);

– estimate (1. 4) is in the spirit of [12, 13] but the dependence with respect toT whenT →0⁺ is completely explicit;

• concerning the assumptions:

– assumption (1. 1) is very practical to understand the behavior of the biorthogonal family: γmin allows us to estimate (σm)_m≥1 from above, whileγmaxallows us to estimate (σm)_m≥1from below;

– assumption (1. 1) is in the spirit of the asymptotic development of the eigenvalues used in Tenenbaum-Tucsnak [37] or Lissy [26, 27]:

(1. 5) λ_n=rn²+O(n).

(Note that (1. 1) and (1. 5) are close, but one can easily find sequences that satisfy one of them without satisfying the other:

∗ for example the family {n², n²+√

n} satisfies (1. 5) with r =

1

4, but does not satisfy (1. 1) (the gapsp

λ_n+1−√

λ_n are not uniformly bounded from below by a positive constant);

∗ and, on the contrary, (1. 5) forces the sequence (

√λn

n )n to con- verge (to√

r), while one can construct sequences satisfying (1. 1) for which (

√λ_n

n )_ndoes not converge, considering, e.g., the recur- rence formulap

λn+1=√

λn+εnwithεn∈ {1,2}, and choosing εn to be constant on sufficiently large intervals, in such a way that 1 and 2 are limits of subsequences of (

√λ_n n )n.) 1.3. Motivations and main results of this paper.

Even though the aforementioned results give a fairly good picture of the prop- erties of the family (σ_m)_m, some delicate issues remain to be analyzed and will be addressed in this paper. For instance, one would like to understand the depen- dence of the family (σ_m)_mwith respect to relevant parameters that come into play.

Typical examples of such problems are the following ones.

• For the 1D degenerate parabolic equation ut−(x^αux)x= 0, x∈(0, `),

the eigenvaluesλα,nof the associated elliptic operator (with suitable bound- ary conditions) can be expressed using the zeros of Bessel functions ([18]) and depend on the degeneracy parameter α∈(0,2). One can then prove (see [5, 7]) that the global gap condition (1. 1) is satisfied only with

γmax(α)≥c(2−α)^2/3,

withc >0, while the asymptotic gap condition (1. 2) is satisfied with γ_max^∗ (α)≤c^∗(2−α),

where c^∗>0, after the rank

N^∗(α) = 1 2−α; in this case

γmax(α)

γ^∗_max(α)−→+∞ asα→2⁻;

hence it is natural to think that the better asymptotic gap (1. 2) could be used to improve the estimate (1. 3) of the associated biorthogonal se- quences, but the fact that

N^∗(α)−→+∞ as α→2⁻ is certainly to be taken into account.

• In 2D problems such as the Grushin equation (see [2, 3]), where the solution is decomposed into Fourier modes, one has to give uniform bounds for a certain sequence of elliptic problems, the eigenvalues of which satisfy (1. 1) and (1. 2) with someγmin(m),γ^∗_min(m) andN^∗(m) such that

γmin(m)

γ_min^∗ (m) −→0 asm→ ∞ and

N^∗(m)−→+∞ asm→ ∞;

once again, it is natural to think that the better asymptotic gap (1. 2) could be used to improve the estimate (1. 4) of the associated biorthogonal sequence, but the fact that N^∗(m) →+∞ as m → ∞ is certainly to be taken into account.

The above discussion motivates the general question whether estimates (1. 3) and (1. 4) can be improved when (1. 1) is combined with the asymptotic condition (1. 2). This is exactly what we prove in this paper: roughly speaking, (1. 3) and (1. 4) hold true replacingγ_minbyγ_min^∗ andγ_maxbyγ^∗_max. Moreover, the fact that the ’good’ gap condition (1. 2) holds true only after theN^∗first eigenvalues has a cost, and we obtain a precise estimate for that cost. Our main results (Theorem 2.1 and 2.2) are the following: under (1. 2), we prove that:

• every family (σm)_m≥1, biorthogonal to (e^λnt)_n≥1 in L²(0, T), satisfies the lower estimate

(1. 6) kσmk²_L2(0,T)≥b^∗_me^−2λmTe

2 T(γ∗

max)2,

where the ’cost’ b^∗_m=b^∗_m(T, γ_max, γ_max^∗ , N^∗, m) is a rational function of T that we determine explicitly, and

• there exists a biorthogonal family that satisfies (1. 7) kσ_mk²_L2(0,T)≤B_m^∗ e^−2λmTe

C0 T(γ∗

min)2e^C0

√λm γ∗

min,

where C0 > 0 is a universal constant and B_m^∗(T, γmin, γ_min^∗ , N^∗, m) is a rational function of T that we determine explicitly.

