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ASYMPTOTIC EXPANSION OF LARGE EIGENVALUES FOR A CLASS OF UNBOUNDED JACOBI MATRICES

We investigate a class of infinite tridiagonal matrices which define unbounded self-adjoint operators with discrete spectrum. Our purpose is to establish the asymptotic expansion of large eigenvalues and to compute two correction terms explicitly.


GENERAL REMARKS
Infinite Jacobi matrices have appeared in many recent papers related to various questions of pure and applied mathematics (see [7,9,14,19,23]). In this paper we consider a Hermitian tridiagonal matrix such that (d(k)) ∞ k=1 is an increasing sequence going to infinity and (a(k)) ∞ k=1 is a complex valued sequence dominated by (d(k)) ∞ k=1 . Then (1.1) defines in 2 a self-adjoint operator J with discrete spectrum (see [8]). Moreover, J is bounded from below and the eigenvalue sequence of J is defined as the non-decreasing sequence 242 Ayoub Harrat, El Hassan Zerouali, and Lech Zielinski (λ n (J)) ∞ n=1 such that Jv n = λ n (J)v n for n ∈ N * and (v n ) ∞ n=1 is an orthonormal basis of 2 .
We begin the discussion of known results by the fundamental paper of J. Janas, S. Naboko [14]. In this paper the authors describe a method of approximative diagonalization and its application to the analysis of large eigenvalues in certain quantum models. The paper [14] gives also explanations why this type of analysis is important in Quantum Physics.
Hypothesis. There exist µ, ρ, C, c, k 0 ∈ (0; ∞) such that for k ≥ k 0 one has It turns out that asymptotic estimates crucially depend on whether ρ > 1 or not.
In this paper we construct an asymptotic expansion of large eigenvalues under additional regularity conditions imposed on the entries and we give explicit expressions for correction terms in the asymptotic formula with error O(n µ−6ρ ). In Theorem 1.1 we assume that the entries have a classical expansion at infinity. The case ρ = 1 2 gives a nice surprise by ensuring a classical expansion of eigenvalues. A similar construction still works if entries satisfy regularity conditions of symbol type (see Section 1.3). The restriction to tridiagonal matrices is not essential in our approach and we can work all the time with band matrices, but tridiagonal matrices make formulas and calculus more simple. We mention as well that this paper can be viewed as a development of [6] (see also [15]).

ASYMPTOTIC EXPANSION OF LARGE EIGENVALUES
We denote by 2 the Hilbert space of square summable complex valued sequences x(k)y(k). For any θ > 0 we denote , (a(k)) ∞ k=1 satisfy the hypothesis (H1)-(H3) for a fixed µ > 0 and ρ > 0. For x ∈ 2,µ we define Jx ∈ 2 by the formula where we assume x(0) = 0 and a(0) = 0. The formula (1.2) defines the operator J on the domain of definition D(J) = 2,µ . Then J is self-adjoint in 2 , bounded from below and has compact resolvent (see [8]). We will prove the following result.
where α i ∈ C, δ i ∈ R for i ∈ N and δ 0 > 0. Then (H1)-(H3) hold and (1.2) defines in 2 the self-adjoint operator J on the domain D(J) = 2,µ . If (λ n (J)) ∞ n=1 is the eigenvalue sequence of the operator J, then where r(n) obeys the asymptotic expansion of the form and c i,j are real coefficients obtained by the induction scheme in Section 5. The explicit values of c 0,0 , c 1,0 , c 2,0 are computed in Section 6.4 and the value of c 0,1 in Section 6.5 (see also (1.11) and (1.14)-(1.15)).
The operator J is a relatively compact perturbation of the diagonal operator diag(d(n)) ∞ n=1 and the special form of this perturbation allows one to deduce from the min-max principle (see Corollary 3.3). It appears that the assumptions of Theorem 1.1 allow one to replace (1.6) by the stronger estimate and the remainder r(n) = λ n (J) − d(n) obeys the asymptotic formula where r 1 (n) is given by (1.9). The quantity r 1 (n) is of order n µ−2ρ and we call it the first correction term. We mention that the expression (1.9) for the first correction term was obtained in [15] under the assumptions (H1)-(H3) with µ ≥ 1, ρ > 1 (and with weaker remainder estimates). The next step of precision is attained in Theorem 6.2 which describes the asymptotic behaviour of r(n) modulo O(n µ−6ρ ). We show the formula (1.12) and give explicit expressions (1.13)-(1.15) for the leading coefficient of the second correction term, which is of order O(n µ−4ρ ).
In order to obtain a complete asymptotic expansion, we use a method of approximative diagonalization. The idea of our approach is similar to the method presented in [14], but we do not need the assumptions µ ≥ 1 and ρ > 1 used in [14].
At the end of this discussion we remark that the result of Theorem 1.1 is inspired by the paper [18]. The main differences between [18] and our work are the following: -we consider the Hilbert space 2 = 2 (N * ) instead of 2 (Z), -we give a simple algorithm of computing the coefficients c i,j adopted to our problem, -our approach allows us to see that (1.5) contains only even powers of n −ρ .

