D R Yafaev
email: yafaev@univ-rennes1.fr

ASYMPTOTIC BEHAVIOR OF ORTHOGONAL POLYNOMIALS WITHOUT THE CARLEMAN CONDITION

Keywords: 2000 Mathematics Subject Classification. 33C45, 39A70, 47A40, 47B39 Jacobi matrices, Carleman condition, difference equations, orthogonal polynomials, asymptotics for large numbers

Our goal is to find an asymptotic behavior as n → ∞ of orthogonal polynomials Pn(z) defined by the Jacobi recurrence coefficients an, bn. We suppose that the off-diagonal coefficients an grow so rapidly that the series a -1 n converges, that is, the Carleman condition is violated. With respect to diagonal coefficients bn we assume that -bn(ana n-1 ) -1/2 → 2β∞ for some β∞ = ±1. The asymptotic formulas obtained for Pn(z) are quite different from the case a -1 n = ∞ when the Carleman condition is satisfied. In particular, if a -1 n < ∞, then the phase factors in these formulas do not depend on the spectral parameter z ∈ C. The asymptotic formulas obtained in the cases |β∞| < 1 and |β∞| > 1 are also qualitatively different from each other. As an application of these results, we find necessary and sufficient conditions for the essential self-adjointness of the corresponding minimal Jacobi operator.

1. Introduction 1.1. Overview. Orthogonal polynomials P n (z) can be defined by a recurrence relation

a n-1 P n-1 (z) + b n P n (z) + a n P n+1 (z) = zP n (z), n ∈ Z + , z ∈ C, (1.1)
with the boundary conditions P -1 (z) = 0, P 0 (z) = 1. We always suppose that a n > 0, b n = bn . Determining P n (z), n = 1, 2, . . ., successively from (1.1), we see that P n (z) is a polynomial of degree n: P n (z) = γ n z n + • • • where γ n = (a 0 a 1 • • • a n-1 ) -1 . We are interested in the asymptotic behavior of the polynomials P n (z) as n → ∞. This is a classical problem investigated under various assumptions including usually the condition

∞ n=0 a -1 n = ∞ (1.2)
introduced by T. Carleman in his book [START_REF] Carleman | Les fonctions quasi-analytiques[END_REF]. We are aware of only one paper [START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF] where the asymptotics of P n (z) was studied without assumption (1.2); this paper is discussed at the end of this subsection. On the contrary, under assumption (1.2) there is an enormous literature on this subject. Here we mention Nevai's approach (see his book [START_REF] Nevai | Orthogonal polynomials[END_REF]) which allowed the authors of [START_REF] Máté | Asymptotics for orthogonal polynomials defined by a recurrence relation[END_REF] to treat the case a n → a ∞ > 0, b n → 0 as n → ∞ in such a way that

∞ n=0 |a n+1 -a n | + |b n+1 -b n | < ∞. (1.
3)

The asymptotics of the polynomials P n (z) for real z in case of the coefficients a n → ∞ but satisfying the Carleman condition (1.2) was studied in the papers [START_REF] Janas | Jacobi matrices with power-like weights -grouping in blocks approach[END_REF] (see Theorem 3.2) and [START_REF] Aptekarev | Measures for orthogonal polynomials with unbounded recurrence coefficients[END_REF] (see Theorem 3). The authors of [START_REF] Janas | Jacobi matrices with power-like weights -grouping in blocks approach[END_REF] solved equations (1.1) for P n (z) successively for n = 1, 2, . . . (the transfer matrix method) which yielded a representation for (P n (z), P n+1 (z)) as a product of n 2 × 2-matrices. The conditions on the coefficients were rather restrictive in [START_REF] Janas | Jacobi matrices with power-like weights -grouping in blocks approach[END_REF]; in particular, it was assumed there that b n = 0 for all n. Under broader assumptions this problem was considered in [START_REF] Aptekarev | Measures for orthogonal polynomials with unbounded recurrence coefficients[END_REF] where Nevai's method was used.

Under very general assumptions on the coefficients a n , b n the asymptotics of the polynomials P n (z) was studied in [START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF] where however it was assumed that z ∈ R and that the coefficients b n are small compared to a n . The Carleman and non-Carleman cases were treated in [START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF] at an equal footing so that the difference in the corresponding asymptotic formulas for P n (z) was not quite visible. Among the results of the present paper, Corollary 4.3 seems to be the closest to the main result, Theorem C, of [START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF].

I thank G. Świderski for useful discussions.

1.2. Main results. Our goal is to find asymptotic formulas for P n (z) as n → ∞ without the Carleman condition (1.2), that is, in the case

∞ n=0 a -1 n < ∞; (1.4)
then the coefficients a n → ∞ faster than n. Astonishingly, the asymptotics of the orthogonal polynomials in this a priori highly singular case is particularly simple and general. Let us briefly describe some of our main results omitting minor technical assumptions. In addition to (1.4), suppose that there exists a finite limit

- b n 2 √ a n-1 a n =: β n → β ∞ where |β ∞ | = 1 (1.5)
as n → ∞. We distinguish two cases:

|β ∞ | < 1 and |β ∞ | > 1. If |β ∞ | < 1, we set φ n = n-1 m=0 arccos β m , n ≥ 1, (1.6)
where the sum is restricted to m such that |β m | ≤ 1. For example, φ n = πn/2 if b n = 0 for all n ∈ Z + . We show in Theorem 4.1 that, for an arbitrary z ∈ C, all solutions F n (z) of the Jacobi equation

a n-1 F n-1 (z) + b n F n (z) + a n F n+1 (z) = zF n (z), n ∈ Z + , (1.7)
have asymptotic behavior

F n (z) = a -1/2 n k + e -iφn + k -e iφn + o(1) , n → ∞, (1.8)
with some constants k ± ∈ C (depending of course on the solution {F n (z)}). The equalities k + = k -= 0 are possible for the trivial solution F n (z) = 0 only. Conversely, for all k ± ∈ C, there exists a solution F n (z) of equation (1.7) with asymptotics (1.8). In particular, formula (1.8) with some constants k ± (z) is true for the orthogonal polynomials P n (z). Note that due to (1.4) and (1.8) all solutions of the equation (1.7) belong to 2 (Z + ).

In the case |β ∞ | > 1 we set

ϕ n = n-1 m=0 arccosh |β m |, n ≥ 1, (1.9)
where the sum is restricted to m such that |β m | ≥ 1. Then for all z ∈ C, we have (see Theorem 4.13) an asymptotic relation P n (z) = k(z)a -1/2 n (sgn β ∞ ) n e ϕn (1 + o(1)).

(1.10)

If the Jacobi operator with the coefficients a n , b n is essentially self-adjoint, then the coefficient k(z) = 0 unless z is an eigenvalue of this operator. Note that e ϕn can be, alternatively, written as a product

e ϕn = n-1 m=0 |β m | + β 2 n -1 , n ≥ 1.
According to (1.8) and (1.10) the asymptotic behavior of the polynomials P n (z) is the same for all z ∈ C, both for real z and for z with Im z = 0. The coefficients k ± (z) and k(z) can be expressed via the Wronskians of the solution {P n (z)} and the Jost solution {f n (z)} (introduced below) of the Jacobi equation (1.7); these coefficients depend on z.

We emphasize that the asymptotic formulas for the cases (1.2) and (1.4) are qualitatively different from each other. This is discussed in Sect. 5.2 and 5.3.

1.3. Scheme of the approach. We are motivated by an analogy of the difference (1.7) and differential -(a(x)f (x, z)) + b(x)f (x, z) = zf (x, z), x > 0, (1.11) equations where a(x) > 0 and b(x) is real. To a large extent, x, a(x) and b(x) here play the roles of the parameters n, a n and b n in the Jacobi equation (1.7). In the case a(x) = 1, b ∈ L 1 (R + ), equation (1.11) has a solution f (x, z), known as the Jost solution, behaving like e i √ zx , Im √ z ≥ 0, as x → ∞. For rather general coefficients a(x), b(x), equation (1.11) has solutions f (x, z) with asymptotics given by the classical Liouville-Green formula (see Chapter 6 of the book [START_REF] Olver | Introduction to asymptotics and special functions[END_REF]). In the case a(x) → a ∞ > 0, b(x) → 0, the Liouville-Green formula was simplified in [START_REF] Yafaev | A note on the Schrödinger operator with a long-range potential[END_REF] which yields solutions f (x, z) of (1.11) with asymptotics

f (x, z) ∼ exp - x 0 b(y) -z a(y) 1/2 dy =: Q(x, z), Re b(y) -z a(y) 1/2 ≥ 0, (1.12)
as x → ∞. Note that the function Q(x, z) (the Ansatz for the Jost solution f (x, z)) satisfies equation (1.11) with a sufficiently good accuracy. Formula (1.12) was modified in [START_REF] Yafaev | Semiclassical asymptotic behavior of orthogonal polynomials[END_REF] for Jacobi equations (1.7) where a n → a ∞ > 0, b n → 0 as n → ∞ and condition (1.3) is satisfied. This permitted to find asymptotics of the orthogonal polynomials P n (z) for such coefficients a n , b n in a very natural way. We are applying the same scheme in the non-Carleman case. Here, we accept conditions (1.4) and (1.5). Let us briefly discuss the main steps of our approach.

A. First, we forget about the orthogonal polynomials P n (z) and distinguish solutions (the Jost solutions) f n (z) of the difference equation (1.7) by their asymptotics as n → ∞. This requires a construction of an Ansatz Q n (z) for the Jost solutions.

B. Under assumption (1.4) this construction (see Sect. 2.3) is very explicit and, in particular, does not depend on z ∈ C. In the case |β ∞ | < 1, we set

Q n = a -1/2 n e -iφn (1.13)
with the phase φ n defined by formula (1.6). In the case |β ∞ | > 1, the Ansatz equals

Q n = a -1/2 n (sgn β ∞ ) n e -ϕn (1.14)
where the phase ϕ n is given by (1.9). It is shown in Sect. 2.4 that in both cases the relative remainder

r n (z) := ( √ a n-1 a n Q n ) -1 a n-1 Q n-1 + (b n -z)Q n + a n Q n+1 , n ≥ 1, (1.15)
belongs to 1 (Z + ). At an intuitive level, the fact that the Ansätzen (1.13) and (1.14) do not depend on z ∈ C can be explained by the fast growth of the coefficients a n which makes the spectral parameter z negligible.

Actually, the Ansätzen we use (especially, the amplitude factor a -1/2 n

) are only distantly similar to the Liouville-Green Ansatz for the Schrödinger equation (1.11).

C. Then we make in Sect. 2.5 a multiplicative change of variables In the case |β ∞ | > 1, a solution g n (z) of (1.7) linearly independent with f n (z) can be constructed by an explicit formula

f n (z) = Q n u n (z) (1.
g n (z) = f n (z) n m=n0 (a m-1 f m-1 (z)f m (z)) -1 , n ≥ n 0 , (1.17)
where n 0 = n 0 (z) is a sufficiently large number. This solution grows exponentially as n → ∞,

g n (z) = 1 2 β 2 ∞ -1 a 1/2 n (sgn β ∞ ) n+1 e ϕn (1 + o(1)).
Since g n (z) is linearly independent with f n (z), the polynomials P n (z) are linear combinations of f n (z) and g n (z) which leads to the formula (1.10). Note that (1.17) is a discrete analogue of formula (1.36) in Chapter 4 of the book [START_REF] Yafaev | Mathematical scattering theory: Analytic theory[END_REF] for the Schrödinger equation. The scheme described briefly above seems to be quite different from [START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF] where the first step was a study of the Turán determinants P n (z) 2 -P n-1 (z)P n+1 (z).

1.4. Jacobi operators. It is natural (see the book [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF]) to associate with the coefficients a n , b n a three-diagonal matrix

J =        b 0 a 0 0 0 0 • • • a 0 b 1 a 1 0 0 • • • 0 a 1 b 2 a 2 0 • • • 0 0 a 2 b 3 a 3 • • • . . . . . . . . . . . . . . . . . .        (1.18)
known as the Jacobi matrix. Then equation (1.1) with the boundary condition P -1 (z) = 0 is equivalent to the equation J P (z) = zP (z) for the vector P (z) = {P n (z)} ∞ n=0 . Thus P (z) is the "eigenvector" of the matrix J corresponding to the "eigenvalue" z.