Let us observe that the presence of the exponential factorse

2 T(γ∗

max)2 ande

C0 T(γ∗

min)2

in (1. 6) and (1. 7) is quite natural and has already been pointed out by Seidman- Avdonin-Ivanov [36], Tenenbaum-Tucsnak [37], and Lissy [26, 27] (see also Ha- raux [22] and Komornik [24] for a closely related context). On the other hand, the precise estimate of the behavior of b^∗_m and B_m^∗ with respect to parameters, that we develop in this paper, is completely new and will be crucial for the sensitivity analysis of control costs to be performed in [7].

Our proofs are based on complex analysis techniques and hilbertian methods developed by Seidman-Avdonin-Ivanov [36] and G¨uichal [20]. We have also used an idea from Tenenbaum-Tucsnak [37] and Lissy [26, 27], based on the introduction of an extra parameter depending on T and the gap conditions.

Finally, as mentioned above, the gap conditions (1. 1)-(1. 2) are practical to obtain such estimates from above and below, but one could investigate more general

conditions, of the form

(1. 8) αnn≤λn+1−λn ≤βnn,

with lim infnαn >0 and lim supβn <∞. This would include the case of assump- tions (1. 1) and (1. 5) and could allow to study more general problems.

1.4. Plan of the paper.

The paper is organized as follows:

• in section 2, we state our results;

• section 3 is devoted to the proof of Theorem 2.1 (construction of a biorthog- onal family and derivation of upper bounds);

• section 4 is devoted to the proof of Theorem 2.2 (lower bounds for biorthog- onal families).

2. Setting of the problem and main results

2.1. Existence of a suitable biorthogonal family and upper bounds.

We will prove the following result, that in some sense precises results of Fattorini and Russell [12, 13] (in short time) and are in the spirit of results of Tenenbaum- Tucsnak [37], Lissy [26, 27] (with a slightly weakened assumption on the eigenval- ues).

Theorem 2.1. Assume that

∀n≥1, λn ≥0, and that there is some 0< γmin< γ_min^∗ such that

(2. 1) ∀n≥1, p

λn+1−p

λn ≥γmin,

and

(2. 2) ∀n≥N^∗, p

λn+1−p

λn≥γ_min^∗ .

Denote

(2. 3) M^∗:= (1−γmin

γ_min^∗ )(N^∗−1).

Then there exists a family (σ_m⁺)_m≥1 which is biorthogonal to the family (e^λnt)_n≥1 in L²(0, T):

(2. 4) ∀m, n≥1,

Z T 0

σ_m⁺(t)e^λntdt=δmn.

Moreover, it satisfies: there is some universal constant C independent of T, γ_min, γ_min^∗ ,N^∗ andmsuch that, for allm≥1, we have

(2. 5) kσ_m⁺k²_L2(0,T)≤e^−2λmTe

C T(γ∗

min)2e^C

√

λm γ∗

minB^∗(T, γmin, γ_min^∗ , N^∗, m), where

(2. 6) B^∗(T, γ_min, γ_min^∗ , N^∗, m)

=





 Cu

_(8M∗)!

(λm(γ^∗_min)²T²)^4M∗ + 1

e^CuM∗e^Cu

λN∗ γmin√

λm(_T3/2¹ +_(γ∗ ¹

min)²T²)ifT ≤ _(γ∗¹ min)²

Cu

(^(γ∗min)8M

∗(8M^∗)!

λ^4M_m ^∗ ) + 1

e^CuM∗e^Cu

λN∗ γmin√

λm((γ^∗_min)²+ (γ_min^∗ )³)if T ≥_(γ∗¹ min)²

.

