Mirna Charif

Lech Zielinski

PERTURBATION SERIES FOR JACOBI MATRICES AND THE QUANTUM RABI MODEL

Keywords: Jacobi matrix, unbounded self-adjoint operators, quasi-degenerate eigenvalue perturbation, perturbation series, Quantum Rabi Model, rotating wave approximation

We investigate eigenvalue perturbations for a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. In particular we obtain explicit estimates for the convergence radius of the perturbation series and error estimates for the Quantum Rabi Model including the resonance case. We also give expressions for coefficients near resonance in order to evaluate the quality of the rotating wave approximation due to Jaynes and Cummings.

1. General presentation 1.1. Introduction. The main motivation of this paper is the Quantum Rabi Model (QRM) which is the simplest physical example of interactions between radiation and matter. We refer to [START_REF] Scully | Quantum optics[END_REF] for physical explanations (see also [START_REF] Braak | Semi-classical and quantum Rabi models: in celebration of 80 years[END_REF]) and to [START_REF] Xie | The quantum Rabi model: solution and dynamics[END_REF] for a list of recent research works in relation with the QRM. It appears (see [START_REF] Tur | Jaynes-Cummings model: solution without rotating wave approximation[END_REF], [START_REF] Boutet De Monvel | On the spectrum of the quantum Rabi model[END_REF]) that the QRM Hamiltonian is unitarily equivalent to the direct sum J ω ∆ (g) ⊕ J ω -∆ (g), where ∆, ω and g are real parameters (see Section 1.5) and J ω s (g) is the self-adjoint operator defined in 2 by the matrix

J ω s (g) =            -s 2 g √ 1 0 0 0 • • • g √ 1 ω + s 2 g √ 2 0 0 • • • 0 g √ 2 2ω -s 2 g √ 3 0 • • • 0 0 g √ 3 3ω + s 2 g √ 4 • • • . . . . . . . . . . . . . . . . . .            (1.1)
The QRM has become a subject of numerous experimental works in the domain of the Cavity Quantum Optics. In practice the value of the coupling constant g is small and it is natural to investigate an eigenvalue by means of the Taylor series with respect to g. Let us notice that all diagonal entries of the matrix (1.1) are distinct if s is not a multiple of ω. Thus the most interesting phenomena appear when s is a multiple of ω. For simplicity, in this paper, we consider the situation s = ω and our analysis will concern the following problems :

(i) to prove that the eigenvalue branches are analytic functions of g and to give an explicit bound for the convergence radius, (ii) to give explicit estimates of the error due to a cut-off of the Taylor's series, (iii) to give a method of computing the coefficients and to express corrections of low order. The results concerning points (i) and (ii) are given in Theorem 1.3. Concerning the point (i), we must control the spectrum with respect to g in order to avoid eigenvalue crossing. The result Date: January 28, 2021.

concerning the point (ii) is obtained in a standard way: the Cauchy's integral formula (see Section 2.3) gives the bounds on the coefficients and clearly the estimates become better when the convergence radius is greater. Theorem 1.3 is preceded by Theorem 1.2 which describes results on perturbations of a simple eigenvalue.

The expressions for coefficients are given in Theorem 1.4 for perturbations of a simple eigenvalue and in Theorem 1.5 for perturbations of a double eigenvalue. The case of the matrix (1.1) with s = ω has been intensively studied in physics literature because of the rotating wave approximation introduced in the famous paper of Jaynes and Cummings [START_REF] Jaynes | Comparison of quantum and semiclassical radiation theories with application to the beam maser[END_REF]. The reason of this popularity has been double: the Jaynes-Cummings model is explicitly solvable (see Section 1.5) and the experimental results had confirmed a high quality of this approximation. However more recent experiments have allowed to enlarge the values of the coupling constant g and have shown limits of this approximation.

Our interest in this problem comes from the paper [START_REF] He | First-order corrections to the rotating-wave approximation in the Jaynes-Cummings model[END_REF], where the authors investigate the quality of the Jaynes-Cummings approximation and propose the corrections for the eigenvalues of (1.1). The authors of [START_REF] He | First-order corrections to the rotating-wave approximation in the Jaynes-Cummings model[END_REF] evoke the difficulties to control an infinite matrix and propose to look at a small block with a hope to obtain correct approximations. In this paper we propose a simple method of reducing the initial problem to an analogical problem for a finite block (see Section 6). In Section 7 we show how to compute the coefficients and in Section 1.5 we comment on the Jaynes-Cummings approximation. Moreover in Section 5 we explain what is the minimal size of the block in order to recover a given coefficient of the Taylor's series. It appears that the coefficients proposed in [START_REF] He | First-order corrections to the rotating-wave approximation in the Jaynes-Cummings model[END_REF] are not correct because the block is too small. The purpose of this paper is to study these questions for a more general class of self-adjoint operators in 2 of the form J(g) = D + gB where D is diagonal and B is tridiagonal (see Section 1.2). Thus our results can be also applied to other models, e.g. to the two-photon version of the QRM (see [START_REF] He | Unified analytical treatments of qubit-oscillator systems[END_REF]).

1.2. Definition of J(g). We denote by 2 the Hilbert space of square summable complex valued sequences with the norm ||(xj) j∈N * || = ∞ j=1 |xj| 2 1/2 and the scalar product x, y = ∞ j=1 xjyj. The canonical basis of 2 is denoted {ei} i∈N * (i.e. ei = (δi,j) j∈N * ) and 2 fin denotes the subspace of finite linear combinations of vectors from {ei} i∈N * . We denote by σ(L) the spectrum of a linear operator L.

Let (di) ∞ i=1 , (bi) ∞ i=1 , (b i ) ∞ i=1 be real valued sequences and g ∈ R. We denote by J(g) the closure of the linear symmetric operator defined on 2 fin by the matrix

       d1 + gb 1 gb1 0 0 • • • gb1 d2 + gb 2 gb2 0 • • • 0 gb2 d3 + gb 3 b3 • • • 0 0 gb3 d4 + gb 4 • • • . . . . . . . . . . . . . . .        (1.2)
i.e. J(g) = D + gB with D and B satisfying

Dei = diei, (1.3)
Bei = b i ei + biei+1 + bi-1ei-1, (1.4)
where by convention bi-1ei-1 = 0 if i = 1.

We make the following assumptions

(H1) there exists ρ0 > 0 such that 2ρ 2 0 < lim inf i→∞ d 2 i b 2 i + b 2 i-1 (H2) the sequence (b i ) ∞ i=1 is bounded (H3) di ---→ i→∞ ∞.
Lemma 1.1. If (H1)-(H3) hold, then the operator J(g) is self-adjoint, bounded from below and has discrete spectrum for all g ∈ [-ρ0, ρ0].

Proof. See [START_REF] Cojuhari | Discreteness of the spectrum for some unbounded Jacobi matrices[END_REF] or [START_REF] Janas | Infinite Jacobi matrices with unbounded entries: asymptotics of eigenvalues and the transformation operator approach[END_REF]. 1.3. Convergence radius and error estimates. Further on (di) ∞ i=1 , (bi) ∞ i=1 , (b i ) ∞ i=1 are real sequences satisfying (H1)-(H3) and J(g) is the corresponding self-adjoint operator defined for g ∈ [-ρ0, ρ0]. Our first result concerns perturbations of a simple eigenvalue of D. For this purpose we fix k ∈ N * and make the assumption Theorem 1.2. We fix k ∈ N * and assume

d k = di for i ∈ N * \ {k}. ( 1
d k = b k = 0. Let ρ > 0 be such that ρ < inf i =k |di| βi + β k , (1.7)
where βi are given by (1.6). We also assume that (H1)-(H3) hold and ρ ≤ ρ0.

(i) If -ρ < g < ρ then the interval [-β k ρ, β k ρ] contains exactly one eigenvalue of J(g), [-β k ρ, β k ρ] ∩ σ(J(g)) = {λ k (J(g))}. (1.8) (ii) The eigenvalue λ k (J(g)) is simple and g → λ k (J(g)) is real analytic, i.e. λ k (J(g)) = ∞ ν=1 c k,ν g ν if -ρ < g < ρ. (1.9)
(iii) The coefficients in (1.9) satisfy the estimates |c k,ν | ≤ β k ρ 1-ν and one has

λ k (J(g)) - 1≤ν≤N c k,ν g ν ≤ β k |g| N +1 (ρ -|g|)ρ N -1 if -ρ < g < ρ.
(1.10)

Proof. See Section 6.

Our second result concerns perturbations of a double eigenvalue of D. For this purpose we fix k ∈ N * and make the assumption

d k+1 = d k = dj for j ∈ N * \ {k, k + 1}.
(1.11)

Without loss of generality we can replace J(g) by J(g)-(d+gb

)I with d = d k , b = (b k +b k+1 )/2
and further on we assume

d k+1 = d k = 0 and b k+1 = -b k . (1.12)
We introduce the quantities

µ k := b 2 k + b 2 k 1/2 , (1.13)
β k-1 := |b k-2 | + |b k-1 |, (1.14)
β k+2 := |b k+2 | + |b k+2 |, (1.15)
γ k := 3 max{|b k-1 |, |b k+1 |} + 2 max{µ k , β k-1 , β k+2 }, (1.16
)

γ k := max b k-1 d k-1 , b k+1 d k+2 (1.17)
and

β i := |bi-1| + |bi| + |b i | if i / ∈ [k -1, k + 2] (1.18)
Theorem 1.3. Assume that (H1)-(H3) hold and k ∈ N * is fixed. Assume moreover that (1.11)-(1.12) hold and 0 < ρ ≤ ρ0 satisfies the conditions

ρ ≤ inf i / ∈{k,k+1} |di| 2(β i + µ k ) , (1.19)
4ρµ k ≤ inf i / ∈{k,k+1}
|di|, (1.20)

ρ γ k γ k e 2ργ k < µ k , (1.21
) where µ k , γ k , γ k , β i are given by (1.13)- (1.18).

(i) If -ρ < g < ρ and g = 0, then [-2µ k ρ, 2µ k ρ] ∩ σ(J(g)) = {λ k (J(g)), λ k+1 (J(g))}, (1.22)
where the eigenvalues λ k (J(g)), λ k+1 (J(g)) are simple and satisfy

λ k (J(g)) < 0 < λ k+1 (J(g)) if g > 0, λ k+1 (J(g)) < 0 < λ k (J(g)) if g < 0 (1.23)
(ii) If j = 0, 1 and µ k+1 := -µ k , then one has

λ k+j (J(g)) = -µ k+j g + ∞ ν=2 c k+j,ν g ν if -ρ < g < ρ.
(1.24)

(iii) The coefficients in (1.24) satisfy the estimates |c k+j,ν | ≤ µ k ρ 1-ν for j = 0, 1 and

λ k+j (J(g)) + µ k+j g - 2≤ν≤N c k+j,ν g ν ≤ µ k |g| N +1 (ρ -|g|)ρ N -1 if -ρ < g < ρ.
(1.25)

Proof. See Section 6.

1.4. Coefficients of the perturbation series.

Theorem 1.4. Let J(g) be as in Theorem 1.2.

(i) If b j = 0 holds for all j ∈ N * , then c k,ν = 0 when ν is odd and (1.9) holds with

c k,2 = - b 2 k d k+1 - b 2 k-1 d k-1 (1.26) c k,4 = b 4 k d 3 k+1 + b 4 k-1 d 3 k-1 + b 2 k b 2 k-1 d 2 k-1 d k+1 + b 2 k b 2 k-1 d 2 k+1 d k-1 - b 2 k b 2 k+1 d 2 k+1 d k+2 - b 2 k-1 b 2 k-2 d 2 k-1 d k-2
.