Let us now consider Jacobi operators defined by matrix (1.18) in the canonical basis e 0 , e 1 , . . . of the space 2 (Z + ). The minimal Jacobi operator J 0 is defined by the formula

J 0 f = J f on a set D ⊂ 2 (Z + ) of vectors f = {f n } ∞
n=0 such that f n = 0 for sufficiently large n. It is symmetric in the space 2 (Z + ), and its adjoint operator J * 0 is given by the same formula J * 0 f = J f on all vectors f ∈ 2 (Z + ) such that J f ∈ 2 (Z + ). The deficiency indices of the operator J 0 are either (0, 0) (the limit point case) or (1, 1) (the limit circle case). If the Carleman condition (1.2) holds, then (see, e.g., the book [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF]) for all b n the operator J 0 is essentially self-adjoint on D so that J 0 has the unique self-adjoint extension J = clos J 0 (the closure of J 0 ). On the contrary, under assumption (1.4) the deficiency indices of J 0 depend on the value of

|β ∞ |. If |β ∞ | < 1,
then it follows from (1.8) that all solutions of equation (1.7) are in 2 (Z + ), and hence the deficiency indices of the operator J 0 are [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF][START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF]. In this case the operator J 0 has a one-parameter family of self-adjoint extensions J ⊂ J * 0 . Their domains can be described explicitly (see Sect. 6.5 of [START_REF] Schmüdgen | The moment problem[END_REF] or §2 of [START_REF] Simon | The classical moment problem as a self-adjoint finite difference operator[END_REF]) in terms of the orthogonal polynomials P n (z) (of first kind) and P n (z) (of second kind). We recall that P n (z) are defined by equations (1.1) where n ≥ 1 with the boundary conditions P 0 (z) = 0, P 1 (z) = a -1 0 ; clearly, P n (z) is a polynomial of degree n -1. In the case |β ∞ | > 1 we show that the operator J 0 is essentially self-adjoint if and only if

∞ n=0 a -1 n e 2ϕn = ∞ (1.19)
where ϕ n is defined by (1.9); otherwise the deficiency indices of J 0 are [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF][START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF]. This result may be compared with the Berezanskii theorem (see, e.g., page 26 in the book [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF]) stating that the Carleman condition (1.2) is necessary for the essential self-adjointness of J 0 provided b n = 0 and a n-1 a n+1 ≤ a 2 n . The spectra of all self-adjoint Jacobi operators J are simple with e 0 = (1, 0, 0, . . .) being a generating vector. Therefore it is natural to define the spectral measure of J by the relation dρ J (λ) = d(E J (λ)e 0 , e 0 ) where dE J (λ) is the spectral family of the operator J. For all extensions J of the operator J 0 , the polynomials P n (λ) are orthogonal and normalized in the spaces L 2 (R; dρ J ):

∞ -∞ P n (λ)P m (λ)dρ J (λ) = δ n,m ;
as usual, δ n,n = 1 and δ n,m = 0 for n = m.

We also note a link with a moment problem

s n = ∞ -∞ λ n dρ(λ)
where s n are given and the measure dρ(λ) has to be found. A moment problem is called determinate if its solution dρ(λ) is unique. In the opposite case it is called indeterminate. Suppose that s n = (J n 0 e 0 , e 0 ). Then

s n = (J n e 0 , e 0 ) = ∞ -∞ λ n dρ J (λ)
for all self-adjoint extensions J of the operator J 0 . It is known (see, e.g., Theorem 2 in [START_REF] Simon | The classical moment problem as a self-adjoint finite difference operator[END_REF]) that the moment problem with the coefficients s n = (J n 0 e 0 , e 0 ) is determinate if and only if the operator J 0 is essentially self-adjoint.

The comprehensive presentation of the results described in this subsection can be found in the books [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF][START_REF] Schmüdgen | The moment problem[END_REF] or the survey [START_REF] Simon | The classical moment problem as a self-adjoint finite difference operator[END_REF]. We rely essentially on a direct study of the difference equation (1.7) and practically do not use operator methods. So, the above information was given mainly to put our results into the right framework.

Ansatz

In this section, we calculate the remainder (1.15) for the Ansatz Q n defined by formulas (1.13) or (1.14). Then we make substitution (1.16).

2.1. Preliminaries. Let us consider equation (1.7). Note that the values of F N -1 and F N for some N ∈ Z + determine the whole sequence F n satisfying the difference equation (1.7).

Let f = {f n } ∞ n=-1 and g = {g n } ∞ n=-1 be two solutions of equation (1.7). A direct calculation shows that their Wronskian {f, g}

:= a n (f n g n+1 -f n+1 g n ) (2.1)
does not depend on n = -1, 0, 1, . . .. In particular, for n = -1 and n = 0, we have

{f, g} = 2 -1 (f -1 g 0 -f 0 g -1 )
and {f, g} = a 0 (f 0 g 1 -f 1 g 0 )

(we put a -1 = 1/2). Clearly, the Wronskian {f, g} = 0 if and only if the solutions f and g are proportional.

It is convenient to introduce a notation

x n = x n+1 -x n (2.2)
for the "derivative" of a sequence x n . Note the Abel summation formula ("integration by parts"):

M n=N x n y n = x M y M +1 -x N -1 y N - M n=N x n-1 y n ; (2.3)
here M ≥ N ≥ 0 are arbitrary, but we have to set x -1 = 0 so that x -1 = x 0 if N = 0. It follows from equation (1.1) that if P n (z) are the orthogonal polynomials corresponding to coefficients (a n , b n ), then the polynomials (-1) n P n (-z) correspond to the coefficients (a n , -b n ). Therefore without loss of generality, we could have assumed that β ∞ ≥ 0.

To emphasize the analogy between differential and difference operators, we often use "continuous" terminology (Volterra integral equations, integration by parts, etc.) for sequences labelled by the discrete variable n. Below C, sometimes with indices, and c are different positive constants whose precise values are of no importance.

In all our constructions below, it suffices to consider the Jacobi equation (1.7) for large n only.

2.2.

Assumptions. In addition to (1.4) and (1.5), we need some mild conditions on a regularity of behavior of the sequences a n and b n as n → ∞. Let us set

κ n = a n+1 a n , k n = κ n-1 κ n = a n √ a n-1 a n+1 . (2.4)
With respect to a n , we assume that

{k n -1} ∈ 1 (Z + ) (2.5)
which implies also the following properties of the numbers κ n .

Lemma 2.1. Under assumption (2.5) there exists a finite limit

lim n→∞ κ n =: κ ∞ , (2.6
)

and κ ∞ ≥ 1 if condition (1.4) is satisfied. Moreover, {κ n } ∈ 1 (Z + ) (2.7)
if both (1.4) and (2.5) are true.

Proof. By definition (2.4) of k n , we have ln

k n = ln κ n-1 -ln κ n whence ln κ n = ln κ 0 - n m=1 ln k m .
It follows from (2.5) that the series on the right converges which implies the existence of the limit (2.6). If κ ∞ < 1, then, by definition (2.4) of κ n , we would have a n ≤ γa n-1 for some γ < 1 and all n ≥ n 0 if n 0 is sufficiently large. Thus, a -1 n ≥ a -1 n0 γ n0 γ -n so that the series in (1.4) diverges. Since

κ n -κ n-1 = κ 2 n κ n + κ n-1 (1 + k n )(1 -k n ), relation (2.7
) is a direct consequence of assumption (2.5).

Example 2.2. Both conditions (1.4) and (2.5) are satisfied for a n = γn p where γ > 0, p > 1 and for a n = γx n q where x > 1, q < 1. In these cases κ ∞ = 1. For a n = γx n , conditions (1.4) and (2.5) are also satisfied but κ ∞ = √ x. On the contrary, condition (2.5) fails if a n = γx n q with q > 1.

With respect to the coefficients b n or, more precisely, the ratios β n defined by (1.5), we assume that

{β n } ∈ 1 (Z + ). ( 2
(Z + ). Put ζ n = κ n Q n+1 Q -1 n
and introduce notation

α n = 1 2 √ a n-1 a n , β n = - b n 2 √ a n-1 a n , n ∈ Z + ;
for definiteness, we set

a -1 = 1/2 and Q 0 = a -1/2 0
. Then

Q n = 1 √ a n ζ 0 • • • ζ n-1 , n ≥ 1, (2.9)
and (1.15) can be rewritten as

r n (z) = ζ -1 n-1 -2β n + k n ζ n -2zα n .
(2.10)

We have to find ζ n such that {r n (z)} ∈ 1 (Z + ). In principle, the numbers ζ n can be successively determined from the equations r n (z) = 0 which leads of course to very complicated expressions. Fortunately the construction of ζ n becomes quite explicit if one neglects in (2.10) the terms from 1 (Z + ). Obviously, the term 2zα n in the right-hand side of (2.10) can be omitted if condition (1.4) is satisfied. Furthermore, under the assumptions below ζ -1 n-1 can be replaced by ζ -1 n and k n -by 1 which allows us to define ζ n from the equation

ζ n + ζ -1 n = 2β n . ( 2
r n (z) = ζ -1 n-1 -ζ -1 n + (k n -1)ζ n -2zα n . (2.12)
Solutions of (2.11) are obviously given by the equalities

ζ n = β n -i 1 -β 2 n = (β n + i 1 -β 2 n ) -1 (2.13) if |β n | ≤ 1 and ζ n = sgn β n (|β n | -β 2 n -1) = sgn β n (|β n | + β 2 n -1) -1 (2.14) if |β n | > 1.
Of course, the numbers ζ -1 n also satisfy (2.11). In the case |β n | ≤ 1, we choose formula (2.13) for definiteness only. On the contrary, in the case

|β n | > 1, the condition |ζ n | ≤ 1 will be important below. It is convenient to set θ n = arccos β n ∈ [0, π] (2.15) for |β n | ≤ 1 and ϑ n = arccosh |β n | = ln |β n | + β 2 n -1 > 0 (2.16) for |β n | > 1. Then ζ n = e -iθn for |β n | ≤ 1 and ζ n = sgn β n e -ϑn for |β n | ≥ 1. It follows from condition (1.5) that ζ n → β ∞ -i 1 -β 2 ∞ =: ζ ∞ = e -iθ∞ if |β ∞ | < 1 (2.17) and ζ n → sgn β ∞ (|β ∞ | -β 2 ∞ -1) =: ζ ∞ = sgn β ∞ e -ϑ∞ if |β ∞ | > 1 (2.18) as n → ∞.
Here θ ∞ and ϑ ∞ are defined by formulas (2.15) and (2.16) where n = ∞, that is,

θ ∞ = arccos β ∞ ∈ (0, π) and ϑ ∞ = ln |β ∞ | + β 2 ∞ -1 > 0. (2.19) Obviously, |ζ ∞ | = 1 and ζ 2 ∞ = 1 if |β ∞ | < 1 and ζ ∞ ∈ (-1, 1) but ζ ∞ = 0 if |β ∞ | > 1. For |β n | ≥ 1 in the case |β ∞ | < 1 and for |β n | ≤ 1 in the case |β ∞ | > 1,
the numbers θ n and ϑ n can be chosen in an arbitrary way; for definiteness, we set θ n = 0 and ϑ n = 0. Now the Ansatz (2.9) can be written as

Q n = 1 √ a n e -iφn if |β ∞ | < 1 (2.20) and Q n = 1 √ a n (sgn β ∞ ) n e -ϕn if |β ∞ | > 1.
(2.21)

Here

φ n = n-1 m=0 θ m , ϕ n = n-1 m=0 ϑ m (2.
φ n = θ ∞ n + o(n) and ϕ n = ϑ ∞ n + o(n) as n → ∞ (2.23)
with θ ∞ and ϑ ∞ given by formulas (2.19). Moreover, we have the following statement.

Lemma 2.4. Assume that ∞ n=0 |β n -β ∞ | < ∞. ( 2

.24)

Then the sequences φ n -θ ∞ n and ϕ n -ϑ ∞ n have finite limits as n → ∞.

Proof. Observe that

φ n -θ ∞ n = n-1 m=0 (θ n -θ ∞ ) (2.25)
where, by definition (2.15),

|θ n -θ ∞ | = | arccos β n -arccos β ∞ | ≤ C|β n -β ∞ |.
Here n is sufficiently large and the condition |β ∞ | < 1 is taken into account. Therefore assumption (2.24) implies that the series in the right-hand side of (2.25) converges. The difference ϕ n -ϑ ∞ n can be considered quite similarly.

ζ -1 n-1 -ζ -1 n = (β n-1 -β n ) 1 -i β n-1 + β n 1 -β 2 n-1 + 1 -β 2 n , |β ∞ | < 1,
and

ζ -1 n-1 -ζ -1 n = (β n-1 -β n ) 1 + sgn β ∞ β n-1 + β n β 2 n-1 -1 + β 2 n -1 , |β ∞ | > 1.
According to (2.26) each of these identities yields an estimate

|ζ -1 n-1 -ζ -1 n | ≤ C|β n-1 -β n |.
(2.27) Therefore (2.12) implies that

|r n (z)| ≤ C|β n-1 -β n | + |k n -1| + 2α n |z| (2.28)
where the constant C does not depend on z. This leads to the following assertion.

Lemma 2.5. Let conditions (1.4) and (1.5) be satisfied. Then estimate (2.28) for the remainder (2.12) is true for all z ∈ C. Under additional assumptions (2.5) and (2.8), we have

{r n (z)} ∈ 1 (Z + ).
(2.29) 2.5. Multiplicative substitution. Let the sequence Q n be given by formulas (2.20) or (2.21). We are looking for solutions f n (z) of the difference equation (1.7) satisfying the condition

f n (z) = Q n (1 + o(1)), n → ∞.
(2.30)

The uniqueness of such solutions is almost obvious.