FURTHER RESULTS
We construct an asymptotic expansion for large eigenvalues under assumptions slightly weaker than used in Theorem 1.1.
If ν ∈ R then S ν 1 denotes the set of p : N * → C such that the estimate holds for every m ∈ N.
Throughout the paper µ > 0, ρ > 0 are fixed and we centre our analysis on the Jacobi operators (1.2) such that and (H1)-(H2) hold. These conditions could be weakened if one wants to obtain the remainder estimate O(n −η ) for a fixed value of the exponent η (see Theorem 4.1), but for simplicity we use (1.7)-(1.8) throughout Section 5 and 6. As before, J is the self-adjoint operator in 2 with D(J) = 2,µ and (λ n (J)) ∞ n=1 denotes its non-decreasing eigenvalue sequence (counted with multiplicities). We consider the asymptotic formula for (λ n (J)) ∞ n=1 with three different degrees of precision.
(i) The asymptotic formula with one correction term. We show that holds with . (1.10) In Section 6.4 we show that in the case δ 0 = 1 one has (1.10) with It is clear that the general case can be reduced to the case δ 0 = 1 by means of the multiplication of J by a suitable constant.
where S * is the adjoint of S. Below we introduce more notations.
According to this notation, the operator S (i) is the linear mapping on 2 satisfying Let i ∈ Z and f : N * → C. Then τ i f denotes the sequence N * → C satisfying We denote by ∂ i f the sequence N * → C given by the formula In particular ∂ 0 f = 0 and ∂ 1 f (k) = ∂f (k) = f (k+1)−f (k) is the derivative introduced in Section 1. We observe that for any fixed i ∈ Z,

OPERATOR VALUED ERRORS
In this section we introduce notations used throughout the paper to control errors of large eigenvalues. As before, S denotes the shift operator in 2 , i.e. Se k = e k+1 . If θ ≥ 0, then Λ θ denotes the self-adjoint operator in 2 such that Λ θ e k = k θ e k and D(Λ θ ) = 2,θ . If θ ≤ 0, then Λ θ is the inverse of Λ |θ| . In order to deal with these operators, we introduce the subspace of fast decaying sequences The space S −∞ defined by (2.3) will be assumed to be invariant for all operators considered further on and we use the fact that this type of operators form an algebra. Clearly S −∞ is invariant for the shifts S, S * and for f (Λ) if f : N * → C is polynomially bounded. We adopt the following convention: for a linear mapping P : D(P ) → 2 we write P ∈ B( 2 ) if and only if the closure of P is a bounded operator 2 → 2 . Let η ∈ R. We introduce the notation Then (2.4) immediately ensures the properties