(1.27)

(ii) In the general case one has

λ k (J(g)) = c 2,k (g) g 2 + c 4,k (g) g 4 + O(g 6 ), (1.28)
where c 2,k (g) and c 4,k (g) are given by using di + gb i instead of di in (1.26)-(1.27).

Proof. The assertions of Theorem 1.4 can be deduced from general formulas given e.g. in Reed-Simon [START_REF] Reed | Methods of Modern Mathematical Physics IV[END_REF]. However we give an independent proof in Section 7.3.

Next we assume that J(g) is as in Theorem 1.3. We denote

d 0 i (g) := di + gb i for i ∈ N * , (1.29) b 1 k (g) := b k + g b 2 k-1 d 0 k (g) -d 0 k-1 (g) , (1.30) b 1 k+1 (g) := b k+1 + g b 2 k+1 d 0 k+1 (g) -d 0 k+2 (g) , (1.31) b 1 k (g) := b k 1 - g 2 b 2 k-1 2(d 0 k (g) -d 0 k-1 (g)) 2 - g 2 b 2 k+1 2(d 0 k+1 (g) -d 0 k+2 (g)) 2
(1.32) and introduce the matrix

B 1 k (g) := b 1 k (g) b 1 k (g) b 1 k (g) b 1 k+1 (g)
.

(1.33) Theorem 1.5. Let J(g) be as in Theorem 1.3,

µ k = b 2 k + b 2 k
1/2 and µ k+1 := -µ k .

(i) Let B 1 k (g) be given by (1.33) and for j = 0, 1, let λ 1 k+j (g) denote the eigenvalue of

B 1 k (g) satisfying λ 1 k+j (g) ---→ g→0 -µ k+j . Then λ k+j (J(g)) = g λ 1 k+j (g) + O(g 4 ).
(1.34)

(ii) If b k = 0, then the estimate (1.34) implies λ k+j (J(g)) = -µ k+j g -g 2 b 2 k-1 2d k-1 + b 2 k+1 2d k+2 + O(g 3 ) (1.35)
(iii) At the end of Section 7.2 we give expressions for λ k+j (J(g)), j = 0, 1, with the error O(g 5 ).

Proof. See Section 7.2.

1.5. Jaynes-Cummings approximation. The simplest interaction between a two-level atom and a classical light field is described by the Rabi model [START_REF] Rabi | On the process of space quantization[END_REF], [START_REF] Rabi | Space quantization in a gyrating magnetic field[END_REF]. The quantized version can be reduced to J ω ∆ (g) ⊕ J ω -∆ (g), where ∆ is the separation energy between two atomic levels and ω is the frequency of the quantized one-mode electromagnetic field. In [START_REF] Jaynes | Comparison of quantum and semiclassical radiation theories with application to the beam maser[END_REF], Jaynes and Cummings proposed to approximate J ω s (g) by

J ω s (g) =               -s 2 0 0 0 0 • • • 0 ω + s 2 g √ 2 0 0 • • • 0 g √ 2 2ω -s 2 0 0 • • • 0 0 0 3ω + s 2 g √ 4 • • • 0 0 0 g √ 4 4ω -s 2 • • • . . . . . . . . . . . . . . . . . .               (1.36)
under the assumption that s ≈ ω and g is small. Since J ω s (g) is the direct sum

s 2 + J ω s (g) = (0) ⊕ m∈N * 2mω + (s -ω) g √ 2m g √ 2m 2mω , (1.37)
we can find explicitly all its eigenvalues. Physical reasons for this approximation in semi-classical and fully quantized version were usually given by means of the time dependent perturbation theory (see [START_REF] Shirley | Solution of the SchrÃűdinger Equation with a Hamiltonian Periodic in Time[END_REF], [1], [START_REF] Frasca | Third-order correction to localization in a two-level driven system[END_REF]). The time independent approach was prosed in [START_REF] He | First-order corrections to the rotating-wave approximation in the Jaynes-Cummings model[END_REF]. Following [START_REF] He | First-order corrections to the rotating-wave approximation in the Jaynes-Cummings model[END_REF] let us consider the case ω = s. In this case, the eigenvalues of (1.36) are

(2m -1 2 )ω ± g √ 2m, m = 0, 1, 2, . . .
On the other hand we can use Theorem 1.5 without the hypothesis d k = d k+1 = 0. The corresponding shift of the diagonal entries in (1.1) gives the expressions

d k ± g|b k | - g 2 2 b 2 k-1 d k-1 -d k + b 2 k+1 d k+2 -d k + O(g 3 )
for the couple λ k (J(g)), λ k+1 (J(g)). In the case of the QRM with

s = ω, one has b k = √ k, d2m -d2m-1 = d2m+2 -d2m+1 = ω and the corresponding eigenvalue couple {λ2m(J ω ω (g)), λ2m+1(J ω ω (g))}, satisfies (2m -1 2 )ω ± g √ 2m - g 2 2ω + O(g 3 ).
We observe that these eigenvalues coincide with the eigenvalues of the Jaynes-Cummings model modulo O(g 2 ). We can also use Theorem 1.5 with b k = 0 in order to cover a situation when the difference s -ω = cg. Reasoning similarly as in Section 2.4 it is also possible to consider the case of entries bi, b i which are analytic functions of g. We have not used this framework in order to simplify the expressions for the convergence radius.

Preliminaries

2.1. Introduction. Sections 2-5 present a finite dimensional perturbation theory. In these sections we denote by L(V ) the set of all linear operators defined on a finite dimensional linear space V and {ei} n i=1 denotes the canonical basis of C n . We assume that J : C → L(C n ) has the form

J(g) = D + gB(g), (2.1)
where D = diag(di) n i=1 , i.e. Dei = diei for i = 1, . . . , n. Sections 2.2 and 2.2 contain two elementary lemmas which are basic ingredients of our further analysis and in Section 2.4 we prove a finite dimensional version of Theorem 1.2. We will use the following

Notation 2.1. (a) For λ ∈ C and ρ ≥ 0 we denote D(λ, ρ) := {λ ∈ C : |λ -λ| ≤ ρ}. If ρ > 0 then D(λ, ρ) := {λ ∈ C : |λ -λ| < ρ} and ∂D(λ, ρ) := {λ ∈ C : |λ -λ| = ρ}. (b) We denote by •, • the scalar product of C n and write B(g) = (bi,j(g)) n i,j=1 with bi,j(g) = ei, B (g)ej .
(c) For ρ > 0 and i = 1, . . . , n, we denote βi(ρ) := sup g∈D(0,ρ) 1≤j≤n

|bi,j(g)|.

2.2. An auxiliary result.

Lemma 2.2. We fix k ∈ {1, . . . , n}. Let ρ > 0 be such that β k (ρ) > 0 and denote

φ k (ρ) := min i =k |d k -di| ρβ k (ρ) + ρβi(ρ) . (2.2) If φ k (ρ) > 1 and |g| < ρ, then ∂D(d k , ρβ k (ρ)) ∩ σ(J(g)) = ∅. (2.3)
Proof. The Gershgorin's theorem (see Th. 3.11 in [START_REF] Saad | Numerical Methods for Large Eigenvalue Problems[END_REF]) ensures σ(J(g))

⊂ D1 ∪ • • • ∪ Dn, where Di := D di + gbi,i(g), j =i |gbi,j(g)| .
Since Di ⊂ D(di, |g| βi(|g|)), it remains to show that for every i one has

|g| < ρ =⇒ ∂D(d k , ρβ k (ρ)) ∩ D(di, |g| βi(ρ)) = ∅. (2.4) Since |g| < ρ =⇒ D(d k , |g| β k (ρ)) ⊂ D(d k , ρ β k (ρ)), it is clear that (2.4) holds if i = k. Assume now that i = k. Since by definition, Φ k (ρ) > 1 implies |d k -di| > ρβ k (ρ) + ρβi(ρ), (2.5)
we deduce (2.4) from the fact that (2.5) ensures

D(d k , ρβ k (ρ)) ∩ D(di, ρβi(ρ)) = ∅.

2.3.

Use of the Cauchy's formula.

Lemma 2.3. We fix β > 0 and ρ > 0.

If η : D(0, ρ) → D(0, βρ) is holomorphic, then |η (ν) (0)| ≤ βρ 1-ν ν! (2.6)
holds for every ν ∈ N. Moreover for every N ∈ N and g ∈ D(0, ρ) one has

η(g) - 0≤ν≤N η (ν) (0) ν! g ν ≤ β|g| N +1 (ρ -|g|)ρ N -1 .
(2.7)

Proof. Denote cν = η (ν) (0)/ν! and take ρ < ρ. Then the Cauchy's formula

cν = 1 2πi |g|=ρ η(g) g -1-ν dg
allows us to estimate |cν | ≤ βρρ -ν and taking the limit ρ → ρ we obtain

|cν | ≤ βρ 1-ν . (2.8)
Using (2.8) we can estimate the left hand side of (2.7) by

ν≥N +1 |cν g ν | ≤ ν≥N +1 βρ(|g|/ρ) ν = βρ |g| N +1 ρ N +1 (1 -|g|/ρ) ,
completing the proof of Lemma 2.3.

2.4.

Finite dimensional version of Theorem 1.2. Let us fix ρ0 > 0 and assume that B is holomorphic D(0, ρ0) → L(C n ), i.e. g → bi,j(g) are holomorphic on D(0, ρ0). We fix k ∈ {1, . . . , n} and assume that

d k = 0 = di if i = k. (2.9)
Assume moreover that 0 < ρ < ρ0 is such that β k (ρ) > 0 and φ k (ρ) > 1. Due to (2.9), d k = 0 is a simple eigenvalue of J(0) = D and (2.3) allows us to define

P k (J(g)) = 1 2πi |λ|=ρβ k (ρ) (λ -J(g)) -1 dλ (2.10)
for g ∈ D(0, ρ). Following Kato [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF] we observe that g → P k (J(g)) is a holomorphic family of eigenprojectors of J(g) satisfying rank P k (J(g)) = rank P k (J(0)) = 1 and λ k (J(g)) = tr J(g)P k (J(g)) is an eigenvalue of J(g) satisfying

σ(J(g)) ∩ D(0, ρβ k (ρ)) = {λ k (J(g))}. Since g → λ k (g) is holomorphic D(0, ρ) → D(0, ρβ k (ρ))
, the estimates (2.6)-(2.7) hold with λ k (g) and β k (ρ) instead of η(g) and β.

3. Quasi-degenerate case in finite dimension

3.1. Introduction. In this section J(g) = D + gB(g) is holomorphic D(0, ρ0) → L(C n ) and D = diag(di) n i=1 .
Moreover we fix n ∈ {1, . . . , n -1} and we make the assumption

i ≤ n < j =⇒ di = dj. (3.1)
We write

C n = V ⊕ V with V := span {ej} 1≤j≤ n , (3.2)
V := span {ej} 1+ n≤j≤n (3.3)
and consider the corresponding decomposition

D = D ⊕ D = diag(dj) n j=1 ⊕ diag(dj) n j=1+ n , (3.4)
where D ∈ L( V ) and D ∈ L( V ) are the restrictions of D to V and V respectively. Then the assumption (3.1) means that D and D have no common eigenvalue.