Lemma 2.6. Equation (1.7) may have only one solution f n (z) satisfying condition (2.30).

Proof. Let f n (z) be another solution of (1.7) satisfying (2.30). It follows from (2.20) or (2.21) that the Wronskian (2.1) of these solutions calculated for n → ∞ equals

{f, f} = a n Q n Q n+1 o(1) = o(1)κ -1 n e -i(φn-1+φn) , |β ∞ | < 1 sgn β ∞ e -ϕn-1-ϕn , |β ∞ | > 1
so that {f, f} = 0 according to (2.6). Thus f = Cf where C = 1 by virtue again of condition (2.30).

Remark 2.7. For the calculation of the Wronskian {f, f}, it is essential that the power of a n in (2.9) equals -1/2.

For construction of f n (z), we will reformulate the problem introducing a sequence

u n (z) = Q -1 n f n (z), n ∈ Z + . ( 2
k n ζ n (u n+1 (z) -u n (z)) -ζ -1 n-1 (u n (z) -u n-1 (z)) = -r n (z)u n (z), n ∈ Z + , (2.33)
for sequence (2.31).

Proof. Substituting expression f n = Q n u n into (1.7) and using the equality

Q n+1 Q n = a n a n+1 ζ n ,
we see that

( √ a n-1 a n Q n ) -1 a n-1 f n-1 + (b n -z)f n + a n f n+1 = a n-1 a n Q n-1 Q n u n-1 -2(α n z + β n )u n + a n a n-1 Q n+1 Q n u n+1 = ζ -1 n-1 u n-1 -2(α n z + β n )u n + k n ζ n u n+1 .
In view of (2.11) the right-hand side here equals

k n ζ n (u n+1 -u n ) -ζ -1
n-1 (u n -u n-1 ) + r n u n with r n defined by (2.12). Therefore equations (1.7) and (2.33) are equivalent.

Modified Jost solutions

In this section, we reduce the Jacobi difference equation (1.7) for Jost solutions f n (z) to a Volterra equation for functions u n (z) defined by formula (2.31) and satisfying the condition (2.32). Solutions u n (z) of the Volterra equation can be constructed by iterations. At this point, there is almost no difference between the cases |β ∞ | < 1 and |β ∞ | > 1. Finally, formula (2.31) yields asymptotics of f n (z) as n → ∞.

3.1. Volterra integral equation. The sequence u n (z) satisfying the difference equation (2.33) will be constructed as a solution of an appropriate "Volterra integral" equation. We set

σ n = ζ n ζ n-1 , S n = σ 1 σ 2 • • • σ n-1 , n ≥ 2, ( 3.1)
and, for n < m,

G n,m = -(κ m-1 ζ m ) -1 S m+1 m p=n+1 κ p-1 S -1 p = -(κ m-1 ζ m ) -1 m p=n+1 κ p-1 σ p • • • σ m . (3.2)
Note that the kernel G n,m does not depend on z. Let us consider an equation

u n (z) = 1 + ∞ m=n+1 G n,m r m (z)u m (z) (3.3)
where the sequence r m (z) is defined by formula (2.12). Our first goal is to estimate the matrix elements G n,m . This is quite straightforward in the case

|β ∞ | > 1.
|G n,m | ≤ C < ∞. (3.4)
Proof. By virtue of (2.18), we have

|σ n | ≤ s ∞ < 1 (3.5) if s ∞ ∈ (ζ 2 ∞ , 1)
and n is sufficiently large, say, n ≥ n 0 . Moreover, {κ n } ∈ ∞ (Z + ) and {κ -1 n } ∈ ∞ (Z + ) according to Lemma 2.1. Therefore it follows from (3.2) that

|G n,m | ≤ C m p=n+1 s m-p+1 ∞ = C 1 -s m-n ∞ s -1 ∞ -1 ≤ C s ∞ 1 -s ∞ , n < m, if n ≥ n 0 . Let now n < n 0 . If n < m ≤ n 0 , then the sum (3.2) consists of at most n 0 terms. If m > n, then G n,m = G n,n0 + G n0,m
is a sum of two bounded terms.

In the case |β ∞ | < 1 we have to "integrate by parts". Proof. As in the previous lemma, we can suppose that n ≥ n 0 where n 0 is sufficiently large. Since |ζ p | = 1 according to (2.13), it follows from definition (3.1) that

|S m+1 S -1 p | = |σ p • • • σ m | = 1, p ≤ m. (3.6)
Relation (2.17) where

ζ 2 ∞ = 1 implies that |σ n -1| ≥ c > 0. (3.7)
By definitions (2.2) and (3.1) we have

(S -1 p ) = S -1 p+1 -S -1 p = (σ -1 p -1)S -1 p ,
and hence integrating by parts (that is, using formula (2.3)), we find that

m p=n+1 κ p-1 S -1 p = m p=n+1 κ p-1 (σ -1 p -1) -1 (S -1 p ) = κ m-1 (σ -1 m -1) -1 S -1 m+1 -κ n-1 (σ -1 n -1) -1 S -1 n+1 - m p=n+1 (κ p-2 (σ -1 p-2 -1) -1 ) S -1 p . (3.8) Note that ((σ -1 p-1 -1) -1 ) = (σ p-1 -1) -1 (σ p -1) -1 σ p-1
where σ p ∈ 1 (Z + ) by virtue of (2.27) and (2.8). Using also (2.7) and (3.7), we see that this sequence belongs to 1 (Z + ) whence

(κ p-1 (σ -1 p-1 -1) -1 ) ∈ 1 (Z + ).
u (k+1) n (z) = ∞ m=n+1 G n,m r m (z)u (k) m (z), k ≥ 0, (3.9)
for all n ∈ Z + . Then the estimates

|u (k) n (z)| ≤ C k k! ∞ m=n+1 |r m (z)| k , ∀k ∈ Z + . (3.10)
are true for all sufficiently large n with the same constant C as in (3.4).

Proof. Suppose that (3.10) is satisfied for some k ∈ Z + . We have to check the same estimate (with k replaced by k + 1 in the right-hand side) for u

(k+1) n . Set R m = ∞ p=m+1 |r p |.
According to definition (3.9), it follows from estimate (3.4) and (3.10) that

|u (k+1) n | ≤ C k+1 k! ∞ m=n+1 |r m |R k m . (3.11)
Observe that

R k+1 m + (k + 1)|r m |R k m ≤ R k+1 m-1
, and hence, for all N ∈ Z + , (k + 1)

N m=n+1 |r m |R k m ≤ N m=n+1 (R k+1 m-1 -R k+1 m ) = R k+1 n -R k+1 N ≤ R k+1 n .
Substituting this bound into (3.11), we obtain estimate (3.10) for u

|u n (z) -1| ≤ e Cεn(z) -1, n ≥ 0, (3.12)
where the constant C does not depend on z ∈ C and

ε n (z) = ∞ m=n+1 |β m-1 -β m | + |k m -1| + α m |z| → 0 as n → ∞. (3.13)
For all n ∈ Z + , the functions u n (z) are entire functions of z ∈ C.

Proof.

Set

u n = ∞ k=0 u (k) n (3.14)
where u

(k) 0 = 1 and u (k)
n , k ≥ 1, are defined by recurrence relations (3.9). Estimate (3.10) shows that this series is absolutely convergent. Using the Fubini theorem to interchange the order of summations in m and k, we see that

∞ m=n+1 G n,m r m u m = ∞ k=0 ∞ m=n+1 G n,m r m u (k) m = - ∞ k=0 u (k+1) n = 1 - ∞ k=0 u (k) n = 1 -u n .
This is equation (3.3) for sequence (3.14). Estimate (3.12) for the sequence (3.14) also follows from (3.10) where r n (z) satisfies (2.28).

According to (2.12) the remainder r m (z) and hence the kernels G n,m r m (z) are linear functions of z. Therefore recurrence arguments show that all successive approximations u Corollary 3.5. We have an estimate

|u n (z) -1| ≤ Cε n (z) (3.15)
where ε n (z) → 0 uniformly on compact subsets of C.

It turns out that the construction above yields a solution of the difference equation (2.33).

Lemma 3.6. Let r n (z) and G n,m be given by formulas (2.12) and (3.2), respectively. Then a solution u n (z) of integral equation (3.3) satisfies also the difference equation (2.33).

Proof. It follows from (3.3) that

u n+1 -u n = ∞ m=n+2 (G n+1,m -G n,m )r m u m -G n,n+1 r n+1 u n+1 . (3.16)
Since according to (3.2)

G n+1,m -G n,m = -κ n (κ m-1 ζ m ) -1 S -1 n+1 S m+1 and G n,n+1 = ζ -1 n+1 S n+2 S -1 n+1
, equality (3.16) can be rewritten as

κ -1 n (u n+1 -u n ) = - ∞ m=n+1 S -1 n+1 S m+1 (κ m-1 ζ m ) -1 r m u m . (3.17)
Putting together this equality with the same equality for n + 1 replaced by n, we see that

κ -1 n (u n+1 -u n ) -σ -1 n κ -1 n-1 (u n -u n-1 ) = ∞ m=n+1 S -1 n+1 S m+1 r m (κ m-1 ζ m ) -1 u m -σ -1 n ∞ m=n S -1 n S m+1 r m (κ m-1 ζ m ) -1 u m .
Since S n+1 = σ n S n , the right-hand side here equals -(κ n-1 ζ n ) -1 r n u n , and hence the equation obtained coincides with (2.33).

The above arguments show also that the functions u n (z) are of minimal exponential type. where m → 0 as m → ∞. On the other hand, it follows from equation (1.7) for function (1.16) that Proof. Let us proceed from expression (3.17) for u n+1 -u n . It follows from (3.1) and (3.5) that

|u n (z)| ≤ C n |z|(|u n+1 (z)| + |u n+2 (z)|) ≤ • • • ≤ C n,k |z| k (|u n+k (z)| + |u n+k+1 (z)|) (3.
|S -1 n+1 S m+1 | = |σ n+1 • • • σ m | ≤ s m-n ∞
for sufficiently large n and m ≥ n whence

|u n+1 -u n | ≤ C ∞ m=n+1 s m-n ∞ |r m u m |.
So we only have to take the sum over n here. Then we use that s ∞ < 1 and {r m u m } ∈ 1 (Z + ).

f n (z) = Q n u n (z) (3.21)
satisfies equation (1.7). This leads to the following results. We state them separately for the cases

|β ∞ | < 1 and |β ∞ | > 1.
Theorem 3.9. Let the assumptions (1.4), (1.5) with |β ∞ | < 1 as well as (2.5), (2.8) be satisfied.

Then, for all z ∈ C, the equation (1.7) has a solution {f n (z)} with asymptotics

f n (z) = 1 √ a n e -iφn 1 + O(ε n (z)) , n → ∞, (3.22)
where φ n and ε n (z) are given by formulas (1.6) and (3.13), respectively. Asymptotics (3.22) is uniform in z from compact subsets of the complex plane C. For all n ∈ Z + , the functions f n (z) are entire functions of z ∈ C of minimal exponential type.

Corollary 3.10. Suppose that b n = 0, at least for sufficiently large n. Let the assumptions (1.4) and (2.5) be satisfied. Then, for all z ∈ C, the equation (1.7) has a solution {f n (z)} with asymptotics

f n (z) = 1 √ a n (-i) n 1 + O(ε n (z)) , n → ∞.
By analogy with the continuous case, the sequence {f n (z)} ∞ n=-1 will be called the (modified) Jost solution of equation (1.7). Additionally, we define the conjugate Jost solution by the formula fn (z) = f n (z).

(z) = 1 √ a n e iφn 1 + O(ε n ) , n → ∞, (3.24)
Using asymptotic formulas (3.22), (3.24) and definition (2.15), we can calculate the Wronskian (2.1) of the solutions f (z), f (z):

{f (z), f (z)} = lim n→∞ κ -1 n (e iθn -e -iθn ) = κ -1 ∞ (e iθ∞ -e -iθ∞ ) = 2iκ -1 ∞ 1 -β 2 ∞ = 0 (3.25)
according to (2.17). Note that the amplitude factors a Thus, in the case |β ∞ | < 1 for all z ∈ C, we have two linearly independent oscillating solutions f n (z) and fn (z). Under assumption (1.4) both of them belong to 2 (Z + ). This leads to the following result.

Corollary 3.11. Under the assumptions of Theorem 3.9 the minimal Jacobi operator J 0 has deficiency indices (1, 1).

Let us show that, even in the case b n = 0, assumption (2.5) cannot be omitted in Theorem 3.9. The example below was considered in [START_REF] Kostyuchenko | Generalized Jacobi matrices and deficiency indices of differential operators with polynomial coefficients[END_REF], Example 2, and in [START_REF] Janas | Spectral properties of Jacobi matrices by asymptotic analysis[END_REF], Lemma 2.3.