Remark 2.1. Theorem 2.1 completes and improves several earlier results, in par- ticular Theorem 1.5 of Fattorini-Russell [13] and [6] expliciting the dependence of the L² bound with respect to γmin, γ^∗_min and in short time. It is useful in sev- eral problems, in which γmin → 0 with respect to some parameter, which occurs is several cases, see, e.g. [14], [2]. We will use the construction that was used by Seidman, Avdonin and Ivanov in [36], which has the advantage to be completely explicit (which is not the case for the construction of [12, 13, 14, 21, 1], since there is a contradiction argument), combined with some ideas coming from the construc- tion of Tenenbaum-Tucsnak [37] and Lissy [26], adding some parameter, in order to obtain precise results.

2.2. General lower bounds.

We generalize a result of G¨uichal [20] to prove the following Theorem 2.2. Assume that

∀n≥1, λ_n ≥0, and that there are 0< γ_min≤γ_max^∗ ≤γ_max such that

(2. 7) ∀n≥1, γmin≤p

λn+1−p

λn ≤γmax, and

(2. 8) ∀n≥N_∗, p

λ_n+1−p

λ_n≤γ_max^∗ .

Then any family(σ⁺_m)m≥1which is biorthogonal to the family(e^λnt)n≥1inL²(0, T) (hence that satisfies (2. 4)) satisfies:

(2. 9) kσ⁺_mk²_L2(0,T)≥e^−2λmTe

2 T(γ∗

max)2 b^∗(T, γ_max, γ_max^∗ , N_∗, λ₁, m)², where b^∗ is rational inT (and explictly given in the key Lemma 4.4).

Remark 2.2. Theorem 2.2 completes a result of G¨uichal [20] and is useful in several problems, in which γ_max → ∞ with respect to some parameter, which occurs is several cases, see, e.g. [14], and [7]. It is to be noted that the behavior with respect to m can perhaps be improved, comparing with Theorem 1.1 of Hansen [21]. It would be interesting to investigate this.

3. Proof of Theorem 2.1 3.1. The general strategy.

It begins with the following remarks: if the family (σ_m⁺)_m≥1 is biorthogonal to the family (e^λnt)_n≥1, then

∀m, n≥1, Z T

0

σ⁺_m(T−t)e^λn(T−t)dt=δmn, hence

∀m, n≥1, Z T

0

σ⁺_m(T−t)e^λmT

e^−λntdt=δ_mn, hence the family (sm)m≥0 defined by

sm(t) :=σ_m⁺(T−t)e^λmT

is biorthogonal to the family (e^−λnt)n≥1in L²(0, T). Now extendsmby 0 outside (0, T), and consider its Fourier transform

∀z∈C, F(sm)(z) :=

Z

R

sm(t)e^−iztdt.

For allm≥1,F(s_m) is the Fourier transform of a compactly supported function, hence it is an entire function overC, and it satisfies

∀m, n≥1, F(s_m)(−iλ_n) =δ_mn, and it is of exponential type:

|F(sm)(z)| ≤Z T 0

|sm(t)|dt e^T|z|; and also

F(sm)(z) = Z T /2

−T /2

s_m(τ+T

2)e^−iz(τ+T2)dτ, hence

F(sm)(−z)e^−izT2 = Z T /2

−T /2

sm(τ+T

2)e^izτdτ, and

|F(sm)(−z)e^−izT2| ≤Z T /2

−T /2

|sm(τ+T 2)|dτ

e^T2|z|.

Now we recall the Paley-Wiener theorem ([39]): iff :C→Cis an entire function of exponential type, such that there exist nonnegative constantsC, Asuch that

∀z∈C, |f(z)| ≤Ce^A|z|, and iff ∈L²(R), then there existsφ∈L²(−A, A) such that

f(z) = Z A

−A

φ(τ)e^izτdτ.

One of the objects of [36] is to prove the existence of a sequence (f_m)_m of entire functions satisfying

(3. 1)







∀m, n≥1, fm(−iλn) =δmn,

∀z∈C, |fm(−z)e^−izT2| ≤Cme^T2|z|,

∀m≥1, fm∈L²(R)