Analyticity results for degenerate eigenvalues were given in [START_REF] Rellich | Perturbation Theory of Eigenvalue Problems[END_REF], but in this paper we will develop an approach of Schrieffer-Wolff [START_REF] Bharadwaj | An Explicit Method for Schrieffer-Wolff Transformation[END_REF] (see also [START_REF] Klein | Degenerate perturbation theory[END_REF], [START_REF] Lödwing | A Note on the Quantum-Mechanical Perturbation Theory[END_REF], [START_REF] Van Vleck | On σ-Type Doubling and Electron Spin in the Spectra of Diatomic Molecules[END_REF], [START_REF] Bharadwaj | An Explicit Method for Schrieffer-Wolff Transformation[END_REF]) in order to prove Proposition 3.1. Assume that (3.1) holds and ε0 > 0 is small enough. If |g| < ε0 then J(g) is similar to J∞(g) ⊕ J∞(g) where J∞ is holomorphic D(0, ε0) → L( V) and J∞ is holomorphic

D(0, ε0) → L( V).
Proof. See Sections 3.2-3.4.

Using Proposition 3.1 in the case n = 2 we get Corollary 3.2. Assume that (3.1) holds with n = 2 and d1 = d2 = 0. Assume moreover that B(0) is self-adjoint and b1,2(0) = 0. If 0 < ρ < min{|dj| : j ≥ 3}, then there exists ε > 0 such that for g ∈ D(0, ε) one has

D(0, ρ) ∩ σ(J(g)) = {λ1(J(g)), λ2(J(g))}, (3.5)
where g → λj(J(g)) are holomorphic on D(0, ε) for j = 1, 2 and

λj(J(g)) = µjg + O(g 2 ), (3.6)
where µ1, µ2 are eigenvalues of B(0).

Proof. Let ε > 0 be small enough. Then the assertion of Proposition 3.1 ensures

σ(J(g)) = σ( J∞(g)) ∪ σ( J∞(g)) (3.7)
and there exists C > 0 such that dist σ( J∞(g)), {d3, . . . , dn} ≤ C|g|.

Due to (3.8), D(0, ρ) ∩ σ( J∞(g)) = ∅ holds for ε > 0 small enough and (3.7) ensures D(0, ρ) ∩ σ(J(g)) ⊂ σ( J∞(g)).

However J∞(g) = g B∞(g) and B∞(g) = B(0

) + g B (1)
∞ (g). The hypotheses that B(0) is self-adjoint and b1,2(0) = 0 ensure that fact that B(0) has two distinct eigenvalues µ1, µ2, hence σ( B∞(g)) = {µ1(g), µ2(g)} holds with g → µj(g) holomorphic in a neighbourhood of 0 due to the usual perturbation theory presented in Section 2.4.

3.2. Schrieffer-Wolff approximation. We will define J∞(g) ⊕ J∞(g) as the limit of a sequence of operators (J l (g)) ∞ l=0 of the form J l (g) = D + gB l (g).

(3.9)

Using induction with respect to l we begin by setting J0(g) := J(g). Assume now that J l (g) is given by (3.9) and

B l (g) = B l (g) R + l (g) R - l (g) B l (g) (3.10)
where

B l : V → V , B l : V → V , R + l : V → V , R - l : V → V . Then B l (g) = B l (g) ⊕ B l (g) + R l (g) (3.11) and R l (g) = (r i,j,l (g)) n i,j=1 satisfies r i,j,l = 0 if (i, j) ∈ [1, n] 2 ∪ [ n + 1, n] 2 . We define Q l (g) = (q i,j,l (g)) n
i,j=1 by the formula

q i,j,l (g) =    i r i,j,l (g) di -dj if (i, j) / ∈ [1, n] 2 ∪ [ n + 1, n] 2 0 otherwise (3.12)
and observe that (3.12) ensures the equality

[D, iQ l (g)] = -R l (g), (3.13)
where

[A, A ] = AA -A A is the commutator of A, A ∈ L(C n ).
Then we define J l+1 (g) = e -igQ l (g) J l (g)e igQ l (g) .

(3.14)

Notation 3.3. If g ∈ C \ {0} and A, Q ∈ L(C n ), then we denote FgQ(A) := e -igQ Ae igQ -A (3.15) FgQ(A) := 1 g e -igQ Ae igQ -A -g[A, iQ] (3.16)
Using these notations we can express the equality (3.14) in the form

J l+1 (g) = D + g B l (g) + [D, iQ l (g)] + F l (g) (3.17) with F l (g) = F gQ l (g) (D) + F gQ l (g) (B l (g)) (3.18)
and combining (3.17) with (3.11)-(3.13) we obtain Proof. Using integral remainders of the Taylor's formula for s → e -isgQ Ae isgQ we get

J l+1 = D + g B l ⊕ B l + F l . ( 3
FgQ(A) = 1 0 e -isgQ [A, igQ]e isgQ ds,
FgQ(A) = 1 0 e -isgQ [[A, iQ], igQ]e isgQ (1 -s) ds.
We complete the proof using ||e isgQ || ≤ e ||gQ|| .

Lemma 3.6. Assume that C0 > 0 is fixed large enough. Then there exists ε0 > 0 such for g ∈ D(0, ε0) and m ∈ N one has

||Rm(g)|| ≤ C 1+3m 0 |g m | (3.25) || Bm(g) ⊕ Bm(g) -Bm-1(g) ⊕ Bm-1(g)|| ≤ C 1+3m 0 |g m | (3.26)
where for m = 0 we take Bm-1 = 0 and Bm-1 = 0 in (3.26).

Proof. First of all we can assume that C0 > 0 is large enough to ensure

||B(g)|| ≤ C0 for g ∈ D(0, ε0) (3
||Rm(g)|| ≤ 2 -m C0 if m ≤ l, (3.29) || Bm(g) ⊕ Bm(g) -Bm-1(g) ⊕ Bm-1(g)|| ≤ 2 -m C0 if m ≤ l. (3.30) Using (3.30) we can estimate || B l (g) ⊕ B l (g)|| by l m=0 || Bm(g) ⊕ Bm(g) -Bm-1(g) ⊕ Bm-1(g)|| ≤ 2C0, hence ||B l (g)|| ≤ || B l (g) ⊕ B l (g)|| + ||R l (g)|| ≤ 3C0.
(3.31) Next we observe that (3.19) gives (3.36)

F l = B l+1 -B l ⊕ B l = B l+1 -B l R + l+1 R - l+1 B l+1 -B l (3.
Let ε0 > 0 be small enough. Due to Lemma 3.6 we can define

B∞(g) = lim l→∞ B l (g), B∞(g) = lim l→∞ B l (g) (3.37)
and the convergence is uniform with respect to g ∈ D(0, ε0). Thus denoting

J∞(g) := D + g B∞(g), J∞(g) := D + g B∞(g),
we find that J l (g) = U 0,l (g) -1 J(g)U 0,l (g) converges to J∞(g) ⊕ J∞(g) uniformly on D(0, ε0).

We still assume g ∈ D(0, ε0). Then

||U m,l (g)|| ≤ exp m≤i<l ||gQi(g)|| ≤ C1
and ||U 0,l (g) -U0,m|| ≤ C1||U m,l (g) -I|| can be estimated by

C2 m≤i<l ||e igQ i (g) -I|| ≤ C3 m≤i<l ||gQi(g)|| ≤ 2 -m C4|g|.
Thus we can define U∞(g) as the limit of U 0,l (g) as l → ∞ and ||U∞(g) -I|| ≤ C|g| holds if C > 0 is fixed large enough. Assuming moreover |g| ≤ 1/(2C) we conclude that U 0,l (g) -1 converges to U∞(g) -1 as l → ∞ and J∞(g) = U∞(g) -1 J(g)U∞(g).

B =        b 1 b1 0 0 • • • b1 b 2 b2 0 • • • 0 b2 b 3 b3 • • • 0 0 b3 b 4 • • • . . . . . . . . . . . . . . .        . ( 4.1)
We fix k ∈ N * and assume

d k+1 = d k = 0 = di for i / ∈ {k, k + 1} and b k+1 = -b k . (4.2)
Next we remark that a suitable permutation of the canonical basis allows us to work in the framework of Section 3 with n = 2, i.e. we can decompose

C n = V ⊕ V with V := span{e k , e k+1 } and V := V ⊥ . (4.3)
Then D = D ⊕ D holds with D = diag(0, 0), D = diag(di) i / ∈{k,k+1} and
B = B R + R -B with B = b k b k b k -b k . (4.4)
We observe that σ( B) = {-µ, µ} holds with

µ := b 2 k + b 2 k 1/2 . (4.5)
Due to Corollary 3.2 the spectrum of J(g) near 0 is composed of two eigenvalues

λ k (J(g)) = -µg + O(g 2 ), λ k+1 (J(g)) = µg + O(g 2 ), (4.6)
holomorphic on D(0, ε) for a certain ε > 0. In the remaining of the section we prove Proposition 4.1. Assume that J(g) = diag(di) n i=1 + gB holds with B given by (4.1) and real entries (di) n i=1 , (bi) n-1 i=1 , (b i ) n i=1 satisfying (4.2). Let µ k = µ be given by (4.5) and let (β i ) n i=1 , (γi) n i=1 , (γ i ) n i=1 be given by (1. Reasoning as in Section 3.2 we denote J l (g) = D + gB l where B0 = B and (3.14) holds with Q l ∈ L(C n ) satisfying (3.13). Then using notations 4.2, we can write

B l ⊕ B l = Π0B l Π0 + Π 0 B l Π 0 . (4.8)
In this section we consider (3.9) with l = 0 and as before F0(g) is given by the equality

J1(g) = D + g B ⊕ B + F0(g) (4.9)
Moreover R0 = B -B ⊕ B satisfies R0 = Π1R0Π1 and has the form

R0 =     0 b k-1 0 0 b k-1 0 0 0 0 0 0 b k+1 0 0 b k+1 0     ⊕ O V ⊥ 1 (4.10)
where

O V ⊥ 1
is the zero map on V ⊥ 1 . Thus Q0 = Π1Q0Π1 has the form

Q0 =     0 qk-1 0 0 q k-1 0 0 0 0 0 0 qk+1 0 0 q k+1 0     ⊕ O V ⊥ 1 (4.11) with q k+1 := -ib k+1 /d k+2 , q k-1 := ib k-1 /d k-1 . (4.12)
In particular we have

||R0|| = max{|b k-1 |, |b k+1 |}. (4.13)
||Q0|| = γ k = max{|q k-1 |, |q k+1 |}. ( 4
However B • = R0 + Π0B • Π0 + Π 0 B • Π 0 holds with Π0B • Π0 = B ⊕ O V ⊥ 0 and Π 0 B • Π 0 = OV 0 ⊕     0 b k-2 0 0 b k-2 b k-1 0 0 0 0 b k+2 b k+2 0 0 b k+2 0     ⊕ O V ⊥ 2 , hence ||Π 0 B • Π 0 || ≤ max{|b k-2 | + |b k-1 |, |b k+2 | + |b k+2 |}. (4.19)
In order to obtain (4.18), we observe that ||B • || can be estimated by Proof. We denote J(g) := D + g B and observe that

||R0|| + ||Π0B • Π0 + Π 0 B • Π 0 || = ||R0|| + max{||Π0B • Π0||, ||Π 0 B • Π 0 ||} (4.
(J1,0(g) -λ) -1 = (g B -λ) -1 ⊕ ( J(g) -λ) -1 . (4.23)
Since B is unitarily equivalent to diag (-µ, µ) and ||•||0 is the euclidean norm, the corresponding operator norm

||(g B -λ) -1 ||0 = max 1 |λ -gµ| , 1 |λ + gµ| . (4.24)
Due to (4.24) and (4.23), the proof of (4.22) will be complete if we show

||( J(g) -λ) -1 || ≤ 1 ρµ , (4.25
( J(g) -λ) -1 = ν∈N -g A(λ) ν ( D -λ) -1 .
In order to prove (4.27) we observe that A(λ) = αi,j(λ) i,j / ∈{k,k+1} is the matrix with