Example 3.12. Suppose that b n = 0 and that a n = n p (1 + c 1 n -1 ) if n is odd and a n = n p (1 + c 2 n -1 ) if n is even for p > 1 and sufficiently large n. Corollary 3.11 does not apply in this case because

a n √ a n-1 a n+1 = 1 + (-1) n c 2 -c 1 n + O( 1 n 2 ), n → ∞, (3.26)
so that condition (2.5) breaks down unless c 1 = c 2 . Actually, it was proven in [START_REF] Kostyuchenko | Generalized Jacobi matrices and deficiency indices of differential operators with polynomial coefficients[END_REF][START_REF] Janas | Spectral properties of Jacobi matrices by asymptotic analysis[END_REF] that the corresponding Jacobi operator J 0 is essentially adjoint if |c 2 -c 1 | ≥ p -1.

In the case |β ∞ | > 1, we have the following result.

Theorem 3.13. Let the assumptions (1.4), (1.5) with |β ∞ | > 1 as well as (2.5), (2.8) be satisfied.

Then, for all z ∈ C, the equation (1.7) has a solution {f n (z)} with asymptotics

f n (z) = 1 √ a n (sgn β ∞ ) n e -ϕn 1 + O(ε n ) , n → ∞, (3.27)
where ϕ n and ε n (z) are given by formulas (1.9) and (3.13), respectively. Asymptotics (3.27) is uniform in z from compact subsets of the complex plane C. For all n ∈ Z + , the functions f n (z) are entire functions of z ∈ C of minimal exponential type.

According to (2.19) and (2.23) the solution (3.27) tends to zero exponentially as n → ∞. It will also be called the (modified) Jost solution of equation (1.7). However, in contrast to the case where γ > 0 and δ, σ are arbitrary real numbers. Then

|β ∞ | < 1, for |β ∞ | > 1 the
-inθ∞ = (β ∞ -i 1 -β 2 ∞ ) n and e -nϑ∞ = (|β ∞ | + β 2 ∞ -1) -n , ( 3
β n = δ 2γ (1 + σn -1 )(1 -n -1 ) -p/2 = δ 2γ 1 + (σ + p 2 )n -1 + O(n -2 )
and β ∞ = δ(2γ) -1 . The assumptions of Theorem 3.9 (of Theorem 3.13) are satisfied if |δ| < 2γ (resp., |δ| > 2γ). Since

β n -β ∞ = (2γ) -1 δ(σ + p/2)n -1 + O(n -2 ), ( 3
θ n = arccos β ∞ + νn -1 + O(n -2 ), ν = - δ(σ + p/2) |δ 2 -4γ 2 | ,
so that according to (2.22) the sequence φ n -n arccos β ∞ -ν ln n has a finite limit as n → ∞. Therefore formula (3.22) in Theorem 3.9 can be replaced by a simpler one

f n (z) = n -p/2-iν (β ∞ -i 1 -β 2 ∞ ) n 1 + o(1) , n → ∞. Quite similarly, in the case |β ∞ | > 1,
(z) = n -p/2+ν sgn δ (sgn δ) n (|β ∞ | + β 2 ∞ -1) -n 1 + o(1) , n → ∞.

Orthogonal polynomials

4.1. Small diagonal elements. Here we suppose that |β ∞ | < 1. Recall that the Jost solution f (z) = {f n (z)} was constructed in Theorem 3.9 and f (z) was defined by relation (3.23). By virtue of (3.25), an arbitrary solution F (z) = {F n (z)} of the Jacobi equation (1.7) is a linear combination of the Jost solutions f (z) and f (z), that is

F n (z) = k + (z)f n (z) + k -(z) fn (z), (4.1)
where the constants can be expressed via the Wronskians:

k + (z) = κ ∞ {F (z), f (z)} 2i 1 -β 2 ∞ , k -(z) = -κ ∞ {F (z), f (z)} 2i 1 -β 2 ∞ . (4.2)
According to (4.1) the following result is a direct consequence of Theorem 3.9. Recall that the phase φ n is defined by (1.6) and the remainder ε n (z) is given by (3.13).

Theorem 4.1. Let the assumptions of Theorem 3.9 be satisfied. Choose some z ∈ C. Then an arbitrary solution F n of the Jacobi equation (1.7) has asymptotics

F n = a -1/2 n k + e -iφn + k -e iφn 1 + O(ε n (z)) , n → ∞, (4.3)
for some k ± ∈ C. Conversely, for arbitrary k ± ∈ C, there exists a solution F n of the equation (1.7) with asymptotics (4.3).

Recall that the polynomials P n (z) are solutions of the Jacobi equation (1.7) satisfying the conditions P -1 (z) = 0, P 0 (z) = 1. Relations (4.1), (4.2) and Theorem 4.1 remain of course true in this case. Moreover, the corresponding asymptotic coeficients z) .

k + (z) = κ ∞ {P (z), f (z)} 2i 1 -β 2 ∞ , k -(z) = -κ ∞ {P (z), f (z)} 2i 1 -β 2 ∞ (4.
k + (z) = κ(z)e iη(z) , η(z) ∈ R/2πZ and k -(z) = κ(z)e -iη(
Observe that the equalities κ(z) = κ(z) = 0 are impossible, since otherwise relation (4.1) for P n (z) would imply that P n (z) = 0 for all n but P 0 = 1. The next result is a particular case of Theorem 4.1.

Theorem 4.2. Let the assumptions of Theorem 3.9 be satisfied. Then, for all z ∈ C, the sequence of the orthogonal polynomials P n (z) has asymptotics

P n (z) = a -1/2 n κ(z)e -iφn+iη(z) + κ(z)e iφn-iη(z) + O(ε n (z)) , n → ∞. (4.5)
Asymptotics (4.5) is uniform in z from compact subsets of the complex plane C.

If z = λ ∈ R, then relation (4.1) for P n (z) reduces to

P n (λ) = 2 Re k + (λ)f n (λ) , (4.6)
and (4.5) yields the following result.

Corollary 4.3. If λ ∈ R, then P n (λ) = 2a -1/2 n κ(λ) cos φ n -η(λ) + O(ε n ) , n → ∞. (4.7)
Asymptotics (4.7) is uniform in λ from compact subsets of the line R.

Observe that κ(λ) = 0, since otherwise (4.6) would imply that P n (λ) = 0 for all n but P 0 = 1.

4.2.

An identity for asymptotic amplitudes. Here we suppose that Im z = 0. Multiplying equation (1.1) for P n (z) by its complex conjugate Pn (z) and taking the sum over n = 0, 1, . . . , N , we find that

N n=0 a n-1 P n-1 (z) Pn (z) + N n=0 a n P n+1 (z) Pn (z) + N n=0 b n |P n (z)| 2 = z N n=0 |P n (z)| 2 . ( 4.8)
Since P -1 (z) = 0, the first sum on the left equals

N n=0 a n P n (z) Pn+1 (z) -a N P N (z) PN+1 (z).
Therefore taking the imaginary part of (4.8), we see that

a N Im P N +1 (z) PN (z) = Im z N n=0 |P n (z)| 2 . (4.9)
According to (4.5) the left-hand side here equals

κ -1 N Im κ(z)e -iφ N +1 +iη(z) + κ(z)e iφ N +1 -iη(z) κ(z)e iφ N -iη(z) + κ(z)e -iφ N +iη(z) + o(1) (4.10)
with κ N defined by (2.4). Observe that the sum

κ(z)e -iφ N +1 +iη(z) κ(z)e -iφ N +iη(z) + κ(z)e iφ N +1 -iη(z) κ(z)e iφ N -iη(z) = 2κ(z)κ(z) cos φ N + φ N +1 -η(z) -η(z)
is real. Therefore taking relations and (2.6), (2.13) and (2.22) into account, we see that expression (4.10) equals

κ -1 N Im κ(z) 2 e -iθ N + κ(z) 2 e iθ N + o(1) = κ -1 ∞ κ(z) 2 -κ(z) 2 sin θ ∞ + o(1)
where sin

θ ∞ = 1 -β 2 ∞ .
Replacing the left-hand side of (4.9) by this expression, we obtain an identity for the asymptotic amplitude κ(z) = |k + (z)| where k + (z) is defined by (4.4). We also recall that the number κ ∞ is given by (2.6).

Theorem 4.4. Under the assumptions of Theorem 3.9 the identity ) where the phase ϕ n is given by (1.9). In particular, f n (z) = 0 for sufficiently large n, say, n ≥ n 0 = n 0 (z). Now we have to construct a solution g n (z) of (1.7) linearly independent with f n (z). We define it by the formula (1.17), that is,

κ(z) 2 -κ(z) 2 = Im z κ ∞ (1 -β 2 ∞ ) -1/2 ∞ n=0 |P n (z)| 2 holds. Corollary 4.5. If Im z > 0, then κ(z) < κ(z). Equivalently, κ(z) > κ(z) if Im z < 0.
g n (z) = f n (z)G n (z) (4.11)
where

G n (z) = n m=n0 (a m-1 f m-1 (z)f m (z)) -1 , n ≥ n 0 . (4.12)
Theorem 4.6. Suppose that a sequence f (z) = {f n (z)} satisfies the Jacobi equation (1.7). Then the sequence g(z) = {g n (z)} defined by formulas (4.11) and (4.12) satisfies the same equation and the Wronskian {f (z), g(z)} = 1. In particular, the solutions f (z) and g(z) are linearly independent.

Proof. First, we check equation (1.7) for g n . Observe that

a n-1 f n-1 G n-1 + (b n -z)f n G n + a n f n+1 G n+1 = a n-1 f n-1 + (b n -z)f n + a n f n+1 G n + a n-1 f n-1 (G n-1 -G n ) + a n f n+1 (G n+1 -G n ).
The first term here is zero because equation (1.7) is true for the sequence f n . According to definition (4.12)

G n+1 = G n + (a n f n f n+1 ) -1 , (4.13)
so that the second and third terms equal -f -1 n and f -1 n , respectively. This proves equation (1.7) for g n .

It also follows from (4.13) that the Wronskian (2.1) equals

{f (z), g(z)} = a n f n (z)f n+1 (z)(G n+1 (z) -G n (z)) = 1,
whence the solutions f (z) and g(z) are linearly independent.

Theorem 4.11 below shows that the solution g n (z) of equation (1.7) exponentially grows as n → ∞. For a proof of this result, we will have to integrate by parts in (4.12). It is convenient to start with a simple technical assertion. Recall that σ m = ζ m-1 ζ m = e -ϑm-1-ϑm , the numbers κ n were defined in formula (2.4) and the sequence u n (z) was constructed in Theorem 3.4.

Lemma 4.7. Let v m (z) = sgn β ∞ κ m-1 (σ -1 m -1) -1 (u m-1 (z)u m (z)) -1 (4.14)
and t m = e ϕm-1+ϕm . (4.15)

Then (a m-1 f m-1 (z)f m (z)) -1 = v m (z)t m . (4.16)
Proof. It follows from formulas (2.21) and (3.21) that

a m-1 f m-1 f m = sgn β ∞ a m-1 √ a m-1 a m u m-1 u m t -1 m , whence (a m-1 f m-1 f m ) -1 = sgn β ∞ κ m-1 (u m-1 u m ) -1 t m .
By definitions (2.22) and (4.15), we have

t m = t m (σ -1 m -1)
. Putting together the last two formulas with the definition (4.14) of v m , we arrive at (4.16).

Using (4.16) and integrating by parts in (4.12), we find that

G n = n m=n0 v m t m = v n t n+1 -v n0-1 t n0 -G n where G n = n m=n0 v m-1 t m . (4.17)
We will see that the asymptotics of sequence (4.11) as n → ∞ is determined by the first term in the right-hand side of (4.17). Let us calculate it.

Lemma 4.8. The asymptotic relation

lim n→∞ √ a n e -ϕn (sgn β ∞ ) n f n (z)v n (z)t n+1 = sgn β ∞ 2 β 2 ∞ -1 κ ∞ (4.18) holds.
Proof. It follows from Theorem 3.13 that

√ a n (sgn β ∞ ) n f n (z) = e -ϕn (1 + o(1)).
Using definition (4.14) of v n , Lemma 2.1 and Theorem 3.4 we find that

v n (z) = sgn β ∞ κ ∞ (ζ -2 ∞ -1) -1 (1 + o(1)
). Since according to (4.15)

t n+1 = e 2ϕn+ϑn = e 2ϕn ζ -1 ∞ (1 + o(1)), the limit (4.18) equals sgn β ∞ κ ∞ ζ -1 ∞ (ζ -2 ∞ -1) -1
which in view of (2.18) coincides with the righthand side of (4.18).

Next we show that the remainder G n (z) in (4.17) is negligible. We use the following technical assertions.

Lemma 4.9. The sequence v n (z) defined by (4.14) satisfies the condition {v n } ∈ 1 (Z + ).