(see Theorem 2 and Lemma 3 in [36]) under some general assumptions on the se- quence (λ_n)_n. If we can apply such a result in our context (hence with our sequence (λ_n)_n), then the two last properties together with the Paley-Wiener theorem will imply that there exists some φm∈L²(−^T₂,^T₂) such that

fm(−z)e^−izT2 = Z T /2

−T /2

φm(τ)e^izτdτ, hence

f_m(z) = Z T

0

φ_m(t−T

2)e^−iztdt, and then

Z T 0

φ_m(t−T

2)e^−λntdt=f_m(−iλ_n) =δ_mn,

hence (φ_m(t− ^T₂))_m will be biorthogonal to the family (e^−λnt)_n, and (σ⁺_m(t))_m defined by

σ_m⁺(t) =φm(T

2 −t)e^−λmT

will be biorthogonal to the family (e^λnt)n in L²(0, T), as desired. Moreover kσ_m⁺k²_L2(0,T)=e^−2λmT

Z T /2

−T /2

φ_m(τ)²dτ ≤Ce^−2λmTkfmk²_L2(R)

using the Parseval theorem.

Now, it remains to construct such entire functionsfm. The idea is to consider the natural infinite product that satisfies the first condition of (3. 1),fm(−iλn) =δmn, and to multiply it by a so-called ’mollifier’, in such a way that the other two conditions of (3. 1) will be also satisfied. Hence one has to estimate the growth of the natural infinite product, and then to choose a choose a suitable mollifier. This is what is performed in [36]. For our problem, our task will be to add the dependency into the parametersγmin,γ_min^∗ andT, and to understand specifically the behaviour of the natural infinite product, the mollifier and at the end of kσ⁺_mkL²(0,T) with respect to γminandT. We will modify a little the construction of [36], in order to obtain optimal results in our context, see Lemma 3.4, and specifically the definition (3. 19) of the mollifier, where the additionnal parameter N⁰ will be chosen of the size _T(γ∗¹

min)², see (3. 30).

3.2. The counting function.

Consider

∀ρ >0, N_n(ρ) := card {k,0<|λ_n−λ_k| ≤ρ}.

We prove the following:

Lemma 3.1. a) Assume that the gap assumption (2. 1) is satisfied; then (3. 2) ∀n≥0,∀ρ >0, Nn(ρ)≤2

√ρ

γ_min.

b) Assume that the gap assumptions (2. 1)-(2. 2) are satisfied; then

• when n=N^∗:

(3. 3) ∀ρ >0, N_N^∗(ρ)≤ ( ^√ρ

γ_min+

√ρ

γ_min^∗ ifρ≤λ_N^∗ N^∗−1 +

√ρ

γ_min^∗ ifρ≥λ_N^∗,

• when n > N^∗:

(3. 4) ∀n > N^∗,∀ρ >0, Nn(ρ)≤











2√ ρ

γ_min^∗ ifρ≤λn−λN^∗

√ρ γ_min +

√ρ

γ_min^∗ ifλn−λN^∗ ≤ρ≤λn

n−1 +

√ρ

γ_min^∗ ifρ≥λn

,

• when n < N^∗

– whenλn≤λN^∗−λn, then

(3. 5) ∀ρ >0, Nn(ρ)≤











2√ ρ

γmin if ρ≤λn

n−1 +

√ρ

γmin if λ_n ≤ρ≤λ_N^∗−λ_n N^∗−1 +

√ρ

γ_min^∗ if ρ≥λ_N^∗−λ_n ,

– whenλn≥λN^∗−λn, then

(3. 6) ∀ρ >0, Nn(ρ)≤











2√ ρ

γ_min if ρ≤λ_N^∗−λ_n

N^∗−n+

√ρ γ_min +

√ρ

γ_min^∗ if λN^∗−λn≤ρ≤λn

N^∗−1 +

√ρ

γ_min^∗ if ρ≥λ_n

.

Remark 3.1. The main point in (3. 3)-(3. 6) is to observe that Nn behaves as

√ρ

γ^∗_min asρ→+∞, and precising the needed additionnal constants.

Proof of Lemma 3.1. Takek > n. Then λk−λn=p

λk 2−p

λn 2

= (p λk−p

λn)(p λk+p

λn),

and the gap assumption (2. 1) insures that pλk−p

λn≥(k−n)γmin, p λk+p

λn ≥(k−n)γmin+ 2p λn, hence

λk−λn≥(k−n)²γ_min² , and

k > nandλk−λn ≤ρ =⇒ k−n≤

√ρ

γ_min. Similarly,

k < nandλn−λk ≤ρ =⇒ n−k≤

√ρ γmin

.