αi,j(λ) = ei, (D -λ) -1 Bej = (D -λ) -1 ei, Bej = ei, Bej di -λ and since V ⊥ 0 is equipped with the norm (3.21), || A(λ)|| ≤ max i / ∈{k,k+1} j / ∈{k,k+1} | αi,j(λ)| = max i / ∈{k,k+1} β i |di -λ| . ( 4
||gA λ (g)|| ≤ ||gF0(g)|| ||(J1,0(g) -λ) -1 || ≤ |g| 2 γ k γ k e 2ργ k /(µ|g|) < 1, (4.32)
hence λ / ∈ σ(J1(g)) = σ(J(g)) follows from the convergence of the resolvent series

(J1(g) -λ) -1 = (J1,0(g) -λ) -1 ν∈N * (-gA λ (g)) ν (4.33) (b) If θ ∈ [0, 2π[ and 0 < t ≤ ρ, then dist ± tµe iθ , e iθ ∂D(±µρ, µρ) = dist ± tµ, ∂D(±µρ, µρ) = tµ. ( 4
P±(g) = 1 2πi ∂D(±ρµe iθ ,ρµ) (λ -J(g)) -1 dλ (4.37)
are two holomorphic families of projectors defined on U ε,θ . However (4.6) ensures the fact that rank

P±(g) = 1 if |g| ≤ ε, hence rank P±(g) = 1 for all g ∈ U ε,θ . Thus g ∈ U ε,θ =⇒ σ(J(g)) ∩ D(±ρµe iθ , ρµ) = {λ±(g)}
where g → λ±(g) = tr J(g)P±(g) are two distinct eigenvalues of J(g) if g = 0. Thus

σ(J(g)) ∩ D(0, 2ρµ) = {λ+(g), λ-(g)} (4.38)
and λ+(g) (respectively λ-(g)) is the holomorphic extension of λ k+1 (J(g)) (respectively λ k (J(g))) defined on D(0, ρ). Since η(g) = λ k+1 (J(g)) -µg is holomorphic D(0, ρ) → D(0, µρ), Lemma 2.3 ensures |c k+j,ν | ≤ µρ 1-ν and (1.25) holds for j = 1. Similarly, using η(g) = λ k (J(g)) + µg we obtain (1.25) for j = 0.

Block approximation

5.1. Introduction. Let J(g) be as in Proposition 4.1 and c k,ν , c k+1,ν the coefficients of the series (1.24) for λ k (J(g)) and λ k+1 (J(g)) respectively. In this section we fix ν ∈ N * and claim that c k,ν and c k+1,ν depend only on the entries

{(d k+i , b k+i )} i∈[-ν/2,1+ν/2] and {b k+i } i∈[-ν/2,ν/2]
. In other words, the computation of c k,ν and c k+1,ν , can be reduced to the computation of the corresponding coefficients for the operator defined by the sub-matrix of the matrix J(g), namely by the block ( e k+i , J(g)e k+j

) (i,j)∈[-ν/2,1+ν/2] 2 .
For this purpose we prove Proposition 5.1. The coefficients c k,ν and c k+1,ν depend only on Π l JΠ l , where l ∈ N is such that 2l + 1 ≥ ν.

Proof. The proof is given in three steps described in Sections 5.2-5.4.

5.2. Generalized Kato-Temple estimate.

Notation 5.2. (a) Since µ and -µ are two distinct eigenvalues of B l (0) = B, choosing ε0 > 0 small enough we ensure the fact that for g ∈ [-ε0, ε0] the matrix B l (g) has two eigenvalues λ 0,l (g), λ 1,l (g), satisfying

λ 0,l (g) = -µ + O(g), λ 1,l (g) = µ + O(g). (5.1)
For j ∈ {0, 1} we denote λ j,l (g) := g λ j,l (g). Thus (5.1) implies

λ 0,l (g) = -µg + O(g 2 ), λ 1,l (g) = µg + O(g 2 ) (5.2) (b) We denote J l,0 (g) := g B l (g) ⊕ ( D+g B l (g)). Thus σ(J l,0 (g)) = {λ 0,l (g), λ 1,l (g)} ∪ σ( D+g B l (g)) (5.3)
holds due to σ(g B l (g)) = {λ 0,l (g), λ 1,l (g)}.

These notations allow us to deduce immediately the following estimate

λ k+j (J(g)) = λ j,l (g) + O(g l+1 ) (j = 0, •1) (5.4) Indeed, if 0 < ρ < min i / ∈{k,k+1}
|di|, then choosing ε0 > 0 small enough we ensure

σ(J l,0 (g)) ∩ [-ρ , ρ ] = {λ 0,l (g), λ 1,l (g)} if g ∈ [-ε0, ε0] (5.5)
and

J l (g) -J l,0 (g) = gR l (g) = O(g l+1 ) implies dist(σ(J l (g))), σ(J l,0 (g))) = O(g l+1 ) (5.6)
due to the min-max principle. If g ∈ [-ε0, ε0] then combining (5.6) with (5.5) and Theorem 1.3, we get

dist({λ k (J(g)), λ k+1 (J(g))}, {λ 0,l (g), λ 1,l (g)}) = O(g l+1 ) (5.7)
and it is clear that (5.4) follows from (5.7) and (5.2). However writing an analogical decomposition in the case of perturbations of a simple eigenvalue, we find that the standard Kato-Temple inequality (see Th. 3.8 in [START_REF] Saad | Numerical Methods for Large Eigenvalue Problems[END_REF]) ensures error estimates O(g 2l+2 ) for the eigenvalue perturbed by the terms of order O(g l+1 ) in the k-th line and k-th column. The following lemma states an analogical result in our framework. Lemma 5.3. For j = 0, 1, let λ j,l (g) be defined as in Notation 5.2. Then λ k+j (J(g)) = λ j,l (g) + O(g 2l+2 ).

(

where B + n , B - n are the following tridiagonal matrices

B ± n :=              b 1 b1 0 0 b1 b 2 b2 0 b2 b 3 . . . . . . . . . b n-3 bn-3 0 0 bn-3 b n-2 bn-2 0 0 bn-2 b n-1 bn-1 0 0 0 bn-1 b n ± |bn|              (6.2)
In order to prove Theorem 1.2 we show that λ k (J(g)) is well approximated by λ k (J ± n (g)) for large n. In order to prove Theorem 1.3 we show moreover that λ k+1 (J(g)) is well approximated by λ k+1 (J ± n (g)) for large n.

6.2. Auxiliary operator inequality. Lemma 6.1. Let B • be the linear map defined on 2 fin by the formula

B • ei = b • i ei+1 + b • i-1 ei-1, (6.3)
where

(b • i ) ∞ i=1 are real and by convention b • i-1 = 0 if i = 1. Then for x ∈ 2 fin one has ± x, B • x ≤ x, diag(|b • i | + |b • i-1 |) ∞ i=1 x . (6.4)
Proof. If x = (xj) ∞ j=1 then B • x = (b • j xj+1 + b • j-1 xj-1) ∞ j=1 and x, B • x = j b • j xj+1 xj + i b • i-1 xi-1 xi. (6.5)
Writing i = j + 1 in (6.5), we can estimate

| x, B • x | by j 2|b • j | |xj+1xj| ≤ j |b • j |(|xj+1| 2 + |xj| 2 ). (6.6)
To complete the proof we write the right hand side of (6.6) in the form

i |b • i ||xi+1| 2 + j |b • j ||xj| 2 = j (|b • j-1 | + |b • j |)|xj| 2 (6.7)
and observe that the quantity (6.7) is equal to the right hand side of (6.4).

6.3.

Applying the min-max principle. Notation 6.2. If L is a self-adjoint, bounded from below operator with discrete spectrum, then ( λi(L)) ∞ i=1 denotes the sequence of eigenvalues of L, enumerated in non-decreasing order, counting multiplicities, i.e. λ1(L) ≤ λ2(L) ≤ . . . . Let n ∈ N be fixed large enough. We decompose bi = bn,i

+ b • n,i , b i = b n,i + b • n,i , where the sequences (bn,i) ∞ i=1 , (b n,i ) ∞ i=1 are the cut-off given by bn,i = bi if i < n 0 if i ≥ n and b n,i = b i if i ≤ n, 0 if i > n Consequently B = Bn + B • n holds if the operators Bn, B • n are given by Bnei = b n,i ei + bn,iei+1 + bn,i-1ei-1, B • n ei = b • n,i ei + b • n,i ei+1 + b • n,i-1 ei-1 and Lemma 6.1 ensures ± x, B • n x ≤ x, D • n x , (6.8)
where we denoted D

• n := diag(d • i,n ) i∈N * with d • i,n =      0 if i < n |bn| if i = n β i if i > n and β i = |b i | + |bi| + |bi-1|. Next we consider J ± n (g) := D + gBn ± gD • n = J ± n (g) ⊕ diag(di ± gβ i ) ∞ i=n+1 (6.9)
where J ± n (g) is the linear map acting on span{e1, . . . , en} by means of the matrix (6.2). If i > n > k + 1 and ρ satisfies (1.19), then

ρβ i ≤ 1 2 di and di ± gβ i ≥ 1 2 di if -ρ < g < ρ, (6.10)
hence J ± (g) are self-adjoint, bounded from below and have discrete spectrum. Moreover

x, J - n (g)x ≤ x, J(g)x ≤ x, J + n (g)x if g ≥ 0 x, J + n (g)x ≤ x, J(g)x ≤ x, J - n (g)x if g ≤ 0 (6.11)
and the min-max principle ensures We claim that J ± n (g) satisfies the assumptions of Proposition 4.1 for the same value ρ. Indeed, the entries of J ± n (g) are the same as the entries of J(g) except the fact that b n is replaced by b n ± |bn|, bi are replaced by 0 for i ≥ n and b i are replaced by 0 for i > n. Thus the values of β i can only decrease after these modifications and the right hand side of (1.19) can only increase. Applying Proposition 4.1 to J ± n (g) we obtain

λi(J - n (g)) ≤ λi(J(g)) ≤ λi(J + n (g)) if g ≥ 0 λi(J + n (g)) ≤ λi(J(g)) ≤ λi(J - n (g)) if g ≤ 0 (6.
[-2µ k ρ, 2µ k ρ] ∩ σ( J ± n (g)) = {λ k ( J ± n (g)), λ k+1 ( J ± n (g))}, (6.13)
where the eigenvalue λ k+j ( J ± n (g)) is simple if -ρ < g < ρ, g = 0, and satisfies

λ k+j ( J ± n (g)) = -µ k+j g + ∞ ν=2 c ± k+j,n,ν g ν (6.14)
with |c k+j,ν | ≤ µ k ρ 1-ν for j = 0, 1. We observe that (6.9) ensures