248
Ayoub Harrat, El Hassan Zerouali, and Lech Zielinski (2.5) holds for all θ ≥ 0 and the notation where (p i (k)) ∞ k=1 is a complex valued sequence (for any i ∈ Z). Definition 2.5. We say that P is a band matrix if and only if P has the form (2.6) and there is i 0 ∈ N such that |i| > i 0 implies p i (k) = 0 for all k ∈ N * . Definition 2.6. We say that P is off-diagonal if and only if p 0 (k) = 0 holds for all k ∈ N * . If P is given by (2.6), then p 0 (Λ) is called the diagonal part of P . Lemma 2.7. We fix ν ∈ R and i 0 ∈ N. Assume that p i (k) = O(k ν ) for all i ∈ Z and |i| > i 0 implies p i (k) = 0 for all k ∈ N * . Then the band matrix P given by (2.6) can be identified with the linear mapping acting on S −∞ according to the formula (2.7) Proof. We observe that (2.7) can be written in the form Taking the adjoint, we get Λ −θ S * Λ θ ∈ B( 2 ).
2.4. CLASS OF OPERATORS FDO ν 1 (N ) Notation 2.8. If ν ∈ R and N ∈ N, then FDO µ 1 (N ) denotes the set of all band matrices P given by (2.6), where p i ∈ S ν 1 (N ) for all i. Further on the elements of FDO ν 1 (N ) are always identified with linear mappings given by (2.8) where Proof. It suffices to prove the formula which ensures that the right-hand side of (2.9) belongs to FDO ν+µ−1 1 (N ). In order to prove (2.9), we observe that (2.2) allows us to express Case j ≥ 0. In this case S (i) S (j) = S (i+j) holds for any i ∈ Z. Case j < 0, i < 0. In this case S (i) S (j) = S (i+j) holds as well.

CONSEQUENCES OF THE MIN-MAX PRINCIPLE
In this section we prove Proposition 3.2 which is the first fundamental tool of our approach. Its purpose is to detect perturbations which give small errors for large eigenvalues. The importance of Proposition 3.2 lies in the fact that it is used as the ingredient of our approach replacing Lemma 2.1 of J. Janas, S. Naboko [14]. We notice that Lemma 2.1 from [14] is used in [7,15,16,22], but we prefer using Proposition 3.2 for several reasons. One of them is the fact that Lemma 2.1 from [14] needs the additional assumption µ ≥ 1 in the hypothesis (H1).
. . are eigenvalues of L, enumerated in non-decreasing order, counting multiplicities.
Let J be the operator defined by and D(J) = 2,µ . Then J is self-adjoint, bounded from below and has compact resolvent in 2 . If R is a symmetric operator such that D(R) ⊃ 2,µ and J + R is a bounded from below operator with compact resolvent and one has Proof. LetC > 0 and denoteD := D +CI. Assume thatC is fixed large enough.
Indeed, the case k = 1 was proved in Step 1 and using induction with respect to k ∈ N * , we fix k ∈ N * and assume that the assertion of Proposition 3.2 holds if η ≤ 2 k µ.

SPECIAL SELF-ADJOINT OPERATORS
Lemma 3.4. Let P be a symmetric operator on S −∞ given by the formula (3.13) Proof. Let θ ≥ 0. Then [iP, Λ 2θ ] is symmetric on S −∞ and we claim that By (3.14) we can find a constant C θ > 0 such that Writing y = Λ θ x, we get Using (3.15) with θ = 1/2 and the fact that we deduce the assertion by Nelson's Commutator Theorem. Indeed, the properties (3.16) and (3.15) with θ = 1 2 allow us to apply Corollary 1.1 in [12] (using P , Λ and S −∞ instead of H, N and C in [12]). Notation 3.5. If P ∈ FDO ν 1 (N ) is given by (2.8), then the adjoint matrix (P † (i, j)) ∞ i,j=1 = (P (j, i)) ∞ i,j=1 satisfies and it is clear that P † ∈ FDO ν 1 (N ). We will use the notation P + hc := P + P † .
will always denote the self-adjoint operator satisfying (3.13).
Proposition 3.6. Assume that p i (k) = O(k) for i = 1, . . . , i 0 and P is the self-adjoint operator given by (3.17). If θ ≥ 0 then 2,θ is an invariant subspace of e itP for any t ∈ R. Moreover there is a constant C θ such that holds for all x ∈ S −∞ and t ∈ R.
Proof. Let D(P ) denote the domain of the self-adjoint operator P equipped with its graph norm. Let ε > 0 and denote holds for all x ∈ D(P ). Indeed, if x ∈ S −∞ then we compute x and by using (εΛ 2θ +I) −1 x instead of x in (3.15), we find that (3.19) holds for x ∈ S −∞ . Since S −∞ is dense in D(P ), the estimate (3.19) holds for x ∈ D(P ) as well.