Proof. According to (2.14) we have

|ζ n | = (|β n | + β 2 n -1) -1 < |β ∞ | -1 < 1 for sufficiently large n. Moreover, it follows from (2.27) that {ζ n } ∈ 1 (Z + ) if {β n } ∈ 1 (Z + ) whence (σ -1 n -1) -1 ∈ 1 (Z + )
. Putting together Theorem 3.4 and Lemma 3.8 we see that

(u -1 n-1 u -1 n ) ∈ 1 (Z + ). Now the inclusion {v n } ∈ 1 (Z + ) is a direct consequence of definition (4.14).
Lemma 4.10. Let G n (z) be defined in formula (4.17). Then

lim n→∞ √ a n e -ϕn f n (z) G n (z) = 0. (4.19)
Proof. Using Theorem 3.13 and definition (4.15), we see that

√ a n e -ϕn |f n G n | ≤ 2e -2ϕn n m=n0 |v m-1 |e ϕm-1+ϕm ≤ 2e -ϕn n0≤m<[n/2] |v m-1 | + 2 [n/2]≤m≤n |v m-1 |.
Since ϕ n → ∞ as n → ∞, relation (4.19) follows from Lemma 4.9.

Let us come back to the representations (4.11) and (4.17). Putting together Lemmas 4.8, 4.10 and taking into account that the term v n0-1 t n0 is negligible, we obtain the asymptotics of g n (z). a n e -ϕn (sgn For an arbitrary ε > 0 and sufficiently large n, it follows from (2.23) that ϑ ∞ n -ε ≤ ϕ n ≤ ϑ ∞ n + ε. Therefore under the assumptions of Theorem 3.13, the operator J 0 is essentially selfadjoint if a n ≤ cn p for some c > 0, p < ∞ and all n ≥ 1. On the contrary, J 0 has deficiency indices (1, 1) if a n ≥ cx n p (1 + o(1)) where x > 1 and p > 1.

β ∞ ) n+1 g n (z) = κ ∞ 2 β 2 ∞ -1 . ( 4
Since the Wronskian {f (z), g(z)} = 1 by Theorem 4.6, we see that

P n (z) = ω(z)f n (z) -Ω(z)g n (z)
where ω(z) = {P (z), g(z)} and

Ω(z) := {P (z), f (z)} = -2 -1 f -1 (z) (4.22)
is the Jost function. Obviously, ω(z) = 0 if Ω(z) = 0. Therefore Theorems 3.13 and 4.11 imply the following result.

Theorem 4.13. Under the assumptions of Theorem 3.13 the relation

P n (z) = -κ ∞ Ω(z) (sgn β ∞ ) n+1 2 β 2 ∞ -1 e ϕn √ a n (1 + o(1)), n → ∞, (4.23)
is true for all z ∈ C with convergence uniform on compact subsets of C. Moreover, if Ω(z) = 0, then lim n→∞ √ a n e ϕn (sgn β ∞ ) n P n (z) = {P (z), g(z)} = 0.

Remark 4.14. Let assumption (2.24) be satisfied. Similarly to Remark 3.14, we can replace e -iφn and e -ϕn by expressions (3.28) in all assertions of this section. In particular, this is true for asymptotic formulas for orthogonal polynomials P n (z) and condition (4.21) reduces to holds. Then the resolvent (J -z) -1 of the Jacobi operator J satisfies the relations

∞ n=0 a -1 n (|β ∞ | + β 2 ∞ -1) 2n < ∞.
((J -z) -1 e n , e m ) = Ω(z) -1 P n (z)f m (z), Im z = 0, (4.24)
if n ≤ m and ((J -z) -1 e n , e m ) = ((J -z) -1 e m , e n ).

Since in view of Theorem 3.13, f n (z) and hence Ω(z) are entire functions of z ∈ C, we can state In view of formula (4.22) and equation (1.7) where n = 0 for the Jost solution f n (z), the equation for eigenvalues of J can be also written as

(b 0 -λ k )f 0 (λ k ) + a 0 f 1 (λ k ) = 0.
It follows from representation (4.24) where n = m = 0 and formula (4.22) that, up to terms regular in a neighborhood of the point λ k ,

((J -z) -1 e 0 , e 0 ) = 2 f 0 (λ k ) ḟ-1 (λ k ) 1 λ k -z , where ḟ-1 (λ) = d f -1 (λ) dλ . (4.25)
On the other hand, since

((J -z) -1 e 0 , e 0 ) = ∞ -∞
dρ(λ) λ -z by the definition of the spectral measure, we see that

((J -z) -1 e 0 , e 0 ) = ρ({λ k }) λ k -z ,
again up to terms regular in a neighborhood of the point λ k . Comparing this expression with (4.25), we arrive to a standard formula

ρ({λ k }) = 2 f 0 (λ k ) ḟ-1 (λ k ) .
Another well known formula

ρ({λ}) = ∞ n=0 P n (λ) 2 -1 (4.26)
for the spectral measure at an eigenvalue λ of a self-adjoint Jacobi operator J can be obtained in the following way. Let Jψ = λψ where the eigenvector ψ = (ψ 0 , . . . , ψ n , . . .) is normalized. Necessarily, ψ n = cP n (λ) and, by the normalization, c 2 P n (λ) 2 = 1. Therefore ρ({λ}) = (ψ, e 0 ) 2 = ψ 2 0 = c 2 is given by formula (4.26). Formula (4.26) is true without any assumptions on the coefficients a n , b n for an arbitrary self-adjoint extension J of the minimal Jacobi operator J 0 . Note however that in the essentially self-adjoint case the condition P n (λ) 2 < ∞ determines eigenvalues of the operator J = clos J 0 . This is not true if the deficiency indices are (1, 1) when P n (λ) 2 < ∞ for all λ.

Discussion

5.1.

Operators with discrete spectrum. We start with a very general result which, in particular, applies to Jacobi operators. An assertion below is essentially known, and we give it mainly for completeness of our presentation. The operator J will be defined via its quadratic form

(Ju, u) = ∞ n=0 b n |u n | 2 + 2 Re ∞ n=0 a n u n ūn+1 (5.1)
where a n are complex and b n are real numbers. for some < 1 and all sufficiently large n. Then the spectrum of the operator J is discrete and is semi-bounded from below (resp., from above).

Proof. Since finite-rank perturbations cannot change the discreteness of spectrum, we can suppose that estimate (5.2) is true for all n. Let B be the operator corresponding to the form

(Bu, u) = ∞ n=0 b n |u n | 2 . (5.3)
Evidently, its spectrum is discrete and is semi-bounded from below (resp., from above). For a proof of the same result for the operator J, it suffices to check that the forms (5.1) and (5.

∞ n=0 a n u n ūn+1 ≤ 2 ∞ n=0 |a n ||u n | 2 ∞ n=0 |a n ||u n+1 | 2 ≤ ∞ n=0 (|a n-1 | + |a n |)|u n | 2 .
Therefore estimate (5.4) is a direct consequence of the condition (5.2).

Proposition 5.1 holds, in particular, for semi-bounded Jacobi operators when a n > 0. It applies directly to the Friedrichs' extension of the operator J 0 , but its conclusion remains true for all extensions J of J 0 because the deficiency indices of J 0 are finite. In particular, condition (5.2) is satisfied with any ∈ (|β ∞ | -1 , 1) under the assumptions of Theorem 3.13. Thus, we recover the result of Corollary 4.16. Moreover, the corresponding operator J is semi-bounded from below (from above) if β ∞ < -1 (if β ∞ > 1, respectively). 5.2. The Carleman case. In this subsection we still consider the case a n → ∞ but assume that the Carleman condition (1.2) holds. Other assumptions on a n and b n are essentially the same as in the main part of this paper. In particular, we suppose that condition (1.5) is satisfied. Recall that under the Carleman condition the minimal Jacobi operator J 0 is essentially self-adjoint. The results stated here will be published elsewhere, but some of them are close to the papers [START_REF] Janas | Jacobi matrices with power-like weights -grouping in blocks approach[END_REF][START_REF] Aptekarev | Measures for orthogonal polynomials with unbounded recurrence coefficients[END_REF][START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF]. Now the term 2zα n in the right-hand side of (2.10) cannot be neglected and the numbers ζ n have to be defined as approximate solutions of the equation

ζ n + ζ -1 n = 2β n + 2zα n .
For simplicity of our discussion, we assume here that

∞ n=0 a -3 n (1 + |b n |) < ∞. (5.5)
Conditions (1.2) and (5.5) admit a growth of the off-diagonal coefficients a n as n p for p ∈ (1/2, 1] and even for p ∈

(1/3, 1] if b n = 0. Suppose first that |β ∞ | < 1.
As before, the Ansatz Q n (z) is defined by formula (2.9), but, instead of (2.13), we now set

ζ n = (β n -i 1 -β 2 n ) exp iz α n 1 -β 2 n . (5.6)
An easy calculation shows that the corresponding remainder (2.10) is in 1 (Z + ). Therefore repeating the arguments of Sect. 2 and 3, we find that equation (1.7) has a solution (the Jost solution) f n (z) distinguished by the asymptotics

f n (z) = a -1/2 n e -iφn+izψn (1 + o(1)), Im z ≥ 0, n → ∞, (5.7)
where the phase φ n is defined by formula (1.6) and

ψ n = n-1 m=0 α m |1 -β 2 m | . ( 5.8)
Observe that ψ n → ∞ as n → ∞ according to (1.2), (1.5), but ψ n = O(n 2/3 ) under assumption (5.5). It now follows from (2.23) that the phase ψ n is negligible compared to φ n . It is easy to see that f n (z) ∈ 2 (Z + ) for Im z > 0. The Jost functions f n (z) are now analytic in the half-plane Im z > 0 and are continuous up to the real line. We set f n (z) = f n (z) for Im z ≤ 0. Then the functions f n (z) are analytic in the complex plane with the cut along R and are continuous up the cut.

If z = λ ∈ R, we have two Jost solutions f n (λ + i0) and f n (λ -i0) = f n (λ + i0) of equation (1.7). Their Wronskian is the same as (3.25) (where κ ∞ = 1):

{f (λ + i0), f (λ -i0)} = 2i 1 -β 2 ∞ = 0.
Therefore the polynomials P n (λ) are linear combinations of f (λ + i0) and f (λ -i0):

P n (λ) = Ω(λ -i0)f n (λ + i0) -Ω(λ + i0)f n (λ -i0) 2i 1 -β 2 ∞ , n = 0, 1, 2, . . . , λ ∈ R, (5.9)
where Ω(z) is defined by formula (4.22). Note that Ω(λ ± i0) = 0 for all λ ∈ R according to (5.9). It now follows from (5.7) for z = λ ∈ R that the polynomials P n (λ) have asymptotics

P n (λ) = -a -1/2 n |Ω(λ + i0)|(1 -β 2 ∞ ) -1/2 sin(φ n -λψ n + arg Ω(λ + i0)) + o(1) , n → ∞. (5.10)
If Im z = 0, then by virtue of Theorem 4.6 a solution g n (z) of the Jacobi equation (1.7) linear independent with f n (z) can be constructed by explicit formula (1.17). Similarly to Theorem 4.11, it can be checked that

g n (z) = 1 2i 1 -β 2 ∞ 1 √ a n
e iφn-izψn (1 + o(1)), Im z > 0, n → ∞, so that g n (z) grows faster than any power of n as n → ∞. Since the Wronskian {f (z), g(z)} = 1, the asymptotics of the orthonormal polynomials can be easily derived from this result:

P n (z) = iΩ(z) 2 1 -β 2 ∞ 1 √ a n
e iφn-izψn (1 + o(1)), Im z > 0, n → ∞.

(5.11)

Now the Jacobi operator J = clos J 0 is self-adjoint, and its resolvent (J -z) -1 is given by the general formula (4.24). Since the Jost solutions f n (z) are continuous functions of z up to the real axis and Ω(λ ± i0) = 0 for λ ∈ R, the spectrum of the Jacobi operator J is absolutely continuous. Moreover, using relation (5.9), it is easy to deduce from (4.24) a representation for the spectral family dE(λ) of the operator J: d(E(λ)e n , e m ) dλ = π -1 1 -β 2 ∞ |Ω(λ + i0)| -2 P n (λ)P m (λ).

In particular, for the spectral measure, we have the expression

dρ(λ) = π -1 1 -β 2 ∞ |Ω(λ + i0)| -2
dλ. It follows that the spectrum of J coincides with the whole real axis.

Formulas (5.10) and (5.11) are consistent with the classical asymptotic expressions for the Hermite polynomials when a n = (n + 1)/2 and b n = 0 (see, e.g., Theorems 8.22.6 and 8.22.7 in the G. Szegő's book [START_REF] Szegő | Orthogonal polynomials[END_REF]). Asymptotics (5.10) was obtained earlier in [START_REF] Janas | Jacobi matrices with power-like weights -grouping in blocks approach[END_REF][START_REF] Aptekarev | Measures for orthogonal polynomials with unbounded recurrence coefficients[END_REF][START_REF] Świderski | Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients[END_REF], but as far as (5.11) is concerned, we are aware only of the paper [START_REF] Rakhmanov | On asymptotic properties of polynomials orthogonal on the real axis[END_REF] where an asymptotics of |P n (z)| as n → ∞ was found in terms of a behavior of the corresponding absolutely continuous spectral measure for |λ| → ∞.