σ(J ± n (g)) = σ( J ± n (g)) ∪ {di ± gβ i : i > n} (6.15) Since (1.19) ensures 2ρµ k < di -ρβ i , one has [-2µ k ρ, 2µ k ρ] ∩ σ(J ± n (g)) = [-2µ k ρ, 2µ k ρ] ∩ σ( J ± n (g)), (6.16)
hence (1.22)-(1.23) hold with J ± n (g) instead of J(g) and λ k+j (J ± n (g)) = λ k+j ( J ± n (g)) for j = 0, 1. Let us fix 0 < g < ρ. Using (1.22)-(1.23) with J + n instead of J, we get

λ l-1 (J + n ) < -2ρµ k ≤ λ l (J + n ) < 0 < λ l+1 (J + n ) ≤ 2ρµ k < λ l+2 (J + n ), (6.17)
where l ∈ N * is such that λ k (J + n (g)) = λ l (J + n (g)) and due to the continuity of g → λ l (J + n (g)),

one has l = 1 + card{i ∈ N * : di < d k }. Similarly λ l-1 (J - n ) < -2ρµ k ≤ λ l (J - n ) < 0 < λ l+1 (J - n ) ≤ 2ρµ k < λ l+2 (J - n ). (6.18)
Thus λ k+j (J ± n (g)) = λ l+j (J ± n (g)) for j = 0, 1, and

λ l-1 (J) < -2ρµ k ≤ λ l (J) < 0 < λ l+1 (J) ≤ 2ρµ k < λ l+2 (J) (6.19)
follows from (6.12). Using (6.19) we obtain (1.22)-(1.23) with λ k+j (J(g)) = λ l+j (J(g)) for j = 0, 1, and (6.14) ensures

λ k+j (J(g)) ≤ λ k+j (J + n (g)) ≤ -µ k+j g + 2≤ν≤N c + k+j,n,ν g ν + CN |g| N +1 , (6.20)
λ k+j (J(g)) ≥ λ k+j (J - n (g)) ≥ -µ k+j g + 2≤ν≤N c - k+j,n,ν g ν -CN |g| N +1 (6.21)
for j = 0, 1. We observe that Proposition 5.1 applied to J + n and

J - n give n > k + ν =⇒ c + k+j,n,ν = c - k+j,n,ν (6.22)
due to the fact that Π l ( J + n -J - n )Π l = 0 holds if k + 1 + l < n. Therefore taking n > k + N in (6.20)-(6.21) we obtain

λ k+j (J(g)) = -µ k+j g + 2≤ν≤N c + k+j,n,ν g ν + O(|g| N +1 ). (6.23) Since Proposition 4.1 ensures |c + k+j,n,ν | ≤ µ k ρ 1-ν , we find that for every N ∈ N * , λ k+j (J(g)) = -µ k+j g + 2≤ν≤N c k+j,ν g ν + O(|g| N +1 ) (6.24)
holds with |c k+j,ν | ≤ µ k ρ 1-ν . Similar inequalities can be written when -ρ < g < 0. Thus g → λ k+j (J(g)) is real analytic, its convergence radius is greater or equal ρ and the remainder estimates (1.25) follow as in the proof of Lemma 2.3.

1 i (g), d 1 i+1 (g) the eigenvalues of Ai(g) := d 0 i (g) gbi(g) gbi(g) d 0 i+1 (g) (7.1) (c) For t ∈ R we denote U(t) := cos t -sin t sin t cos t . If di = di+1 then d 0 i (g) = d 0 i+1 (
g) holds for small |g| and the direct calculation gives

d 1 i (g) = d 0 i (g) -ri(g), d 1 i+1 (g) = d 0 i+1 (g) + ri(g) (7.2) with ri(g) = g 2 b 2 i d 0 i+1 (g) -d 0 i (g) - g 4 b 4 i (d 0 i+1 (g) -d 0 i (g)) 3 + O(g 6 ). (7.3) Moreover U(gθi(g))Ai(g)U(gθi(g)) -1 = diag(d 1 i (g), d 1 i+1 (g)) (7.4) holds with θi(g) := 1 2g arctan 2gbi d 0 i (g) -d 0 i+1 (g) (7.5)
Using arctan t = sin t + O(t 3 ) = t + O(t 3 ) and cos t = 1 -t 2 2 + O(t 4 ), we obtain

θi(g) = bi d 0 i (g) -d 0 i+1 (g) + O(g 2 ) (7.6) sin(gθi(g)) = gbi d 0 i (g) -d 0 i+1 (g) + O(g 3 ) (7.7) cos(gθi(g)) = 1 - g 2 b 2 i 2(d 0 i (g) -d 0 i+1 (g)) 2 + O(g 4 ).
(7.8) 7.2. Proof of Theorem 1.5. We consider the first similarity transformation using

e igQ 0 =     c -s 0 0 s c 0 0 0 0 ċ - ṡ 0 0 ṡ ċ     ⊕ I V ⊥ 1 , (7.9)
where s(g) := sin gθ k-1 (g)) ), c(g) := cos gθ k-1 (g)) , (7.10) ṡ(g) := sin gθ k+1 (g)) , ċ(g) := cos gθ k+1 (g)) (7.11) with θi(g) given by (7.5). The corresponding similarity transformation allows us to diagonalize the blocks A k-1 (g) and A k+1 (g). Indeed, the direct calculation shows that the matrix J1(g) = e -igQ 0 (g) J(g)e igQ 0 (g) equals

        d 0 k-2 gb k-2 c -gb k-2 s 0 0 0 * d 0 k-1 -r k-1 0 gb k s ċ -gb k s ṡ 0 * * d 0 k + r k-1 gb k c ċ -gb k c ṡ 0 * * * d 0 k+1 -r k+1 0 gb k+2 ṡ * * * * d 0 k+2 + r k+1 gb k+2 ċ * * * * * d 0 k+3         (7.12)
where the stars correspond to the symmetric entries and we have not written the terms which are the same as in J(g). Then J1(g) = D + g( B1 ⊕ B1 + R1) holds with

gR1(g) =         0 0 -gb k-2 s 0 0 0 * 0 0 gb k s ċ 0 0 * * 0 0 -gb k c ṡ 0 * * * 0 0 gb k+2 ṡ * * * * 0 0 * * * * * 0         ⊕ O V ⊥ 2 (7.13) and g B1(g) = d 0 k + r k-1 gb k c ċ gb k c ċ d 0 k+1 -r k+1 = gB 1 k (g) + O(g 4 ) (7.14) where B 1 k (g) is given by (1.33). Indeed, if b 1 k , b 1 k , b 1
k+1 are given by (1.30)-(1.32), then using (7.8) we find b k c ċ = b 1 k + O(g 3 ) and using (7.3) we find

d 0 k + r k-1 = gb 1 k + O(g 4 ), d 0 k+1 -r k+1 = gb 1 k+1 + O(g 4 ).
Due to (7.14) the difference between eigenvalues of g B1(g) and gB 1 k (g) is O(g 4 ) and Theorem 1.5 follows from (5.8) with l = 1. (iii) Let gR1 be given by (7.13) and Q1 obtained from (3.12). Then (5.9) gives (7.17)

J2 = J1 + [D, igQ1] + [g( B1 ⊕ B1 + R1), igQ1] + 1 2 [[D, igQ1], igQ1] + O(
Then (5.8) ensures λ k+j (J(g)) = λj,2(g) + O(g 6 ), where {λj,2(g)}j=0,1 are the eigenvalues of g B2 and using (7.15)-(7.17) we obtain λj,2(g) with the error O(g 5 ).

7.3. Proof of Theorem 1.4. Let J(g) be as in Theorem 1.4. We apply the approach of Section 3 in the case n = 1. Consider first the case when b i = 0 for all i. Under this assumption we can check by induction that the functions g → ei, J l (g)ej are even when i -j is even and odd when i -j is odd. Since λ k (J(g)) = ei, J l (g)ej + O(g 2l ), it is clear that g → λ k (J(g)) is even. Further on we consider a general case. + O(g 6 ).

Using s = gθ k-1 + O(g 3 ), (7.21) and d k,1 = O(g 2 ), we get

d k,2 = d k,1 - g 4 b 2 k-1 b 2 k-2 d 0 k-1 (g) 2 d 0 k-2 (g) - g 4 b 2 k b 2 k+1 d 0 k+1 (g) 2 d 0 k+2 (g) + O(g 6 ). (7.25)
If we express d k,1 := r k-1 -r k using (7. [START_REF] Schrieffer | Relation between the Anderson and Kondo Hamiltonians[END_REF]) and (7.23), we find that the quantity (7.25) gives the right hand side of (1.28). To complete the proof it remains to observe that ||gR2(g)|| = O(g 3 ) ensures λ k (J(g)) = d k,2 (g) + O(g 6 ) either by the usual Kato-Temple estimate or by repeating the proof of Lemma 5.3 in this case.

. 5 )

 5 Since σ(J(g) -(d + gb)I) = σ(J(g)) -(d + gb), our analysis of J(g) can be reduced to an analysis of J(g) -(d + gb)I and in particular we can use d = d k , b = b k . Thus without any loss of generality we can assume d k = b k = 0. Moreover we denote βi := |bi-1| + |bi| + |b i |.(1.6)

.19) 3 . 3 .Lemma 3 . 5 .

 3335 Norm estimates. Notation 3.4. (a) We denote by || • ||0 the norm defined on V by ||x||0 = |x k | 2 + |x k+1 | 2 ) 1/2 . (3.20) (b) We denote by || • || the norm defined on V by We denote by || • || the norm defined on C n by the formula || x + x|| := max{ || x||0, || x|| } for x ∈ V , x ∈ V , (3.22) and ||A|| = sup{||Ax|| : ||x|| = 1} is the corresponding operator norm. If A and Q ∈ L(C n ), then ||FgQ(A)|| ≤ e 2||gQ|| ||[A, gQ]||, (3.23) || FgQ(A)|| ≤ 1 2 e 2||gQ|| ||[[A, Q], gQ]||. (3.24)

  .27) for a certain ε0 > 0 and ||Q l (g)|| ≤ C0||R l (g)||. (3.28) Let l ∈ N be such that the estimates (3.25)-(3.26) hold if m ≤ l. Further on we assume that |g| ≤ ε0 and 2C 3 0 ε0 ≤ 1. Then one has C 1+3m 0 |g m | ≤ 2 -m C0 and

  32) and in order to prove that (3.25)-(3.26) hold for m = l + 1 it suffices to check the estimate ||F l (g)|| ≤ C 4+3l 0 |g l+1 |. (3.33) However using A = D and [D, igQ l ] = -gR l in (3.24), we can estimate|| FgQ l (D)|| ≤ 1 2 e 2||gQ l (g)|| ||[R l (g), gQ l (g)]|| ≤ e ||R l (g)|| ||gQ l (g)|| (3.34) due to 2||gQ l (g)|| ≤ 2ε0C0||R l (g)|| ≤ 2ε0C 2 0 ≤ 1 and ||[A, A ]|| ≤ 2||A|| ||A ||. Similarly, (3.23) allows us to estimate ||FgQ(B l (g))|| ≤ 2e ||B l (g)|| ||gQ l (g)||. (3.35) Combining (3.34), (3.35) with (3.18) and assuming C0 ≥ 7e, we get ||F l (g)|| ≤ e(||R l (g)|| + 2||B l (g)||)|g| C0||R l (g)|| ≤ ≤ 7eC 2 0 |g| ||R l (g)|| ≤ C 3 0 |g| ||R l (g)||. Thus (3.25) for m = l gives (3.33), completing the proof of (3.25)-(3.26) for m = l + 1. 3.4. End of the proof of Proposition 3.1. For m, l ∈ N satisfying m < l we denote U m,l (g) = e igQm . . . e igQ l-1 .