Corollary 3.7. Let P be as in Lemma 3.4. Then
(i) S −∞ is an invariant subspace of e itP for any t ∈ R, Let us fix N ∈ N * and t 0 > 0. Then the family (x ε,θ t ) 0<ε≤1 is bounded in C N ([−t 0 , t 0 ]; 2 ) due to (3.18), hence using the Ascoli Theorem we can find a sequence (ε n ) ∞ n=0 which converges to 0 and (x εn,θ . We conclude that the pointwise limit t → Λ θ e itP x is a function belonging to C N −1 ([−t 0 , t 0 ]; 2 ).

APPLICATION OF THE TAYLOR FORMULA
Let P be the operator satisfying the assumptions of Lemma 3.4 and assume Q = O(Λ µ ). We will denote F tP (Q) := e −itP Qe itP for t ∈ R. Let x ∈ S −∞ and η ∈ R. Then the function t → Λ η F tP (Q)x is smooth R → 2 due to Corollary 3.7. Denoting and by using the Taylor formula in t = 1, we obtain on S −∞ the equality

It is clear that Proposition 3.6 ensures
on the domain D(A) = 2,µ . Then A is a self-adjoint and bounded from below operator with compact resolvent in 2 . Moreover, holds with .

SOLUTION OF AN AUXILIARY MATRIX EQUATION
We observe that the assumptions (H1) and (H2) concern the sequence (d(k)) ∞ k=k0 , where k 0 ∈ N is fixed. Since any modification of {d(k) : k < k 0 } can be viewed as a perturbation R satisfying R = O(Λ −∞ ), the corresponding error of the n-th eigenvalue is O(n −∞ ) due to Proposition 3.2. Neglecting such errors we may assume that (H2) holds with k 0 = 1, i.e.

PROOF OF THEOREM 4.1(i)
Step 1. Let A be given by (4.1). Then (1) be as in Section 4.2. According to Section 3.2 the operator P is self-adjoint and we claim that (0) due to Lemma 2.9, hence [A 0 , P ] = O(Λ µ−2ρ ) and the Taylor formula gives (3.22).
Step 2. We claim that we obtain and Step

PROOF OF THEOREM 4.1(ii)
Step 1. As before we write A = D + A 0 and P is defined as in Section 4.2. Then follows from Lemma 2.9 and we claim that the diagonal part of [A 0 , iP ] satisfies where r 1 (k) is given by (4.2). Indeed, for i = 1, . . . , i 0 , we have and Step 2. We claim that Indeed, Lemma 2.9 ensures Step 3. We claim that where we denoted Then D 1 = d 1 (Λ) is the diagonal part of A 1 and we can write (1) is off-diagonal. Thus Theorem 4.1(i) can be applied to A 1 with d 1 (k) = d(k) + r 1 (k) instead of d(k) and 2ρ instead of ρ. We obtain which completes the proof due to (4.13).
Clearly Lemma 2.4 holds with S ν 1 instead of S ν 1 (N ) and the proof of Lemma 2.9 ensures The structure of (J where k (l) N ∈ N * is fixed large enough. holds with ρ 0 = ρ. Next we fix l ∈ N and make the induction hypothesis that J (l) has the properties given in Section 5.1. Our purpose is to construct J (l+1) having analogical properties for l+1 instead of l. At the beginning we assume that A (l) ∈ FDO µ−ρ l 1 is off-diagonal, i.e.
Let k (l) ∈ N * be fixed large enough. Then ∂d (l) (k) ≥ 1 2 ck µ−1 holds for k ≥ k (l) (see (5.6)), hence we can define p and p (l) (l) . Similarly as in Section 4.2 we find that the operator and it is easy to see that using induction, we obtain Next we choose m l ∈ N * large enough and write the Taylor formula holds due to (5.11), we can rewrite (5.13) in the form where the estimate of last term follows from ad m l −1 iP Let A (l+1) be the off-diagonal part ofÃ (l) and let r l+1 (Λ) be its diagonal part. Theñ due to (5.17) and ρ l+1 = 2ρ l . Thus setting we find i.e.