The where ϕ n and ψ n are defined by formulas (1.9) and (5.8), respectively. Now the functions f n (z) are analytic in the whole complex plane. The second solution g n (z) of the Jacobi equation (1.7) is again given by equality (1.17) which leads to the asymptotic formula

P n (z) = -Ω(z) (sgn β ∞ ) n+1 2 β 2 ∞ -1
e ϕn+zψn √ a n (1 + o(1)), n → ∞.

(5.12)

As in Sect. 4.4, the resolvent of J is determined by formula (4.24), but, in contrast to the case |β ∞ | < 1, its singularities are due to zeros of the denominator Ω(z) only. Therefore the spectrum of J is discrete which is consistent with Proposition 5.1.

5.3.

The non-Carleman case versus Carleman one. Let us now compare the results of this paper for the non-Carleman case (1.4) with those described in the previous subsection. Suppose first that |β ∞ | < 1. According to Theorem 3.9 all solutions of equation (1.7) for every z ∈ C belong to 2 (Z + ) so that the Jacobi operator J 0 has a one parameter family of self-adjoint extensions J. In this case formula (1.7) for the resolvents of the operators J makes no sense. Nevertheless the scalar products ((J -z) -1 e 0 , e 0 ), that is, the Cauchy-Stieltjes transforms of the corresponding spectral measures dρ J (λ), can be expressed via the orthogonal polynomials of the first P n (z) and of the second P n (z) kinds by the Nevanlinna formula obtained by him in [START_REF] Nevanlinna | Asymptotische Entwickelungen beschränkter Funktionen und das Stieltjessche Momentenproblem[END_REF] (see, e.g., formulas (2.31), (2.32) in the book [START_REF] Akhiezer | The classical moment problem and some related questions in analysis[END_REF]). This remarkable formula implies, in particular, that the spectra of all self-adjoint extensions J of J 0 are discrete. The spectral measure at eigenvalues of J is given by formula (4.26). Our construction is quite independent of the Nevanlinna theory, but if some link with this theory exists, it would be desirable to find it.

Asymptotic formulas (4.7) and (5.10) look rather similar although the phase in (5.10) contains an additional term λψ n . The amplitude factor a -1/2 n in (4.7) and (5.10) is the same. However, under assumption (1.2) the sequence (5.10) never belongs to 2 (Z + ) while under assumption (1.4) the sequence (4.7) belongs to this space for all λ ∈ R.

The difference between the Carleman and non-Carleman cases is even more obvious for Im z = 0. According to Theorem 4.1 in the non-Carleman case, all solutions of the Jacobi equation oscillate as n → ∞, but they are in 2 (Z + ) due to the amplitude factor a -1/2 n . In the Carleman case, the solution f n (z) exponentially decays while the solution g n (z) exponentially grows as n → ∞.

In the case |β ∞ | > 1, the asymptotic formulas (4.23) and (5.12) for orthogonal polynomials are close to each other, but the first of them contains an additional factor κ ∞ while (5.12) contains a factor e zψn . We emphasize that in the Carleman case (1.2) the Jacobi operators J 0 are essentially self-adjoint for all sequences b n while in the non-Carleman case (1.4) the Jacobi operators J 0 are essentially self-adjoint if and only if condition (1.19) is satisfied (see Corollary 4.12).

Lemma 3 . 1 .

 31 Let the assumptions (1.4), (1.5) where |β ∞ | > 1 and (2.5) be satisfied. Then kernel (3.2) is bounded uniformly in m > n ≥ 0:

Lemma 3 . 2 .

 32 Let the assumptions (1.4),(1.5) where |β ∞ | < 1 and (2.5), (2.8) be satisfied. Then kernel (3.2) is bounded uniformly in m > n ≥ 0, that is, estimate (3.4) holds.

Let us multiply identity ( 3 . 8 ) 3 . 2 .Lemma 3 . 3 .

 383233 by S m+1 . According to(3.6) and (3.7) all three terms in the righthand side of the equality obtained are bounded for n ≥ n 0 . Successive approximations. Let us come back to the Volterra equation (3.3). Lemma 2.5 ensures inclusion (2.29), and Lemmas 3.1 or 3.2 show that the kernels G n,m are bounded. This allows us to estimate iterations of equation(3.3) and then solve it. Let the assumptions (1.4), (1.5) as well as (2.5), (2.8) be satisfied. Set u

.Theorem 3 . 4 .

 34 Now we are in a position to solve equation (3.3) by iterations. Let the assumptions (1.4), (1.5) as well as (2.5), (2.8) be satisfied. Then equation (3.3) has a unique bounded solution u n (z). Moreover,

n

  (z) are analytic functions of z ∈ C. The same assertion is of course true for the series (3.14).

Lemma 3 . 7 .

 37 Under the assumptions of Theorem 3.4, for an arbitrary > 0 and some constants C n (ε) (that do not depend on z ∈ C), every function u n (z) satisfies an estimate|u n (z)| ≤ C n ( )e |z| . (3.18) Proof. It suffices to check (3.18) for |z| ≥ 1. According to (3.12) and (3.13) we have an estimate |u m (z)| ≤ e m|z| (3.19)

  equation (1.7) because the coefficients a n and b n are real, and it has the asymptotics fn

-1/ 2 n

 2 in (3.22) and (3.24) are quite natural (cf. Remark 2.7) because their product should cancel a n in the Wronskian (3.25).

  .28) respectively. In this case the remainders O(ε n ) should be replaced by o(1). D. R. YAFAEV 3.4. Example. Suppose that (for sufficiently large n) a n = γn p and b n = -δn p (1 + σn -1 ), p > 1,

4 )

 4 satisfy the relations k -(z) = k + (z) because P n (z) = P n (z). Of course, k + (z) and k -(z) are analytic functions of z ∈ C. Set κ(z) = |k + (z)|. Then

4. 3 .

 3 Large diagonal elements. Here we consider the case |β ∞ | > 1. Choose an arbitrary z ∈ C. By Theorem 3.13, the sequence f n (z) defined by equality (3.21) satisfies equation (1.7), and it has exponentially decaying asymptotics (3.27

Theorem 4 . 11 .

 411 Under the assumptions of Theorem 3.13 the solution (1.17) of the Jacobi equation (1.7) satisfies for all z ∈ C the relation lim n→∞ √

Corollary 4 . 16 .

 416 The spectrum of the operator J is discrete, and its eigenvalues λ 1 , • • • , λ k , . . . are given by the equation Ω(λ k ) = 0. The resolvent (J -z) -1 is an analytic function of z ∈ C with poles in the points λ 1 , • • • , λ k , . . .. A much more general result of this type is stated below as Proposition 5.1.

Proposition 5 . 1 .

 51 Suppose that b n → ∞ (or b n → -∞) and that |a n-1 | + |a n | ≤ |b n | (5.2)

a n u n ūn+1 ≤ ∞ n=0 |b n ||u n | 2 ( 5 . 4 )

 254 for some < 1. Let us use the following obvious inequality 2

n β 2 n - 1 . 2 n(

 12 cases |β ∞ | > 1 and |β ∞ | < 1 are technically rather similar, but the asymptotic behavior of orthogonal polynomials and spectral properties of the Jacobi operators are quite different in these cases. If |β ∞ | > 1, we set (cf. (5.6)) ζ n = sgn β n (|β n | -β 2 n -1) exp -z α Then again the corresponding remainder (2.10) is in 1 (Z + ). Therefore repeating the arguments of Sect. 2 and 3, we find that equation (1.7) has a solution (the Jost solution) f n (z) distinguished by the asymptotics f n (z) = a -1/sgn β ∞ ) n e -ϕn-zψn (1 + o(1)), n → ∞,

  [START_REF] Yafaev | Mathematical scattering theory: Analytic theory[END_REF] which permits us to reduce the difference equation (1.7) for f n (z) to a Volterra "integral" equation for the sequence u n (z). This equation depends of course on the parameters a n , b n . In particular, it is somewhat different in the cases |β ∞ | < 1 and |β ∞ | > 1. However in both cases this Volterra equation is standardly solved by iterations in Sect. 3.1 and 3.2. This allows us to prove the existence of solutions u n (z) of this equation such that u n (z) → 1 as n → ∞. According to(1.13) or(1.14) and(1.16) this yields (see Sect. 3.3) asymptotics of the Jost solutions f n (z). The functions f n (z) turn out to be analytic in z ∈ C D. We come back to the orthogonal polynomials P n (z) in Sect. 4. First, we observe that the sequence fn (z) = f

n (z) also satisfies the equation (1.7). In the case |β ∞ | < 1, the solutions f n (z) and fn (z) are linearly independent. Therefore it follows from (1.13) that all solutions of the Jacobi equation (1.7) have asymptotic behavior

(1.8)

. In particular, formula (1.8) is true for the orthogonal polynomials P n (z).

  Lemma 2.8. Let z ∈ C, let ζ n be defined by formulas (2.13) or (2.14), and let the remainder r n (z) be given by formula (2.12). Then equation (1.7) for a sequence f n (z) is equivalent to the equation

			.31)
	Then (2.30) is equivalent to the condition
	lim n→∞	u n (z) = 1.	(2.32)
	Let us derive a difference equation for u n (z).

  20)for every k = 1, 2, . . .. For a given > 0, choose k such that 2 n+k ≤ , 2 n+k+1 ≤ . Then putting estimates(3.19) and (3.20) together, we see that|u n (z)| ≤ 4C n,k |z| k e |z|/2 . Since |z| k ≤ c k ( )e |z|/2, this proves (3.18).In the case |β ∞ | > 1, we also need estimates on u n .

Lemma 3.8. Let |β ∞ | > 1.

Under the assumptions of Theorem 3.4, we have {u n } ∈ 1 (Z + ).

  construction of Theorem 3.13 yields only one solution f n (z) of the Jacobi equation (1.7).

	Remark 3.14. Suppose that instead of (2.8), a stronger assumption (2.24) holds and use
	Lemma 2.4. Then taking into account relations (2.15) and (2.16), we can replace e -iφn in (3.22)
	and e -ϕn in (3.27) by
	e

  it follows from definitions (2.16), (2.22) and formula (3.29) that the sequence ϕ n -n arccosh |β ∞ | -ν sgn β ∞ ln n has a finite limit as n → ∞. Therefore formula (3.27) in Theorem 3.13 can be replaced by a simpler one f n

  In this case all solutions of equation (1.7) are in 2 (Z + ). Therefore we can state Corollary 4.12. Under the assumptions of Theorem 3.13 the minimal Jacobi operator J 0 is essentially self-adjoint if and only if condition (1.19) is satisfied. Otherwise, that is under assumption (4.21), the operator J 0 has deficiency indices (1, 1).

		.20)
	Convergence here is uniform on compact subsets of C.
	It follows from (4.20) that {g n (z)} ∈ 2 (Z + ) (for all z ∈ C) if and only if
	∞
	a -1 n e 2ϕn < ∞.	(4.21)
	n=0

  4.4.Resolvent. If condition(1.19) is satisfied, then the closure J of the minimal Jacobi operator J 0 is self-adjoint. In this case Ω(z) = 0 if and only if z is an eigenvalue of J. The resolvent of the operator J can be constructed by the standard (cf. Lemma 2.6 in[START_REF] Yafaev | Analytic scattering theory for Jacobi operators and Bernstein-Szegő asymptotics of orthogonal polynomials[END_REF] or Lemma 5.1 in[START_REF] Yafaev | Semiclassical asymptotic behavior of orthogonal polynomials[END_REF]) formulas. Recall that e n , n ∈ Z + , is the canonical basis in 2 (Z + ), f n (z) is the Jost solution of the equation (1.7) constructed in Theorem 3.[START_REF] Simon | The classical moment problem as a self-adjoint finite difference operator[END_REF] and Ω(z) is the Wronskian (4.22). Proposition 4.15. Under the assumptions of Theorem 3.13 suppose also that condition(1.19)

Supported by project Russian Science Foundation 17-11-01126.


Xusheng Hu

Feng Zhu

Xiaolu Gong
email: gong@utt.fr

Numerical investigation of the effects of heating and contact conditions on the thermal charging performance of composite phase change material

Keywords: Metal foam, Phase change material, Heating and contact conditions, Numerical simulation, Volume-average method

Phase change material (PCM) saturated in metal foam is a promising candidate for thermal energy storage (TES). However, there are some potential factors affecting the thermal charging performance of composite PCM, such as heating and contact conditions. In this paper, paraffin and copper foam are selected as PCM and metal matrix, respectively. Heating conditions are top, left and bottom heating. The contact gaps are set to 0, 0.4 and 0.8 mm to represent different contact conditions. The numerical investigation on thermal charging performance of composite PCM under different heating and contact conditions is performed. Based on volume-averaged method, the numerical model is developed to study the thermal characteristics of composite PCM including solid-liquid interface, liquid fraction, velocity, total melting time and average heat storage rate. Results show that there are the comprehensive effects of heating and contact conditions on thermal performance. The heating condition can affect phase change heat transfer mode, and thus has a significant influence on thermal charging performance. The contact conditions have the different effects on thermal behavior of composite under different heating conditions, e.g., under top heating condition, the total melting time of composite with 0 mm and 0.8 mm contact gap is almost the same. Whereas, under left heating condition, the total melting time of composite with 0.8 mm contact gap is 2.6 times that of composite with 0 mm contact gap. The study of the effect mechanism of heating and contact conditions is of great significance for practical application of composite PCM in TES system.