4 . 3 4. 1 .

 431 Finite dimensional version of Theorem 1.Introduction. In this section we consider J(g) = D + gB ∈ L(C n ), assuming that D = diag(di) n i=1 and B is a tridiagonal matrix,

2 k + b 2 k 1 / 2 , 5 . 4 . 2 .

 2212542 14)-(1.18). Assume moreover that ρ > 0 satisfies (1.19)-(1.21). (i) If -ρ < g < ρ and g = 0, then (1.22) and (1.23) hold (ii) the expansion formula (1.24) holds with µ k+1 = -µ k = -b (iii) the coefficients satisfy |c k+j,ν | ≤ µ k ρ 1-ν for j = 0, 1 and (1.25) holds. Proof. The proof is given in four steps described in Sections 4.2-4.First step of the proof of Proposition 4.1. Notation 4.2. (a) For m ∈ N we define the linear subspace Vm := span{e k+i } i∈[-m,m+1] . (4.7) We observe that dim Vm = min{2m + 2, n} and V0 = V . (b) We define Πm as the orthogonal projector on Vm and Π m := I -Πm.

  ) where we used || • || to denote the operator norm induced by the norm (3.21). Since (1.20) ensures 2ρµ ≤ |di| -2ρµ ≤ |di| -|λ| ≤ |di -λ| for all i / ∈ {k, k + 1}, ||( D -λ) -1 || = max A(λ) := ( D -λ) -1 B. We claim that (4.25) follows from ||g A(λ)|| ≤ 1 2 . (4.27) Indeed, we obtain (4.25) using (4.26)-(4.27) to estimate the norm of the resolvent series

( a )

 a If |g| ≤ ρ, then ∂D(0, 2ρµ) ∩ σ(J(g)) = ∅ (4.29) (b) Assume that θ ∈ [0, 2π[ and 0 < t ≤ ρ. If g = te iθ then ∂D(±µρe iθ , µρ) ∩ σ(J(g)) = ∅. (4.30) Proof. (a) If |λ| = 2µρ then |λ ± gµ| ≥ µρ ≥ µ|g| and Lemma 4.4 ensures ||(J1,0(g) -λ) -1 || ≤ 1/(|g|µ). (4.31) Denote A λ (g) := F0(g)(J1,0(g) -λ) -1 . Then (4.31), (4.15), (1.21) and |g| ≤ ρ, ensure

4 . 5 .

 45 .34) If λ ∈ ∂D(±µρe iθ , µρ) = e iθ ∂D(±µρ, µρ) and g = te iθ , then |λ ± gµ| ≥ |g|µ holds due to (4.34). Then Lemma 4.4 gives (4.31), (4.32)-(4.33) and λ / ∈ σ(J1(g)) = σ(J(g)). End of the proof of Proposition 4.1. Lemma 4.5(a) allows us to define on D(0, ρ) the projectors P (g) = 1 2πi ∂D(0,2ρµ) (λ -J(g)) -1 dλ (4.35) with rank P (g) = rank P (0) = 2, i.e J(g) has at most two eigenvalues in D(0, 2ρµ). Our next step is to show that for κ ∈ {1, -1} one has card σ(J(|g|e iθ )) ∩ D(κρµe iθ , ρµ) = 1 if 0 < |g| < ρ. (4.36) Let us choose ε > 0 small enough. Then the property (4.36) holds if 0 < |g| ≤ ε due to (4.6). Due to Lemma 4.5(b), the property (4.30) holds for g ∈ K ε,θ := [ε, ρ]e iθ . Since K ε,θ is compact, K ε,θ has an open connected neighbourhood U ε,θ such that the property (4.30) still holds for g ∈ U ε,θ . Thus

6. 5 . 7 .

 57 End of the proof of Theorem 1.2. If ρ satisfies (1.7), then (6.10) should be replaced by the fact that one can choose a constant C = C(ρ) large enough to ensure i ≥ C ⇒ di ± gβi ≥ cdi (6.25) where c = c(ρ) > 0 and g ∈ [-ρ, ρ]. It remains to fix n0 ∈ N large enough and use a similar reasoning under the additional assumption that n ≥ n0. Computations of coefficients 7.1. Introduction. To begin we recall well known situation of matrices 2 x 2. Notation 7.1. (a) Further on we denote d 0 i (g) := di + gb i . (b) We denote by d

g 5 )

 5 due to R1 = O(g) and Q1 = O(g). Moreover [D, igQ1] = -gR1 allows us to simplifyJ2 = D + g( B1 ⊕ B1) + [g( B1 ⊕ B1), igQ1] + 1 2 [gR1, igQ1] + O(g 5 )and B2 -B1 depends only on 1 2 [gR1, igQ1] + O(g 5 ) due to (5.10). Finally we find thatg B2(g) = d 0 k + r k-1 + p k gb k c ċ gb k c ċ d 0 k+1 -r k+1 + p k+1 + O(g 5 ) (7.15)holds withp k := e k , 1 2 [gR1, igQ1]e k = -g 2 s 2 b 2 k-2 /d k-2 -g 2 ṡ2 b 2 k /d k+2 ,(7.16)p k+1 := e k+1 , 1 2 [gR1, igQ1]e k+1 = -g 2 s 2 b 2 k /d k-1 -g 2 ṡ2 b 2 k+1 /d k+3 .

Notation 7 . 2 . 1 . 1

 7211 (a) We write d 0 i (g) := di + gb i and recall the assumption d 0 k = 0. (b) For l ∈ N we denote V l := span{e k+j } -l≤j≤l and V := V0 = Ce k .We begin by diagonalizing A k-1 (g) (seeNotation 7.1). For this purpose we use U (g) := with c and s given by (7.10). Then U (g) -1 J(g)U (g) equals (g)3 + O(g 6 ).

(7. 19 )g 2 b 2 k c 2 d 0 k+1 -r k- 1 - g 4 b 4 k c 4

 1914 The next step consists in diagonalizing the block r k-1 gb k c * c := cos(g θ), s := sin(g θ) with that J1 := Ü -1 U -1 JU Ü has the form  (d 0 k+1 -r k-1 ) 3 + O(g 6 ).

(7. 23 )and c 2 = 1 -g 2 θ 2 k- 1 + O(g 4 ), we get c 2 d 0 k+1 -r k- 1 = 1 - 1 + O(g 4 ) 2 - g 2 b 2

 2321214111422 We claim that the quantity(7.23) can be written in the formg g 2 (b k-1 /d 0 k-1 ) 2 + O(g 4 ) d 0 k+1 + g 2 b 2 k-1 /d 0 k-1 + O(g 4 )and multiplying this expression by g 2 b 2 k we get the first three terms of (7.24). Let us denote d k,1 := r k-1 -r k . Then reasoning similarly as before we can use two rotations to diagonalize the blocksd 0 gives J2 = D + gB2 with B2 = (d k,2 ) ⊕ B2 + R2 and d k,2 = d k,1 + g 2 b 2 k-2 s 2 c2 d k,1 -d 0 k-k+1 s2 d 0 k+2 -d k,1

  12) 6.4. End of the proof of Theorem 1.3. Let us fix ρ > 0 satisfying (1.19)-(1.21) and assume that n > k + 2.

Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville EA 2597, Université du Littoral Côte d'Opale, F-62228 Calais, France, E-mail: lech.zielinski@univlittoral.fr

Proof. Step 1. We claim that B l+1 = B l + O(g 2l+1 ).

Since J l = D + g( B l ⊕ B l + R l ) holds with R l (g) = O(g l ) and J l+1 can be expressed by e -igQ l J l e igQ l = J l + ig[J l , Q l ] -

with [D, iQ l ] = -R l and Q l (g) = O(g l ), we obtain J l+1 = D + gB l+1 with

Therefore

and we obtain

(5.10)

To complete the proof of (5.10) we observe that

where the last equality follows from the fact that Π0Q l Π0 = 0.

Step 2. We claim that B l+i = B l + O(g 2l+1 ) holds for all i ∈ N.

Indeed, reasoning by induction we can repeat the proof from Step 1.

Step 3. To complete the proof of (5.8) we observe that (5.4) ensures

and choosing ε0 > 0 small enough we obtain

for -ε0 < g < ε0 due to the min-max principle.

5.3.

Second step of the proof of Proposition 5.1.

Lemma 5.4. For every l ∈ N one has

)

Proof. If l = 0, then (5.12) holds due to J0 = J and (5.13) holds due to (4.10). Reasoning by induction we fix l ≥ 1 and assume that (5.12)-(5.13) hold with l -1 instead of l. However

where the last equality is due to (5.12) with l -1 instead of l. However a tridiagonal matrix J satisfies JΠ l+1 = Π l JΠ l+1 and

(5.16)

Combining (5.15) with (5.16) we get (J l -J)Π l+1 = 0. Similarly we get Π l+1 (J l -J) = 0, hence (5.12) holds and it remains to prove that (5.12) implies (5.13).

Since B l = (J l -D)/g, it is clear that (5.12) implies

(5.17)

and (B l -B)Π l+1 = 0 we get holds for a certain i ∈ N. Then (1.24) holds and similarly

(5.20)

It remains to prove that (5.19) implies (5.21). Using induction we will prove that

holds for l = 0, . . . , i. Since J0 = J and J • 0 = J • , (5.22) holds for l = 0 due to (5.19). Let us assume that (5.22) holds for a certain l ≤ i -1. Then

follows from (5.22) due to B l -B • l = g -1 (J l -J • l ). Moreover, using (5.23) and

we get Πi(R l -R • l )Πi = 0. Therefore Lemma 5.4 and l ≤ i -1 ensure In both cases we use the min-max principle and deduce the estimates for coefficients of the Taylor series using a finite dimensional block and Proposition 5.1. For this reason we assume g ∈ R.

Our approach uses the operators J + n (g) and J - n (g) ∈ L(C n ), given by the formula J ± n (g) := diag(di) n i=1 + g B ± n (6.1)


Annie-Claude Pérez
email: annie-claude.perez@univ-tln.fr

Claude Jauffret
email: jauffret@univ-tln.fr

Denis Pillon
email: pillon_denis@orange.fr

Keywords: Target motion analysis, tracking, range-only, observability, constant turn motion, constant acceleration motion, sonar, radar, electronic support measurement

this paper is the companion-paper of another paper presented in FUSION'17 concerning bearings-only target motion analysis (BOTMA). In this one, bearing data are replaced by range data: we study observability in range-only target motion analysis (ROTMA). When the observer is in constant turn motion, the target's trajectory is observable, as in BOTMA. If the observer is in constant acceleration motion, necessary and sufficient conditions of observability in ROTMA are proven to be not the same than in BOTMA, although the equality of the ranks of the respective Fisher information matrices. As in BOTMA, the rendezvous routes of the observer and the target play a crucial role. When the system is not observable, ghost-targets (the ones at the same range than the true target) are identified.