Hence

Nn(ρ)≤2

√ρ γmin

.

This proves (3. 2).

Now we prove (3. 3)-(3. 6): let us introduce

N_n⁺(ρ) := card{k > n, λk−λn ≤ρ}, N_n⁻(ρ) := card{k < n, λn−λk≤ρ}.

We distinguish the three cases.

• Whenn=N^∗: from the previous study, we see that N_N⁺∗(ρ)≤

√ρ

γ_min^∗ , and N_N⁻∗(ρ)≤ ( ^√_ρ

γ_min ifρ≤λN^∗

N^∗−1 ifρ≥λ_N^∗; this gives that

N_N^∗(ρ)≤ ( ^√ρ

γ_min +

√ρ

γ^∗_min ifρ≤λN^∗

N^∗−1 +

√ρ

γ^∗_min ifρ≥λN^∗

,

which gives (3. 3).

• Whenn > N^∗: now we have N_n⁺(ρ)≤

√ρ

γ_min^∗ , and

N_n⁻(ρ)≤







√ρ

γ_min^∗ ifρ≤λn−λN^∗

√ρ

γ_min ifλn−λN^∗≤ρ≤λn

n−1 ifρ≥λ_n

,

which gives (3. 4).

• Whenn < N^∗: now we have N_n⁻(ρ)≤

( ^√_ρ

γ_min ifρ≤λ_n n−1 ifρ≥λ_n, and

N_n⁺(ρ)≤ ( ^√ρ

γ_min ifρ≤λN^∗−λn

N^∗−n+

√ρ

γ^∗_min ifρ≥λN^∗−λn

,

hence whenλn≤λN^∗−λn we have N_n(ρ)≤











2√ ρ

γmin ifρ≤λ_n n−1 +

√ρ

γ_min ifλ_n≤ρ≤λ_N^∗−λ_n N^∗−1 +

√ρ

γ^∗_min ifρ≥λN^∗−λn

,

which gives (3. 5), and similar estimates when λn≥λN^∗−λn, which give

(3. 6).

Before going further, let us give another estimate of the counting function, which reveals to be more practical and more natural, since it gives a better understanding of the role of the different parameters:

Lemma 3.2. Assume that the gap assumptions (2. 1)-(2. 2) are satisfied; then

• when n=N^∗:

(3. 7) ∀ρ >0, NN^∗(ρ)≤ ( ^√ρ

γ_min +

√ρ

γ^∗_min ifρ≤λN^∗

(1−^γ_γ^min∗ min

)(N^∗−1) + 2

√ρ

γ_min^∗ ifρ≥λN^∗

,

• when n > N^∗: (3. 8)

∀ρ >0, N_n(ρ)≤











2√ ρ

γ_min^∗ ifρ≤λn−λN^∗

√ρ γmin +

√ρ

γ_min^∗ ifλ_n−λ_N^∗≤ρ≤λ_n (1−^γ_γ^min∗

min

)(N^∗−1) + 2

√ρ

γ^∗_min ifρ≥λ_n

,

• when n < N^∗: (3. 9)

∀ρ >0, N_n(ρ)≤ (₂^√_ρ

γmin ifρ≤max{λn, λN^∗−λn}

(1−^γ_γ^min∗ min

)(N^∗−1) + (1 +√ 2)

√ρ

γ_min^∗ ifρ≥max{λn, λ_N^∗−λ_n}, and also

(3. 10) ∀ρ >0, N_n(ρ)≤ (2√

ρ

γ_min ifρ≤λN^∗

(1−^γ_γ^min∗

min)(N^∗−1) + 2

√ρ

γ_min^∗ ifρ≥λN^∗

.

Remark 3.2. Lemma 3.2 enlightens the role of the quantity (1−^γ_γ^min∗

min)(N^∗−1) (denoted M^∗ in (2. 3)); whenγmin =γ_min^∗ or if N^∗ = 1, this quantity is equal to zero, and we logically find estimates similar to the ones of Lemma 3.1 (i.e. the ’1 gap condition’); in the more interesting case whereγmin < γ^∗_min andN^∗>1, this quantity measures the increase of the counting function with respect to the ’1 gap condition’.

Let us note also that we expect that (3. 9) holds true with 2 instead of 1 +√ 2, however we could not prove it in full generality.