Nomenclature

Introduction

With the rapid development of human society and economy, environment pollution and energy issue are becoming more and more serious in recent years. Researchers pay extensive attention to the development of renewable energy resources [START_REF] Sharma | Review on thermal energy storage with phase change materials and applications[END_REF], in which thermal energy storage (TES) plays an important role in energy storage and release [START_REF] Oro | Review on phase change materials (PCMs) for cold thermal energy storage applications[END_REF][START_REF] Reddy | Review of latent heat thermal energy storage for improved material stability and effective load management[END_REF]. Due to the high energy storage density and good thermal stability, phase change materials (PCMs) are widely used in TES system [START_REF] Yang | Thermal response of annuli filled with metal foam for thermal energy storage: An experimental study[END_REF], such as solar thermal power plant [START_REF] Khan | Evaluation of solar collector designs with integrated latent heat thermal energy storage: A review[END_REF], energy efficient building [START_REF] De Gracia | Phase change materials and thermal energy storage for buildings[END_REF] and waste heat recovery system [START_REF] Miro | Thermal energy storage (TES) for industrial waste heat (IWH) recovery: A review[END_REF]. However, the relatively lower thermal conductivity hinders widespread utilization of PCMs in TES system. To enhance the thermal behavior of PCM, some methods have been studied, which include encapsulated PCMs [START_REF] Zhao | Review on microencapsulated phase change materials (MEPCMs): Fabrication, characterization and applications[END_REF][START_REF] Marchi | Characterization of panels containing micro-encapsulated Phase Change Materials[END_REF], adding high thermal conductivity additive [START_REF] Zou | Thermal performance enhancement of composite phase change materials (PCM) using graphene and carbon nanotubes as additives for the potential application in lithium-ion power battery[END_REF][START_REF] Zhang | Preparation and thermal properties of short carbon fibers/erythritol phase change materials[END_REF], inserting metal matrix and pin-fin [START_REF] Hu | Pore-scale numerical simulation of the thermal performance for phase change material embedded in metal foam with cubic periodic cell structure[END_REF][START_REF] Arshad | Thermal performance of phase change material (PCM) based pin-finned heat sinks for electronics devices: Effect of pin thickness and PCM volume fraction[END_REF], and embedding metal foam [START_REF] Zhao | Heat transfer enhancement for thermal energy storage using metal foams embedded within phase change materials (PCMs)[END_REF][START_REF] Li | Experimental and numerical studies on melting phase change heat transfer in open-cell metallic foams filled with paraffin[END_REF][START_REF] Hu | Experimental and numerical study on the thermal behavior of phase change material infiltrated in low porosity metal foam[END_REF]. A mass of investigations demonstrated that embedding metal foam can effectively heighten the thermal behavior of PCM.

Yang et al. [START_REF] Yang | Role of porous metal foam on the heat transfer enhancement for a thermal energy storage tube[END_REF][START_REF] Yang | Design and operating evaluation of a finned shell-and-tube thermal energy storage unit filled with metal foam[END_REF] carried out the numerical studies of the role of metal foam on the enhancement in thermal behavior of TES system using PCM. Results proved that embedding metal foam can significantly enhance the thermal behavior of TES unit and the uniformity of temperature was also improved. Xiao et al. [START_REF] Xiao | Effective thermal conductivity of open-cell metal foams' impregnated with pure paraffin for latent heat storage[END_REF][START_REF] Xiao | Preparation and thermal characterization of paraffin/metal foam composite phase change material[END_REF] prepared the paraffin/metal foam composite PCMs and experimentally investigated the thermal characteristics of composite PCMs, especially effective thermal conductivity. Lafdi et al. [START_REF] Lafdi | Experimental study on the influence of foam porosity and pore size on the melting of phase change materials[END_REF] investigated the effects of porosity and pore density on the thermal behavior of PCM saturated in aluminum foam using the experimental approach. The numerical and experimental investigations on phase change process of PCMs impregnated in graded porosity metal foam were performed in [START_REF] Yang | Solidification of fluid saturated in open-cell metallic foams with graded morphologies[END_REF][START_REF] Yang | Numerical analysis on thermal behavior of solid-liquid phase change within copper foam with varying porosity[END_REF]. Effect of orientation of copper foam embedded in PCM on thermal performance was experimentally examined by Baby and Balaji [START_REF] Baby | Experimental investigations on thermal performance enhancement and effect of orientation on porous matrix filled PCM based heat sink[END_REF]. The visual experiments of phase transition process were designed to analyze the thermal characteristic of paraffin/copper foam composites [START_REF] Zhang | Melting heat transfer characteristics of a composite phase change material fabricated by paraffin and metal foam[END_REF][START_REF] Jin | A pore-scale visualized study of melting heat transfer of a paraffin wax saturated in a copper foam: Effects of the pore size[END_REF], where the local thermal non-equilibrium problem between the metal ligament and PCM was emphatically studied. Boomsma and Poulikakos [START_REF] Boomsma | On the effective thermal conductivity of a three-dimensionally structured fluid-saturated metal foam[END_REF] proposed the tetrakaidecahedron model taking into account the intricate configuration of metal foam. Feng et al. [START_REF] Feng | Pore-scale and volume-averaged numerical simulations of melting phase change heat transfer in finned metal foam[END_REF] performed the pore-scale and volume-average numerical simulation. They found that two numerical models have similar predictions and one-temperature volume-average model can be employed to imitate composite PCMs. It can be concluded from the above literature that most studies of metal foam/paraffin composite PCM mainly focus on the following aspects: the preparation of composite PCM, heat transfer mechanism, development of theoretical model and effects of configuration parameters (e.g. porosity, pore size and porosity gradient) on thermal behavior of composite PCMs.

Yataganbaba and Kurtbas [START_REF] Yataganbaba | Effect of Heating Position on Thermal Energy Storage in Cavity With/Without Open-cell Metallic Foams[END_REF] conducted an experimental investigation of the effects of heating positions and embedding metal foam on the thermal performance of pure PCM. Results showed that heating positions significantly affected the melting rate and temperature response of pure paraffin and melting of PCM was heightened by embedding metal foam. For real applications of composite PCM, such as solar thermal storage and thermal management of electronic devices, heating positions are different, which may affect its thermal performance. Moreover, for the same TES device, the studies of heating condition are also very necessary and significant. For instance, the heating position may have an important influence on the energy storage rate of composite PCM used in TES. However, the study on the effect of heating condition on the thermal performance of composite PCM is not enough.

In practical utilization, PCM needs to be encapsulated in containers [START_REF] Ferrer | Corrosion of metal containers for use in PCM energy storage[END_REF]. The junction types between metal foam and container wall can result in the difference of contact condition. We should select the natural contact or bonding metal foam to the wall via sintering or brazing, which is still an open question. Hence, the effect of contact condition on thermal behavior of composite PCM employed in TES should be particularly investigated, which is conducive to determining which kind of the junction type is chosen to achieve high heat storage rate in a real application. To this end, the aim of this paper is to investigate the effects of heating and contact conditions on thermal charging performance for composite PCM. Based on the volume-average method, numerical simulation is implemented to investigate thermal characteristics of composite PCMs during the phase transition process, including solid-liquid interface, velocity, liquid fraction, melting time and heat storage rate. Nine different conditions are established to investigate the melting evolvement and heat transfer mechanism of composite PCMs subjected to different heating and contact conditions.

Problem description

Fig. 1 presents the common contact condition of composite PCM used in TES. To easily assemble, there may be a small clearance fit between metal foam and wall. A contact gap of the width of δ yields between metal skeleton and container wall. The gap can be filled with PCM due to the flow of liquid PCM. The additional thermal resistance caused by the gap may have an effect on thermal behavior of composite PCM. Actually, due to different processing and assembly processes, the gap may occur in each pair of contact surfaces (i.e. top, bottom and sides). Besides, the heating condition also plays a vital role in the energy storage of composite PCM. To investigate the comprehensive effects of heating and contact conditions on the thermal performance of composite PCM, nine different heating and contact condition models are developed, as displayed in Fig. 2. For instance, Conds. 1, 4 and 7 are used for investigating the effect of heating position (blue wall is heated at a constant temperature, and grey represents PCM), and Conds. 4-6 are employed to study the influence of contact condition on thermal performance when heating condition is the same. The dimension of metal foam is the same (i.e. ) for the nine different cases, in which foam is filled with PCM. The contact gaps with size of 0, 0.4 and 0.8 mm are set to represent different contact conditions. Since copper foam possesses not only high thermal conductivity but also large specific area, copper foam with 95% porosity and 5 PPI pore density is selected as metal matrix. Paraffin has the ability to store a large amount of heat by latent heat and is widely applied in TES, so it is used as PCM in this paper. The thermophysical properties of paraffin and copper foam are displayed in Table 1.

Numerical simulation

Physical model

The computational domain of composite PCM is simplified to a two-dimensional model employed for the volume-averaged simulation. Fig. 3 presents the schematic diagram of physical model with the left heating condition. A square computational domain with size of 50 50

× mm 2 is used to represent the composite PCM, in which PCM is saturated in metal foam. The bad contact condition is considered (i.e. there is a gap with the width of δ between hot wall and composite PCM) to investigate the effect of contact condition, and we assume that the gap is completely filled with paraffin wax. The hot wall is heated with the fixed temperature Tw = 350 K. The other walls are set to be adiabatic. The initial temperature of model is Ti = 298 K.

Mathematical method

The necessary assumptions should be made in mathematical method for the present physical model, as follows: (1) Metal foam is assumed to be homogenous and isotropic; (2) the flow of fusion paraffin within metal foam is treated as impressible and Newtonian; (3) The thermal property of paraffin is considered as constant and same at solid and liquid state, except density which is subject to the Boussinesq approximation.

The volume-averaged method is employed to simulate phase change process within composite PCM. Based on the assumption of local thermal equilibrium, one-temperature energy model is used to solve heat transfer between PCM and metal skeleton. Darcy-Brinkman-Forchheimer model is employed for considering influences of metal foam on the flow of fusion PCM. According to the above assumptions and method, the volume-averaged governing equations can be given by as follows:

Continuity equation:

( ) 0 f f U t ρ ρ ∂ + ∇⋅ = ∂ r (1) Momentum equations 2 2 (
)

f f f f C U U u u u K K p u Su t x ρ µ ρ ρ µ ϕ ϕ ϕ     ⋅∇ + ∇ +     ∂ ∂ + = - - + ∂ ∂ r r (2) 2 2 ( ) ( ) f f f f f m C U U v v v g T T K K p v Sv t y ρ µ ρ γ ρ ρ µ ϕ ϕ ϕ     ⋅∇ + ∇ + + -     ∂ ∂ + = - - + ∂ ∂ r r (3)
The term S in Eqs. ( 2) and ( 3) is the source term of damping force, and defined as the following equations [31]:

2 3 (1 ) m S A β β λ - = - + (4)
Where Am is a mushy zone constant and set as 10 5 . λ is a small constant and set at 10 -3 . β is the liquid fraction within the pore and defined by: ( ) ( )

1 2 1 0 / 1 m m m T T T T β   = - -    1 1 2 2 < < < > m m m m T T T T T T T (5)
The relationship between β and ϕ can be written as:

= ϕ ε β ⋅ (6)
Energy equation for PCM and metal foam:

( ) 2 (1 ) f f s s f f eff f T c c c U T k T L t t β ερ ε ρ ερ ερ ∂ ∂   + - + ⋅ ∇ = ∇ -   ∂ ∂ r (7)
The pore size dp can be determined by: The permeability K and inertial coefficient C are required for the calculation of liquid flow within porous medium, which are complex parameters and difficult to be obtained by the experimental test. K and C are finally determined by using the model proposed in Ref. [START_REF] Calmidi | Transport Phenomena in High Porosity Fibrous Metal Foams[END_REF] and written by the following equations:

K d e ε ε ε π - - --   -  = -     -     (9)
1.63 0.132 ((1 )/0.04)

1 1 0.00212(1 ) 1.18 3 1 C e ε ε ε π - - --   -  = -     -     (10)
For the one-temperature model, the effective thermal conductivities of composite PCMs are an indispensable parameter to numerically simulate phase change heat transfer. Many models are employed to determine the effective thermal conductivities, which are proposed in previous literature. The theoretical model developed by Bhattacharya et al. [START_REF] Bhattacharya | Thermophysical properties of high porosity metal foams[END_REF] is very close to actual configuration of metal foam. Thus, Bhattacharya's model is employed in the present study. The effective thermal conductivity can be defined as: 0.65 0.35(