I. INTRODUCTION

HIS PAPER addresses the observability analysis in rangeonly target motion analysis (ROTMA) when the target is assumed to be in constant velocity (CV) motion whereas the observer maneuvers smoothly. It completes another one [START_REF] Jauffret | Bearings-Only Target Motion Analysis: Observability when the Observer Maneuvers Smoothly[END_REF] in which we analyze observability in bearing-only target motion analysis in the same context. These both papers are the continuation of [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] where the trajectory of the observer is simpler. Here, we will prove that when the observer is in constant turn (CT) motion, the target is observable in ROTMA. This maneuver was previously used in ROTMA [START_REF] Ristic | Target Motion Analysis Using Range-Only Measurements: Algorithms, Performance and Application to ISAR Data[END_REF], [START_REF] Sathyan | Multiple Hypothesis Tracking with Multiframe Assignment Using Range and Range-rate Measurements[END_REF]. When the observer is in constant acceleration (CA) motion, several situations can occur: the target can be observable or not. This proves the existence of ineffective maneuvers in ROTMA, as in BOTMA [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF]. We propose a criterion of observability, based upon range measurements. When the trajectory of the target is not observable, the set of ghost-targets is given.

The paper is organized as follows: In Section II, the problem formulation and notations are presented.

In section III we recall the observer kinematic models and give condition on range of rendezvous route when the observer is in CA motion. In section IV, we prove observability of the target when the observer is in CT motion. Section V is devoted to observers in CA motion. For this type of scenario, the trajectory of the target is not necessarily observable. Therefore we give necessary and sufficient observability conditions. In unobservable cases, we identify the set of ghost-targets. Illustrative examples are given. The conclusion follows.

II. PROBLEM FORMULATION AND NOTATIONS A. Definitions and notations

In a plane given Cartesian system, a target (T) moves in CV motion and an observer (O) maneuvers smoothly.

    T T T T T T y x t y t x t X   ) ( ) (  .
The trajectory of the target relative to the observer is given by its relative position      

            T t y t x t y t x t X t X t X OT OT OT OT O T OT   ) ( ) (    .
The vector

         r r r OT h h v V cos sin 0 . The polar coordinates of   0 OT V are then   r r h v , .

B. What is observability ?

The previous notation are extended in order to emphasize this dependence:  

      OT OT f X X X t r X t r T t      , , , , 0
. Otherwise, the trajectory is said to be unobservable: at least one vector

  T OG OG OG OG OG y x y x X   ) 0 ( ) 0 (  (defining a CV motion) different from OT X exists such that     OT OG X t r X t r , ,  . The vector O OG G X X X  
defines the "virtual" trajectory of a "ghost-target", denoted G. Each "ghost-target" is defined by

        T 0 0 0 G G G y x P  , T ] [ G G G y x V    ,     T )] ( ) ( [ 0 t y t x V t P t P G G G G G   
, and

    T G G G G G y x t y t x t X   ) ( ) (  , with the convention   0 G G X X  .
Observability analysis has two aims: a) give a necessary and sufficient condition to have unicity of OT X (the trajectory of the target is observable) and the way to know that from measurements, b) when the trajectory is unobservable, characterize the set of the ghost-targets, that is OG X . This will conduct our paper.

Due to the length of the paper, some proofs given in [START_REF] Jauffret | Bearings-Only Target Motion Analysis: Observability when the Observer Maneuvers Smoothly[END_REF] and [START_REF] Jauffret | Observability: Range-Only vs. Bearings-Only Target Motion Analysis when the Observer Maneuvers smoothly[END_REF] are omitted here.

III. OBSERVER CINEMATIC MODELS

Let us start our study by defining the two types of maneuvers considered here: the constant turn motion, that is the observer travels in an arc of a circle at constant speed, and the constant acceleration motion. We detail hereafter the equations of these two motions.

A. CT motion

The observer turns around a fixed point

  T C C C y x P  at range 0  
, with a constant turn rate 0   (positive if the motion of the observer is clockwise) and an "initial phase"  relative to North, at 0  t . Its speed is constant. As a consequence, at time t , the location of the observer is given by

                   

O and T are on an RDVR if and only if

              0 0 0 0 0 0 2 2 OT OT OT OT OT OT x y x y x y y       .
See [START_REF] Jauffret | Bearings-Only Target Motion Analysis: Observability when the Observer Maneuvers Smoothly[END_REF], for the proof.

Proposition 4: Criterion on range of RDVR-I

The target and the observer are on RDVR-I if and only if a scalar r exist such that, for

  f T t , 0  , either r t r t t r x     0 2 2 1 ) (  or r t r t t r x     0 2 2 1 ) (  and . 2 0 2 x r r    
The proof is given in [START_REF] Jauffret | Observability: Range-Only vs. Bearings-Only Target Motion Analysis when the Observer Maneuvers smoothly[END_REF].

IV. OBSERVABILITY WHEN OBSERVER IS IN CT MOTION

The following result is established when the noise-free range are continuously available during

  f T t , 0 

Proposition 5: Observability when observer is in CT motion

If the observer is traveling along an arc of a circle, then any target moving with a constant velocity is observable by range measurements only.

Proof: Suppose now that another target G moving with a constant velocity, say G V , is at the same range as the target of interest T. The square of the range at any time t is

                          2 2 2 2 2 t y t y t x t x t y t y t x t x t r O G O G O T O T                                  . cos 0 sin 0 cos 0 sin 0 2 2 2 2 t t y t y t x t x t y t y t x t x G G G G T T T T                                 or, equivalently,                                                         . , cos sin 2 cos 0 sin 0 2 0 0 2 0 0 cos sin 2 cos 0 sin 0 2 0 0 2 0 0 2 2 2 2
G G G G G G G G G G G G T T T T T T T T T T T T                                                                
This implies the following five equalities:

                                                                             . cos sin cos sin cos 0 sin 0 cos 0 sin 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 2 2                 t y t x t y t x t y t x t y t x y y x x y y x x y x y x y x y x t T T G G T T G G T T T T G G G G T T G G T T G G            
The last two equations are equivalent to

          t P P t t T G            0 0 0 cos sin T     and       t V V t t T G            0 cos sin T     . Since                 t t cos sin spans the whole two-dimensional space,     0 0 T G P P  and T G V V  . QED.
More generally, if the observer's trajectory contains at least one arc of a circle, then the trajectory of any target having a constant velocity is observable in ROTMA, as in BOTMA [START_REF] Jauffret | Observability: Range-Only vs. Bearings-Only Target Motion Analysis when the Observer Maneuvers smoothly[END_REF].

V. OBSERVABILITY WHEN OBSERVER IS IN CA MOTION

In a first time we propose a necessary and sufficient observability condition. In a second time we propose to define the set of 'ghost targets' when de trajectory of the target is not observable.

A. Necessary and sufficient observability condition

The following analysis will be conducted for the relative motion with respect to the observer's trajectory. The question is to identify vectors

  T y x y x X    such that   t t P t y t y x t x OT             , 2 2   (1)
We can already identify two solutions:

          T 0 0 0 0 OT OT OT OT OT y x y x X   
, and

          T 0 0 0 0 OT OT OT OT S OG y x y x X      (since 0  y 
).

We will show below that most of the time, other ghost-targets can exist.

Lemma 3

The set of solutions  1) is defined by the following equations:

   T y x y x   of (
                    ) 5 ( 0 ) 4 ( 0 0 0 ) 3 ( 0 0 0 0 ) 2 ( 0 0 2 2 2 2 2 2 2 2 x OT x x OT OT OT x OT OT OT OT OT OT x x x y x x y x y y x x y y x x y x y x                           or equivalently by   ) 9 ( sin ) 8 ( sin cos ) 7 ( cos ) 6 ( 0 0 2 2 2 2 0 0 2 0 2 2 r x r x x r r x r r h v x r h v x y x h v r y y x x r y x                      Note that (5) implies   0 OT x x    .
The next proposition gives us a guide to find all the solutions.

Proposition 6

The vectors

          T 0 0 0 0 OT OT OT OT OT y x y x X    and           T 0 0 0 0 OT OT OT OT S OG y x y x X     
are solution of [START_REF] Jauffret | Observability in Passive Target Motion Analysis[END_REF]. Any other solution

  T y x y x X OG    satisfies   0 2 sin 2 0 0 2 0 2 2      r h v r x v x r x r x r    and   0 OT x x    .
Proof: We exploit here the four equations of Lemma 3. In order to render the proof lighter, we introduce the following notations:

2 0 r a  ,   r r h v r b   0 0 cos  , and 0 0 2 sin   x r r v c   .
From [START_REF] Jauffret | Bearings-Only Target Motion Analysis: Observability when the Observer Maneuvers Smoothly[END_REF]

      . ( 10
)
Eq. ( 7) implies

    2 2 2 0 OT x x b y y    
. Inserting this into (10), we get

    2 2 2 2 2 0 cy y x x x b x y x OT        . (11)
Finally, using Eq. ( 6), we replace

            2 2 2 2 2 0 0 x a c x a x x x b x x a x OT OT           . (12)
The cubic equation (12) has at most three real roots (one of them is

  0 OT x
). Let us denote the three solutions as

  2 , 1 , 0  i x i
(in some cases, only one or two roots exist).

For convenience, the root   0 OT x will be denoted 0

x .

To compute the two other roots, we first develop (12):

          0 cst 0 2 0 0 2 2 2 3          x OT OT OT x a x b x c x x x x          0 cst 0 2 2 3       x OT x a x b x c x x    . ( 13
)
Since 0

x is a root, we have

    0 cst 0 2 0 2 0 3 0      x OT x a x b x c x x    . ( 14
)
The difference (13) -( 14) is

          0 0 2 0 2 0 2 3 0 3        x OT x a x b x x c x x x x    , or, equivalently,             ) 15 ( 0 0 2 0 0 2 0 2 0         x OT x a x b c x x x x x x x x    Dividing (15) by   0 x x  , we get         0 0 2 0 0 2 0 2        x OT x a x b c x x x x x x    (16),
which is a quadratic equation.

Rearranging the terms of (16), we end up with

      0 0 2 0 2 0 0 2        x OT x x x a x b c x x c x x x      or equivalently   0 0 2 0 0 0 2              a x b c x x c x x x OT x x x x x       . Now,
      r h h v r v r v x x r r x r x r x r          0 sin cos 2 sin 2 0 0 0 2 0 2 2        r h h v r v x x r r x r x r       0 2 sin 2 0 0 2 0 2 2       r h v r v x x r x r x r    . QED
This proposition tells us that the number of ghost is finite and the observability must be studied by means of the polynomial function

  2 0 0 2 r S r x x x Q       , with x r v   2  and   r h S 2 sin 0   
. Note that  is negative. By convention, the heading r h is zeroed when 0  r v . Now, we are able to give a necessary and sufficient observability condition.

Proposition 7: Observability criterion when observer is in CA motion

Assuming that the observer is traveling with a constant acceleration vector, the trajectory of the target is observable from at least four range measurements acquired at different times if and only if O and T are on an RDVR-I.

Proof:

Firstly, suppose that the trajectory of the target is observable. Then,

S OG OT X X  , which implies     0 0 0   OT OT y y  , that is   0 OT P ,   0 OT V
, and  are collinear; so

     0 OT P and      0 OT V . In terms of coordinates, this yields       0 0 , 0 , 0    OT x OT x OT y x x      , and
  0 0  OT y  . Hence   x OT x   0 2   ,    , 0 sin sin 0 0 0 0 0 OT x r r S r          and consequently           0 0 0 0 2 2 2 2 OT x OT OT x OT x x x x x x x Q         .
Formally, the roots of

  0  x Q are   0 OT x x  and     x OT OT x x x  0 0 2    
. Since the trajectory of the target is observable, either

      x OT OT OT x x x  0 0 0 2     , in other words,   0 OT x is a double root (this corresponds to 2 2     ), or     x OT OT x x  0 0 2    is an unacceptable physical solution; that is     2 0 2 2 0 0 r x x x OT OT           . Because   0 2 2 0 OT x r  , this inequality is equivalent to   2 2  
. The target and the observer are on an RDVR-I from Proposition 2.