Proof of Lemma 3.2.

• Whenn=N^∗, it is sufficient to note that pλN^∗≥γmin(N^∗−1), hence, whenρ≥λ_N∗, we have

N^∗−1 = (1−γmin

γ^∗_min)(N^∗−1) +γmin

γ_min^∗ (N^∗−1)

≤(1−γmin

γ_min^∗ )(N^∗−1) +

√λN^∗

γ^∗_min ≤(1−γmin

γ_min^∗ )(N^∗−1) +

√ρ γ^∗_min; this and (3. 3) imply (3. 7).

• Whenn > N^∗: whenρ≥λn, we have pλ_n≥p

λ_N^∗+ (n−N^∗)γ^∗_min≥(N^∗−1)γ_min+ (n−N^∗)γ_min^∗ , hence

n−1≤

√λn

γ_min^∗ + (1−γmin

γ_min^∗ )(N^∗−1)≤

√ρ

γ_min^∗ + (1−γmin

γ_min^∗ )(N^∗−1);

this estimate and (3. 4) imply (3. 8).

• When n < N^∗, we obtain (3. 10) proceeding in the same way: when ρ≤ λN^∗, then clearlyNn(ρ) is less than the number of terms that would be at both sides, for which the gap of their square root would beγmin, hence

N_n(ρ)≤2

√ρ γ_min;

when ρ ≥ λN^∗, then clearly one has all the N^∗−1 first terms, and the others, for which the gap of their square root isγ_min^∗ , hence

Nn(ρ)≤N^∗−1 +

√ρ γ_min^∗ ; but then

N^∗−1 = (1−γmin

γ^∗_min)(N^∗−1) +γmin

γ_min^∗ (N^∗−1)

≤(1−γ_min

γ_min^∗ )(N^∗−1) +

√λ_N^∗

γ^∗_min ≤(1−γ_min

γ_min^∗ )(N^∗−1) +

√ρ

γ^∗_min, which gives (3. 10);

• finally we prove (3. 9): in the same way, ifρ≤max{λn, λN^∗−λn}one has immediately

N_n(ρ)≤ 2√ ρ γmin

;

whenρ≥max{λn, λ_N^∗−λ_n}, then we already know from (3. 5) and (3. 6) that

N_n(ρ)≤N^∗−1 +

√ρ

γ_min^∗ , hence

Nn(ρ)≤(1−γmin

γ_min^∗ )(N^∗−1) +γmin

γ_min^∗ (N^∗−1) +

√ρ γ^∗_min; since

pλ_N^∗≥γ_min(N^∗−1) and

ρ≥max{λn, λN^∗−λn} =⇒ ρ≥ 1 2

λn+ (λN^∗−λn)

= 1 2λN^∗, we deduce that

γmin(N^∗−1)≤p

λN^∗≤p 2ρ, hence

Nn(ρ)≤(1−γmin

γ_min^∗ )(N^∗−1) + (1 +√ 2)

√ρ

γ_min^∗ , which is (3. 9).

This concludes the proof of Lemma 3.2.

3.3. A Weierstrass product.

Motivated by [36], we consider

(3. 11) Fm(z) :=

∞

Y

k=1,k6=m

1−iz−λm

λk−λm

² .

Then the growth in kof λ_k ensures that this infinite product converges uniformly over all the compact sets, hence F_m is well-defined and entire overC. Moreover

F_m(−iλ_n) =

∞

Y

k=1,k6=m

1−λn−λm

λ_k−λ_m 2

=

(0 ifm6=n, 1 ifm=n, ,

hence

(3. 12) ∀m, n≥1, Fm(−iλn) =δmn.

We are going to estimate the growth ofFm. We prove the following

Lemma 3.3. a) Assume that the gap assumption (2. 1) is satisfied. Then the function Fmsatisfies the following growth estimate: there is some uniform constant Cu (independent of m,γmin, andz) such that

(3. 13) ∀z∈C, |Fm(z)| ≤e^γminCu

√λ_m

e^γminCu

√

|z|.

b) Assume that the gap assumptions(2. 1)-(2. 2)are satisfied. Then the function F_m satisfies the following growth estimate: there is some uniform constant C_u (independent of m,γ_min,γ_min^∗ andz), such that

(3. 14) ∀m≥1,∀z∈C, |Fm(z)| ≤Bmqm(|z|)e^Cu

√|z|

γ∗ min, with

(3. 15) ∀m≥1, Bm≤e

Cu γmin√λN∗

λm e

Cu γ∗

min

√λ_m

and

(3. 16) ∀m≥1, q_m(|x|)≤

3 + 2|z|² λ²_m

^M∗ ,

where M^∗ has been defined in (2. 3).