)

eff f s f s k k k k k ε ε ε ε = + - + - + (11)

Initial and boundary conditions

The initial velocity for solid PCM is zero, and the initial temperature is at a constant temperature of 298K, the initial condition can be given by:

0 ≤ x ≤ 50, 0 ≤ y ≤ 50, 0, 298K i u v T = = = (12)
The boundary conditions required for governing equations can be also found in Fig. 3 and defined by the following equations:

x = 0, 0 ≤ y ≤ 50, 0, 350K w u v T = = = (13) x = 50, 0 ≤ y ≤ 50, 0, 0 T u v x ∂ = = = ∂ (14) y = 0, 0 ≤ x ≤ 50, 0, 0 T u v y ∂ = = = ∂ (15) y = 50, 0 ≤ x ≤ 50, 0, 0 T u v y ∂ = = = ∂ (16)

Numerical procedure and validation

For the present numerical simulation, commercial CFD software FLUENT 18.0 is adopted to solve the governing equations along with the initial and boundary conditions based on finite volume method. The second order upwind differencing scheme is recommended to discretize the governing equation for momentum and energy. PRESTO is applied to compute the pressure equation, which is recommended to calculate the liquid flow within the porous medium. PISO algorithm is used for coupling velocity-pressure field. The convergence of solution is checked at every time step for numerical simulation at unsteady state, and the convergence criteria of scaled residual are set to 10 -6 to monitor calculation stability of energy and momentum equations. Structure grids are generated in the computational domain of the two-dimension model. The grid and time independence tests are performed for guaranteeing the validity of numerical simulation. Three different numbers of grids (43k, 63k and 77k cells) and three different time steps (0.5 s, 1 s and 2 s) are examined. The liquid fraction of composite PCM with Cond.1 is tested using the above three grids and time steps. The comparison results are presented in Fig. 4(a) and (b). It is observed that results exhibit good agreement and the relative error is within 2%. To ensure the calculation accuracy and reduce computation time, the medial grid 63k cells and time step 1 s are enough for present numerical simulation.

The model in the present study is validated with the experimental results extracted in the literature. The same copper foam and paraffin is used in numerical simulation for comparison with Zheng et al. [START_REF] Zheng | Thermal performance of copper foam/paraffin composite phase change material[END_REF], and the same initial and boundary condition is adopted. The evolvement comparison of solid-liquid interface is exhibited in Fig. 5. Blue represents solid phase for composite PCM. It can be found that the numerical results are good in agreement with experimental photos. Hence, the present model can be applied to numerically simulate phase change heat transfer of copper foam saturated with paraffin.

Results and discussion

Evolvement of phase change charging

Fig. 6 shows the evolvement of phase change of composite PCM under Conds. 1-3. The solid-liquid interface contours for every condition are extracted at 20%, 40% and 80% liquid fraction, respectively. The solid-liquid interfaces are flat and parallel with the top face, which indicates that natural convection within composite PCM is distinctly suppressed under top heating condition. The numerical results of Hu et al. [START_REF] Hu | Modeling phase change material in micro-foam under constant temperature condition[END_REF] also showed the flat melting interface, which indicated that heat transfer of composite PCM with top heating was dominated by heat conduction. It can be found that the velocity value is very small (order of magnitude 10 -6 ) and can be almost ignored during the melting process, which also demonstrates that natural convection of liquid PCM is restrained and heat conduction dominants heat transfer during the whole phase change process. The melting time of composite PCMs is almost the same at the same liquid fraction. Fig. 6. Evolvement of phase change for composites with top heating at 20%, 40% and 80% liquid fraction Fig. 7 presents the evolution of phase change of composite PCM under Conds. 4-6. The solid-liquid interface is mildly inclined at the initial stage of melting, which owes to the effect of natural convection. As melting progresses, liquid paraffin flows upward and collides with solid paraffin at the interface location. It is noted from the vector diagrams that the velocity value is the greater close to hot face and interface during phase change process. The melting for PCM within top region is quicker than bottom region and melting interface is inclined, which is ascribed to role of natural convection. This indicates that natural convection plays a dominant role in phase change heat transfer. It is noted that the velocity decreases with the increase of contact gap at the same liquid fraction. This phenomenon demonstrates that natural convection can be restrained as contact conditions grow worse (i.e. the gap size increases). As a result, the melting time is longer for the larger contact gap when the liquid fraction is the same, e.g., the melting time of Cond. 6 is 1387s and almost prolonged about 2.8 times as much as that of Cond. 4 at the 80% liquid fraction. Fig. 7. Evolvement of phase change for composites with left heating at 20%, 40% and 80% liquid fraction Fig. 8 shows the evolvement of phase change for composites under Conds. 7-9. At the beginning of melting process, the solid-liquid interface is flat and parallel to bottom face. Moreover, velocity is extremely weak in the liquid phase domain. It reveals that heat conduction is in a dominant position at the early period of phase transition. As melting continues, solid-liquid interface is wavy, which is attributed to the influence of natural convection. It can be found from experimental results of Yang et al. [START_REF] Yang | Effect of inclination on the thermal response of composite phase change materials for thermal energy storage[END_REF] that a similar phenomenon was showed in melting evolution of composite PCM with bottom heating (i.e. there is a wavy melting interface). It is demonstrated that the heat conduction and natural convection have comprehensive influences on thermal charging process. It can be seen that the melting of composites in the middle region is quicker than other regions and the velocity gradually increases, which confirms that natural convection becomes stronger and gradually dominates the melting process of composites in the late stage of melting. We can find from the velocity vector diagrams that velocity declines with the increase of contact gap. As a consequence, melting time of composites with the larger contact gap is lengthened at the same liquid fraction, which is owing to the influence of contact condition on natural convection. To study the influence of heating condition, numerical results of Conds. 1, 4 and 7 are compared and analyzed. It can be found that heat transfer mechanism is significantly different under different heating conditions, e.g., heat conduction dominates the thermal charging process under top heating condition, whereas natural convection plays a dominant role in phase transition heat transfer under left heating condition. The melting time of composites under three heating conditions is different when liquid fraction is the same. For instance, the melting time of composite with left heating is the shortest and reduced by about 43% compared to composite with top heating, which indicates that the natural convection conduces to the enhancement of thermal charging performance of composite PCM. It is concluded that heating condition has an obvious influence on thermal charging performance by affecting phase change heat transfer mode.

Liquid fraction and melting time

The variation of liquid fraction of composites with 0 mm gap under three heating conditions is analyzed as shown in Fig. 9. It is observed from liquid fraction curves that the melting rate of composite with left heating is the fastest and almost remains the value until the end of phase transition. In addition, it can be found that the melting rate of composite under the top and bottom heating conditions is almost the same in the initial period of phase change. The melting rate for top heating is nearly invariable in the whole melting process, since natural convection is suppressed and heat conduction dominates heat transfer as mentioned before. Whereas, the melting rate for bottom heating is visibly heightened, which results from the occurrence of natural convection in the mid stage of melting process. Fig. 9. Variation of liquid fraction under three heating conditions Fig. 10 presents the variation of liquid fraction of composites with left heating under three contact conditions. It is noted that the liquid fraction curves for different contact conditions show a similar variation tendency but different slope (i.e. melting rate). For example, the melting rate of composite PCM with ideal contact condition (i.e. gap is 0 mm) is the fastest and 2.6 times of that with contact gap of 0.8 mm. Furthermore, results show that the melting rate of composites decreases as contact condition worsens, which owes to additional thermal resistance caused by the contact gap between the composite PCM and hot wall. The total melting time of composite PCMs with nine different conditions is presented in Fig. 11. The total melting time for left and bottom heating presents a linear increase with the contact gap. Whereas, the total melting time of composite for top heating hasn't a visible change with the contact gap. This phenomenon is because contact thermal resistance increases with contact gap between composite and wall but the contact resistance can be ignored compared to thermal material resistance of composite PCM. Hence, the poor contact condition has a very small effect on thermal behavior of composite in which heat transfer is dominated by heat conduction. For the ideal contact condition (i.e. δ = 0 mm), the total time is almost the same for left and bottom heating conditions, which is almost cut in half compared with top heating condition. This phenomenon is attributed to different heat transfer mode induced by the heating condition. For instance, natural convection dominates heat transfer of composite under left heating condition, which conducive to the improvement of thermal behavior, whereas, natural convection is restrained for top heating in the whole melting process, as analyzed in Section 4.1.

Heat storage rate

The heat storage rate is a vital criterion for estimating thermal charging performance of composite PCMs. Hence, we perform the study of average heat storage rate (i.e. per unit time per unit mass of composite PCM stores energy during the whole melting process). Fig. 12 displays the average heat storage rate of composite PCM under nine different conditions. Results show that the heat storage rate of composite with left heating is the largest under the ideal contact condition (i.e. δ = 0 mm), which is almost twice of that with top heating. The heat storage rate of composite with left and bottom heating visibly decreases with the increase of contact gap. In contrast, the heat storage rate of composite under top heating condition hasn't the evident reduction with the increase of contact gap. Furthermore, it is seen that the average heat storage rate of composite with top heating is larger than that with left and bottom heating under the worse contact condition (i.e. δ = 0.8 mm). It is demonstrated from the above results that the effects of heating and contact conditions on thermal charging performance are significant and integrated.

Conclusion

In this paper, the numerical investigation is conducted to examine the effects of heating and contact conditions on thermal charging performance of composite PCM. A two-dimensional model based on volume-averaged method is built to investigate heat transfer mechanism of composite PCM in the melting process. Nine different cases are designed to compare the effect of heating and contact conditions on thermal characteristic. The following conclusions can be extracted:

(1) The effect of heating condition can be investigated under the same contact condition. It is found from numerical results that natural convection is suppressed in the whole thermal charging process for top heating. In contrast, the natural convection plays a dominant role in heat transfer for left and bottom heating. It is demonstrated that the heat transfer mechanism is different under different heating conditions, which results in a remarkable impact on thermal charging performance of composite PCM. For instance, the total melting time of left and bottom heating is almost the same and shorten by about 50% compared to top heating when the contact gap is 0 mm.

(2) The contact condition has a significant impact upon thermal performance of composite PCM under left and bottom heating conditions, e.g., the total melting time for left and bottom heating presents a linear increase versus the contact gap. It is further confirmed that contact condition has an obvious influence on thermal characteristic of composite PCM in which heat transfer is dominated by natural convection. Conversely, the influence of contact condition under top heating condition is very small, e.g., the total melting time for top heating is almost the same under three contact conditions (i.e. contact gap is 0, 0.4 and 0.8 mm), which indicates that the influence of contact condition on thermal behavior of composite PCM (in which heat transfer is dominated by heat conduction) can be neglected and it is not necessary for metal foam to be bonded to the container wall via brazing, sintering, etc.

(3) It is found from numerical results that the heating and contact conditions have a combined effect on the thermal charging performance of composite PCM. The heat storage rate of left and bottom heating is almost the same and higher than that of top heating when the contact condition is good (i.e. δ = 0 mm). Hence, left and bottom heating are recommended in the practical application of TES due to the high heat storage rate. The heat storage rate of three heating conditions is almost the same under middle contact condition (i.e. δ = 0.4 mm).

Whereas, the heat storage rate of top heating is higher than that of left and bottom heating as contact condition grows worse (i.e. δ = 0.8 mm).

(4) This study can provide theoretical guidance for the selection of heating and contact conditions of thermal storage system using composite PCM, such as solar thermal storage, latent heat storage device and thermal battery for taking advantage of off-peak electricity. For future work, the experimental investigation will be conducted to study the effects of contact and heating conditions on thermal charging performance of composite PCM, and optimized design of TES device using composite PCM will be performed considering role of heating and contact conditions.

  velocity in x and y direction (m/s) w wall

Fig. 1 .

 1 Fig. 1. Schematic diagram of contact condition of composite PCM for TES

Fig. 2 .

 2 Fig. 2. Different heating and contact conditions

Fig. 3 .

 3 Fig. 3. Physical model of composite PCM

Fig. 4 .Fig. 5 .

 45 Fig. 4. Independence tests of (a) grids and (b) time step

Fig. 8 .

 8 Fig. 8. Evolvement of phase change for composites with bottom heating at 20%, 40% and 80% liquid fraction

Fig . 10 .

 10 Fig . 10. Variation of liquid fraction under three contact conditions

Fig. 11 .

 11 Fig. 11. Total melting time of composite PCM

Fig. 12 .

 12 Fig. 12. Average heat storage rate

Table 1

 1 Thermophysical properties of materials

	Parameters		Materials
		Paraffin	Copper foam
	c (J/kg K)	2300	386
	ρ (kg/m 3 )	900	8900
	k (W/m K)	0.3	380
	µ (kg/m s) γ (1/K)	0.00324 0.0005
	L (J/kg)	148800
	Tm1 (K)	323
	Tm2 (K)	331

Acknowledgements

The authors thank China Scholarship Council (CSC) for providing a scholarship to conduct this study.