Conversely, suppose that the target and the observer are on an RDVR-I; that is

     0 OT P and      0 OT V , and . 2 2     We have             T T 0 0 0 0 0 0 x x OT OT OT OT y x y x        .
Under this assumption, the scalars a, b, and c (defined in the proof of Proposition 6) take the following values:

  2 2 2 2 2 , , x x x c b a            
. So, the set of equations ( 6), (7), and (9) becomes

  ) 19 ( ) 18 ( ) 17 ( 2 2 2 2 2 2 2 2 2 2 x x x x x x x y y y x y x                        which is equivalent to     ) 22 ( ) 21 ( ) 20 ( 2 2 2 2 2 x x x x x x y x y y x y                 
Taking the square of (21), we obtain

  2 2 2 2 2 x y y x x        .
Then using ( 20) and ( 22), we get

     2 2 2 2 2 2 x x x x x x x x              , which can be factorized into       2 2 2 x x x x x x x              (23). If x x    , then (23) is equivalent to   x x x     2    , from which we derive   x x    2   
. Reporting this result into (20), we have

    2 2 2 2 2 x y        ; that is,   2 2 2 2 2 x y        . As a consequence,   0 2 2 2      . Let

QED.

We face the apparent paradox encountered in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF]: the ranks of the FIM's in ROTMA and in BOTMA are equal then less than 3 since the bearings are constant [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF]. The trajectory of the target is observable in ROTMA although the FIM is rankdeficient.

B. Unobservable case: construction of the set of ghosts

We start with the following proposition:

Proposition 8: Existence condition of only one ghost in ROTMA for a CA motion

Assume that the observer is in CA motion. One unique ghost exists if and only if O and T are on an RDVR-II. Moreover, its trajectory is defined by S OG X .

Proof:

First, assume that O and T are on an RDVR-II; that is,

    0 2 1 0 0 2    x c OT c OT t x t x   and     0 0 0   OT c OT y t y  .
Any ghost is also on an RDVR-II with the target. Consequently, we have

0 2 1 2    x c c t x t x   and 0   y t y c  . We deduce that     0 0 OT c OT c x t x x t x      . Since   0 OT x x    , we conclude that   0 OT x x 
, and

  0 OT y y   and   0 OT y y     .
Conversely, assume now that one unique ghost exists. We recall that its trajectory is defined by

  T y x y x X    and that   0 OT x x    .
First of all, let us prove that

  0 OT x x  . Indeed, if   0 OT x x  , then 0   y y 
(otherwise, another ghost exists whose trajectory is defined by

  T y x y x X      
, which is in contradiction with the unicity of the ghost). At this point, we necessarily have

    T 0 0 0 OT x x X   . Suppose that   0 0  OT y or   0 0  OT y  . Then, the vector          T 0 0 0 0 OT OT OT y x y x    
defines the trajectory of a ghost. Since the ghost is unique, we have

              T T 0 0 0 0 0 0 0 OT OT OT OT OT y x y x x x X        , which is a contradiction. Hence     0 0 0   OT OT y y  .
To summarize, we have obtained

      T 0 0 0 0 OT OT OT x x X   ,
and

    T 0 0 0 OT x x X   . Consequently,   0 OT x x  ,
which is in contradiction with the assumption. Hence,

  0 OT x x  . Therefore,     0 0  OT x Q . Let us develop the expression of     0 OT x Q :         2 0 0 2 0 0 0 r S r x x x Q OT OT OT       , with x r v   2  and     . cos sin cos 2 2 sin 2 cos sin 2 sin 0 2 2 0 0 r r r r r h h h h h S                   r r r r x r x r OT OT h h h h v r v r y x Q cos sin cos 2 2 sin 2 cos sin sin 0 0 0 2 2 0 2 0 2 0 0 2                r r x r r r x r OT h h v r h h v r y cos sin cos 2 2 sin 2 cos 1 sin 0 0 2 0 2 2 2 0 0 2             r r x r x r r OT h h v r v h r y cos sin cos 2 2 cos sin 2 0 0 2 0 2 2 0 0 2                     0 0 0 2 0 0 2 0 2 2 OT OT x OT OT x OT OT y x y y x y                            0 0 0 0 0 0 2 0 0 2 OT OT OT OT OT OT x OT y x y x y y x Q         .
According to Proposition 3, the target and the observer are on an RDVR-II.

QED.

We plan now to identify all the ghosts; for that, we need a tool given by the following lemma.

Lemma 4

The equation

0 2 0 0 2     r S r x x   has two real roots defined by 2 1      x and 2 2      x , with     2 2 0 2 0 1 4 2 S r S r       . Moreover, 2 0 1 0 x r x r     .
Lemma 4 will conduct our study according to the following table: The following Table II is consistent with Propositions 6, 7 and 8. We have to emphasize upon the particularity of Case 1 and Case 2, when the number of ghosts is equal to 2: we gave as a condition that 2 0    for the target; but this condition has no reason to be satisfied for one ghost. We must hence extend this condition to the two ghosts by writing: for Cases 1 and 2, if the initial bearing of one ghost (or of the target) is equal to 2  , then two ghosts exist.

Table I: Values of   2 1 , x x 1 x 2 x Case 1 0 1 r x   0 2 r x  Case 2 0 1 0 r x r    0 2 r x  Case 3 0 1 r x  0 2 r x  Case 4 0 1 r x   0 2 r x  Case 5 0 1 0 r x r    0 2 r x  Case 6 0 1 r x 
                           r x OT h r x r r X 2 0 0 0 sin 2 0 2         , with 1    ; else                             r x OT h r x r
                           2 1 2 0 1 0 0 2 1 2 0 1 0 cos 0 x r x x h v r x x r x X OT r r OT      , with 1 , 1     ,

Table II: Summarized results

Necessary condition to have ghosts 0 (Observability case: RDVR-I)

1 2 3 Case 1 2 0     never 2 0    2 0     Case 2 never never 2 0    2 0    Case 3 2 0    never 2 0   

C. Examples

In this paragraph, we give one example for each case. The initial position of the observer is

    T 0 0 0  O P .
In the next figures, the trajectories of the observer and target are shown by thick lines, whereas those of the ghosts are shown by thin lines. The capital letters ("O" for observer, "T" for target, and "G" for ghost) designate the moving objects.

1) One example of observable case (case 1)

The target starts at   T 0 5000  (m) with a velocity of   T 2 20 (m/s). The initial velocity of the observer is

    T 2 10 0  O V (m/s). Its acceleration is   T 0 0416 . 0    (m/s 2

2) Three examples of unobservable cases (cases 2 & 5)

In the three following examples, the initial observer velocity is the same as previously, and the duration of the scenarios is 360s. The first example corresponds to case 5 with a RDVR-II. [START_REF] Jauffret | Bearings-Only Target Motion Analysis: Observability when the Observer Maneuvers Smoothly[END_REF]). The observer and the target are on a rendezvous route of type II. We can see in Fig. 2. that one ghost exists, and it is also in a rendezvous route with the observer (see Proposition 8). . Consequently, the observer and the target are not on a rendezvous route from Proposition 4. The bearings are constant (their value is 90°), two ghosts exist, as announced in Table II. and depicted in Fig. 3. Note that if the role of T and G1 (for example) are inverted (G1 becomes the target and T becomes a ghost), then the bearings of this new target are not constant, but two ghosts only exist.

VI. CONCLUSION

In this paper, observability in ROTMA started in [START_REF] Pillon | Observability: Range-Only vs. Bearings-Only Target Motion Analysis for a Leg by Leg Observer's Trajectory[END_REF] for a target in constant velocity motion have been extended to a smooth observer's maneuver (constant turn motion and constant acceleration motion). We ended up with the following results: When a part of the displacement of the observer is in an arc of a circle, observability is guaranteed. When the observer is in constant acceleration motion, the trajectory of the target is observable if, and only if the trajectory of the target relatively to the observer is in a straight line and the target and the observer are in a rendezvous route. In any other scenario, the target is not observable. This demonstrates that even if the observer kinematic is of an order greater than the kinematic of the target, observability is not guaranteed. We proved that in this case, the set of ghosttargets is finite and contains less than three ghost-targets; we gave also the way to construct them from noise-free range measurement. We proposed a range-based observability criterion to know if the target is observable or not.

  for the proof.Definition: The two types of RDVRThe RDVRs are called rendezvous routes of type I (RDVR-I), when noncollinear as well, the RDVRs are called rendezvous routes of type II (RDVR-II).Note that for the RDVR-II, following three propositions give sufficient and necessary conditions of the two types of RDVD.

Proposition 3 :

 3 Condition of RDVR-II.

  we replace the terms , , b a , and c by their respective values. The equation to be solved is hence

  The target is never observable. At most, three ghosts exist, given by

  and S OG X . This case includes the scenarios where T and O are on an RDVR-II.

(

  Proposition 4 is satisfied). Fig.1. depicts the scenario which lasts 250 s.

Figure 1 .

 1 Figure 1. Observable case in ROTMA, RDVR-I.

Figure 2 . 2 The

 22 Figure 2. Unobservable case in ROTMA, RDVR-II. The next example corresponds to case 2

Figure 3 .

 3 Figure 3. Unobservable case in ROTMA, with constant bearings and no rendezvous route. In the last example, three ghosts exist. It corresponds to case 5. The target starts at   T 3464 2000 (m) with a velocity of

Figure 4 .

 4 Figure 4. Unobservable case in ROTMA, general case (no

Proposition 1: General properties of RDVR

  

										At any time t, the position of the observer is
										O P	  t		O P	  0		t	O V	    2 0 2 t	O , where   0 V	is the initial velocity
										and					x		y	 T	is the (non-zero) acceleration vector. The
										relative position of the target with respect to the observer is
												OT P	  t		OT P	  0		t	OT V	    2 0 2 t	(1)
										Without loss of generality (and in order to simplify the
										coming computations), we will assume that			0	and
												x
												0	. Indeed, a suitable rotation of the entire scenario
										y
										allows us to be in this case.
										When the target is in CA motion, analyzing observability
										conducts us to consider special scenarios: the rendezvous
										routes whose definition is now given
										The target and the observer are said to be on a rendezvous
										route (RDVR), when they are collocated at a time	t .
												c
										From now, we will assume that	c t is not in   f T , 0	.
										If O ( in CA motion) and T are on an RDVR, then
										-either	OT P	  0	,	OT V	  0	and  are collinear,
										-or	OT P	  0	and  are noncollinear, and
				sin		t			. Without loss of generality,
	P O	t	P C	cos		t
	we will assume that	x		y		0	.
					C		C
	B. CA motion

  us consider the two following cases: (i)

																	0	. Then
				0	and			y		0	; we deduce from (17) that	x		 		.
															x
	Reporting this value into (22), we get	y 	2		 		2	. We
															x
	deduce that			0	. Hence			0	, which is incompatible with
	the fact that   0  t r	, for	t			, 0	T		. (ii)			0	. Then
	2 	2   	0	, but by assumption,	2 	2   	0	. Hence
	2 	2   	0	. We have	y		0	. Now, from (21), we get
	x				, which is impossible. From this discussion, we
					x
	conclude that the case	x				must be rejected.
															x
	Hence,	x					and consequently	y		0	and	y 		0	. The
												x
	trajectory of the target is observable.