Remark 3.3. The main point in Lemma 3.3 is to obtain estimates of the growth ofF_min e^Cu

√|z|

γ∗

min under (2. 1)-(2. 2), with explicit constants (given in (3. 15) and (3. 16)), that will help us in the following. Comparing with (3. 13), this gives a better idea of the improvement brought by ’large’ gapγ_min^∗ and of the price to pay due to the ’small’ gap γmin for theN^∗ first eigenvalues. In fact we will first prove the following better estimates: (3. 14) holds true with

(3. 17) B_m=









 e⁽

8 γmin+_γ^Cu∗

min

)√

λm+_γmin⁸ √

λ_N∗−λm

ifm < N^∗ e⁽

4 γmin+^4+Cu_γ∗

min

)√

λ_N∗

ifm=N^∗ e⁽

4 γmin+^8+Cu_γ∗

min

)√

λm−_γmin⁴ √

λm−λ_N∗

ifm > N^∗ and

(3. 18) q_m(|z|) =















1 + 2_max(λ^|z|2+λ2m

m,λ_N∗−λm)²

M^∗

ifm < N^∗

1 + 2^|z|2_(λ^+(λN∗)2

N∗)²

M^∗

ifm=N^∗ 1 + 2^|z|2_λ^+λ₂ ^2m

m

^M∗

ifm > N^∗ ,

and this easily implies (3. 15) and (3. 16).

Proof of Lemma 3.3. Note that Fm(z−iλm) =

∞

Y

k=1,k6=m

1 + z² (λk−λm)²

,

hence (following [36])

ln|Fm(z−iλm)|=

∞

X

k=1,k6=m

ln

1 + z² (λk−λm)²

≤

∞

X

k=1,k6=m

ln

1 + |z|² (λk−λm)²

= Z ∞

0

ln 1 +|z|²

ρ²

dN_m(ρ) = 2 Z ∞

0

N_m(ρ) ρ

|z|²

|z|²+ρ²dρ Then we distinguish several cases:

• Under only (2. 1) we deduce from (3. 2) that

2 Z ∞

0

Nm(ρ) ρ

|z|²

|z|²+ρ²dρ≤ 4 γ_min

Z ∞ 0

√1 ρ

|z|²

|z|²+ρ²dρ= 4 γ_min

Z ∞ 0

√1 s

1

1 +s²dsp

|z|.

Then changingz−iλminto z, ln|Fm(z)| ≤ 4

γmin

Z ∞ 0

√1 s

1 1 +s²ds

(p

|z|+p λ_m), which gives (3. 13).

• Under (2. 1)-(2. 2) and whenm=N^∗, we derive from (3. 7) that 2

Z ∞ 0

N_N^∗(ρ) ρ

|z|²

|z|²+ρ²dρ

= 2 Z λ_N∗

0

N_N^∗(ρ) ρ

|z|²

|z|²+ρ²dρ+ 2 Z ∞

λ_N∗

N_N^∗(ρ) ρ

|z|²

|z|²+ρ²dρ

≤2 Z λ_N∗

0

√ρ( 1

γ_min + 1

γ_min^∗ ) |z|²

ρ(|z|²+ρ²)dρ+ 2 Z ∞

λ_N∗

(M^∗+ 2√ ρ

γ_min^∗ ) |z|² ρ(|z|²+ρ²)dρ

≤2( 1

γ_min + 1 γ_min^∗ )p

|z|

Z λ_N∗/|z|

0

√ 1

s(1 +s²)ds + 2M^∗

Z ∞ λ_N∗/|z|

1

s(1 +s²)ds+4p

|z|

γ_min^∗ Z ∞

λ_N∗/|z|

√ 1

s(1 +s²)ds

≤2( 1 γmin

+ 1 γ_min^∗ )p

|z|2√ λ_N^∗

p|z| +2M^∗[ln s

√1 +s²]^∞_λ

N∗/|z|+4p

|z|

γ_min^∗ Z ∞

0

√ 1

s(1 +s²)ds

≤4p

λN^∗( 1 γmin

+ 1

γ_min^∗ ) +M^∗ln(1 + |z|²

(λN^∗)²) +cu

p|z|

γ^∗_min. Then changingz−iλN^∗ into z,

ln|FN^∗(z)| ≤4p

λ_N^∗( 1 γmin

+ 1

γ_min^∗ ) +M^∗ln(1 + 2|z|²+λ²_N∗

(λN^∗)² ) +c_u

p|z|+√ λN^∗

γ_min^∗ , which gives (3. 14) with theB_mandq_mgiven in (3. 17) and (3. 18).

• Under (2. 1)-(2. 2) and when m > N^∗, applying the same method, we derive from (3. 8) that

2 Z ∞

0

Nm(ρ) ρ

|z|²

|z|²+ρ²dρ≤4( 1

γ_min + 2 γ_min^∗ )p

λm− 4 γ_min

pλm−λN^∗

+M^∗ln(1 +|z|² λ²_m) +cu

p|z|

γ^∗_min. Then changingz−iλm intoz, we obtain (3. 14) with the relatedBmand qmgiven in (3. 17) and (3. 18).

• Under (2. 1)-(2. 2) and when m < N^∗, applying the same method, we derive from (3. 9) that

2 Z ∞

0

Nm(ρ) ρ

|z|²

|z|²+ρ²dρ≤ 8 γ_min(p

λ_m+p

λ_N^∗−λ_m) +M^∗ln(1 + |z|²

max{λm, λN^∗−λm}²) +cu

p|z|

γ^∗_min. Then changingz−iλm intoz, we obtain (3. 14) with the relatedBmand

qmgiven in (3. 17) and (3. 18).

3.4. A suitable mollifier.

Motivated by [36], we made in [6] the following construction: consider T⁰ >0, N⁰≥1,a_k:= ^CN_k⁰₂^,T0 with

C_N⁰_,T⁰ := T⁰ 2P∞

k=N⁰ 1 k²

,

in order that

∞

X

k=N⁰

ak= T⁰ 2 , and finally

(3. 19) PN⁰,T⁰(z) :=e^izT

0 2

∞

Y

k=N⁰

cos(akz).

Then we have the following Lemma 3.4. ([6])

(1) The regularity and the growth of P_N⁰_,T⁰ over C: The function P_N⁰_,T⁰ is entire over Cand satisfies

(3. 20)







PN⁰,T⁰(0) = 1,

∀z∈Csuch that =z≥0, |PN⁰,T⁰(z)| ≤1,

∀z∈C, |e^−izT20P_N⁰_,T⁰(z)| ≤e^|z|T20.

(2) The behaviour ofPN⁰,T⁰overR: there existθ0>0,θ1>0, both independent of N⁰ andT⁰ such thatPN⁰,T⁰ satisfies

(3. 21)





 _C

N0,T0|x|

θ₀

1/2

+ 1≥N⁰ =⇒ ln|PN⁰,T⁰(x)| ≤ −^θ₂¹3

_C

N0,T0|x|

θ₀

1/2

, _C

N0,T0|x|

θ₀

1/2

+ 1≤N⁰ =⇒ ln|P_N⁰_,T⁰(x)| ≤ −_(N^θ10)³

_C

N0,T0|x|

θ₀

2

.

(3) The behaviour of PN⁰,T⁰ overiR+: there is some constantθ2>0, indepen- dent of N⁰ andT⁰, such that PN⁰,T⁰ satisfies

(3. 22) ∀x∈R+, PN⁰,T⁰(ix)≥e^−θ2

√C_N0,T0x

.

The Proof of Lemma 3.4 follows by elementary analysis techniques. In the following we are going to use the mollifierPN⁰,T⁰ to construct the biorthogonal family.

3.5. A sequence of holomorphic functions satisfying (3. 1).

Consider

(3. 23) ∀m≥0,∀z∈C, f_m,N⁰_,T⁰(z) :=F_m(z)PN⁰,T⁰(−z) P_N⁰_,T⁰(iλ_m). We will make the following choices: