Cluster expansion for continuous particle systems interacting via an attractive pair potential and subjected to high density boundary conditions
aa r X i v : . [ m a t h - ph ] F e b Cluster expansion for continuous particle systems interacting via anattractive pair potential and subjected to high density boundaryconditions
Paula M. S. Fialho, Bernardo N. B. de Lima, Aldo Procacci
Departamento de Matem´atica UFMG 30161-970 - Belo Horizonte - MG Brazil
February 5, 2021
Abstract
We propose a method based on cluster expansion to study the low activity/high temper-ature phase of a continuous particle system confined in a finite volume, interacting through astable and finite range pair potential with negative minimum in presence of non free boundaryconditions.
In the area of rigorous statistical mechanics from the very beginning a great effort has been spentin order to outline the possible influence of the boundary conditions on systems confined in afinite (but possibly arbitrarily large) volume. It has been clear soon (see for instance [25, 6] andreferences therein) that in the regime in which more phases may coexist the presence of suitableboundary conditions may force the system in one of those phases. This has been rigourouslyestablished and put on firm ground for a large class of bounded spin systems in a lattice interactingvia a finite range potential. A classical example is the nearest-neighbor Ising model in two ormore dimensions. Similar results can be obtained for large classes of bounded spin systems withfinite range interaction for which a very robust and effective tool, the Pirogov Sinai theory, isavailable (see, e.g. [29] and references therein). The effect and influence of boundary conditions onspin systems which are unbounded or interact with infinite range potential appears to be a moredelicate issue to be treated rigorously and results in the literature are quite rarer. Problems relatedto existence and uniqueness of the infinite volume measure for unbounded spin-systems has beendiscussed e.g. in [15], [4] and [20], while analyticity of free energy and correlations for such systemssubjected to rather general boundary conditions has been treated via cluster expansion in [1] and[24].The situation is even less clear as soon as one considers continuous systems formed by classicalparticles in R d interacting via a pair potential (such as the Lennard-Jones potential, the Morsepotential or even much simpler potentials, e.g. finite range). In this case the only phase whichhas been rigourously analyzed is the low densitity/high temperature phase and no proof on theexistence of phase transitions has been furnished nor a consistent and rigorous treatment of suchsystems outside the low density/high temperature region has been provided with the sole exceptionof the result obtained by Mazel, Lebowitz and Presutti in 1999 [17].1he relation between the boundary condition and the macroscopical behavior of continuousclassical particle systems could in principle be studied rigorously at least in the low density/hightemperature regime where the powerful tool given by the cluster expansion is available. Thisproblem is incidentally mentioned by the classical texts on rigorous statistical mechanics (see e.g.[25, 6]), where however the computations related to the analyticity of the pressure of the gas inthe low density/high temperature regime are always performed assuming free boundary conditions.Although it is widely believed that the macroscopic behavior of continuous particle systems in theregion of parameters λ and β where the cluster expansion converges is not affected by (reasonablywell behaved) boundary conditions, we are not aware of any rigorous result about this issue as soonas one considers pair potentials with a negative (attractive) tail. In particular, the independence ofconvergence radius of the cluster expansion from (reasonable) boundary conditions can be estabil-shed only by assuming that the pair potential v is non-negative, in addition to stable and regular(see e.g. the remark at the end of pag. 3 in [27]). Of course the assumption v ≥ d dimensions confined in afinite volume Λ in the grand canonical ensemble at fixed inverse temperature β and fixed fugacity λ . We assume that these particles interact via a stable pair potential v which, for simplicity, weimpose to be finite range. On the other hand, we allow v to have a negative tail. In other wordsthe stability constant of v may be strictly positive. This system would have a fixed mean density ρ ∅ Λ ( λ, β ) when submitted to free boundary conditions. We then fix a boundary configuration ω outside Λ (i.e. in R d \ Λ) allowing a density ρ ω which may be much larger than ρ ∅ Λ ( β, λ ) but hasto be uniformly bounded. Note that we allow ρ ω to be arbitrary so that we are actually allowingboundary condition with arbitrarily large (but bounded) densities. With these assumptions weshow that the Mayer series of the pressure of the system in presence of the boundary condition ω can be written as the sum of two terms. The first series, the bulk term, has a radius of analyticityin the activity λ that coincides with the free boundary condition convergence radius. The secondseries, the boundary term, has an ω -dependent radius of analyticity decreasing exponentially with ρ ω , but it tends to zero as Λ goes to infinity. Moreover, we show that the bulk term of the finitevolume pressure in presence of boundary conditions ω tends to the Mayer series of the pressurecalculated with free boundary conditions. We consider a system of classical continuous particles confined in a bounded compact region Λ of R d interacting via a translational invariant pair potential v . We will suppose hereafter that Λ is a2ube of size 2 L centered at the origin and lim Λ → ∞ means simply that L → ∞ . We will denoteby | Λ | = (2 L ) d the volume of Λ and in general if U is a compact in R d we denote by | U | its volume.We denote by x i ∈ R d the position vector of the i th particle of the system and by k x i k its Euclideannorm.We will further suppose that our system is subjected to a boundary condition ω which is typicallya locally finite countable set of points in R d (not necessarily distinct), representing the positionsof a set of fixed particles in R d . Namely, ω is a set such that for any compact subset C ⊂ R d , ω ∩ C ) < + ∞ (here ω ∩ C ) is the cardinality of the set ω ∩ C ). As usual, we will supposethat each particle inside Λ, say at position x ∈ Λ, feels the effect of boundary condition ω throughthe potential energy generated by the particles of the configuration ω which are in Λ c = R d \ Λ.We are interested in studying the behavior of the systems in the limit Λ → ∞ for a fixed boundarycondition ω and how eventually this limit may be influenced by ω , having in mind that as thevolume Λ invades R d the fixed particles of ω entering in Λ are disregarded and only those boundaryparticles outside Λ influence particles inside Λ. Assumptions on the pair potential
The pair potential v is supposed to be a translational invariant, symmetric and Lebesgue measur-able function. Therefore it is completely defined by a function v ( x ) in R d with values in R ∪ { + ∞} such that v ( x ) = v ( − x ) for all x ∈ R d . We further assume that(ii) v is finite range : there exists R > v ( x ) = 0 for all k x k ≥ R. (2 . . (i) v is stable : namely, v is such that for some constant C ≥
0, for all n ∈ N and for all x , . . . , x n ∈ R d X ≤ i
0. Let us denote by v − thenegative part of v , namely, for r ∈ [0 , + ∞ ), v − ( r ) = max { , − v ( r ) } . By (2.2), the potential v is bounded below by − B v and hence v − is bounded above by 2 B v .Therefore, by (2.1), we have that Z R d v − ( x ) dx ≤ B v V d ( R ) , (2 . deF where V d ( R ) is the volume of the d -dimensional ball of radius R .3e will suppose hereafter that R d is partitioned in elementary cubes ∆ of suitable size δ > R dδ the set of all these cubes and, given x ∈ R d , we will denoteby ∆( x ) the cube of R dδ to which x belongs. Moreover, for sake of simplicity we assume that Λ is sochosen in such a way Λ and Λ c are both the union of elementary cubes in R dδ (in other words, forany ∆ ∈ R dδ , either ∆ ⊂ Λ or ∆ ⊂ Λ c ). We denote by Λ δ (respectively Λ cδ ) the set of elementarycubes whose union is Λ (respectively Λ c ) and of course by construction Λ δ ∪ Λ cδ = R dδ . Given ∆ ∈ R dδ and x ∈ R d we let dist(∆ , x ) = inf y ∈ ∆ k x − y k and we will suppose that δ is chosen suitably smallin such a way so that for any x ∈ R d δ d X ∆ ∈ R dδ dist(∆ ,x ) ≤ R ≤ V d ( R ) . (2 . tusf Assumptions on boundary conditions
Given the partition of R d in elementary cubes ∆ of size δ > ω , we define the density of ω as the function ρ ωδ : R d → [0 , + ∞ ) x ρ ωδ ( x )with ρ ωδ ( x ) = ω ∩ ∆( x )) δ − d , then, by definition, ρ ωδ ( y ) is constant for all y ∈ ∆( x ). Since ω is locally finite, ρ ωδ ( x ) is everywherefinite.Our assumption on the set of allowed boundary conditions is as follows.(iii) ω is admissible: namely, there exists a finite positive number ρ ω such that, for all elemen-tary cubes ∆ ⊂ R dδ , sup ∆ ⊂ R dδ ω ∩ ∆) | ∆ | ≤ ρ ω . We call Ω ρ the space of all locally finite configurations of particles in R d with maximal density ρ and we set Ω ∗ = ∪ ρ ≥ Ω ρ . Note that the free boundary condition ω = ∅ is obviously in Ω ρ , for all ρ > ∂ Λ denotes the boundary of Λ and let us define, for x ∈ Λ fixed, d Λ x = dist( x, ∂ Λ) = inf y ∈ ∂ Λ k x − y k . For a fixed volume Λ and a fixed boundary condition ω , let us define the function w ω Λ : Λ → R asfollows w ω Λ ( x ) = X y ∈ ω ∩ Λ c v ( x − y ) . (2 . fa This function represents the potential energy felt by a particle sitting in the point x ∈ Λ dueto the fixed particles of the boundary condition ω sitting in points outside Λ. Note that by the4ssumptions (ii) on the pair potential and (iii) on the admissible boundary conditions we have that w ω Λ ( x ) is different from zero only in the frame inside Λ constituted by the points at distance lessthan R to the boundary ∂ Λ.The partition function of the system in the grand canonical ensemble at fixed inverse temperature β > λ > ω Λ ( λ, β ) = ∞ X n =0 λ n n ! Z Λ n dx . . . dx n e − β " P ≤ i 21 if n = 1 (3 . urse where G n is the set of all connected graphs g with vertex set [ n ] . = { , , . . . , n } and edge set E g .The Mayer series of the (finite volume) pressure in presence of non free boundary conditions ω is defined as the power series (3.3) divided by | Λ | , namely, βp ω Λ ( β, λ ) . = 1 | Λ | log Ξ ω Λ ( λ, β ) = ∞ X n =1 c ωn ( β, Λ) λ n , (3 . mayer where c ωn ( β, Λ) = 1 | Λ | n ! Z Λ dx . . . Z Λ dx n Φ T ( x , . . . , x n ) f ω Λ ( x ) . . . f ω Λ ( x n ) . Note that we can also write log Ξ ω Λ ( λ, β ) = λ Z Λ dx f ω Λ ( x )Π ωx , Λ ( β, λ ) , (3 . . .c where Π ωx , Λ ( β, λ ) = ∞ X n =0 c ωn ( x , β, Λ) λ n (3 . . and c ωn ( x , β, Λ) = 1( n + 1)! Z Λ dx . . . Z Λ dx n Φ T ( x , x , . . . , x n ) f ω Λ ( x ) . . . f ω Λ ( x n ) , (3 . cex with Φ T ( x , x , . . . , x n ) = P g ∈ G n Q { i,j }∈ E g (cid:2) e − βv ( x i − x j ) − (cid:3) if n ≥ 11 if n = 0 (3 . urseb where G n denotes now the set of all connected graphs g with vertex set [ n ] . = { , , , . . . , n } andedge set E g . We agree that c ωn ( x , β, Λ) = 1 if n = 0.8sing the above notations, the pressure of the system at finite volume can also be written as βp ω Λ ( β, λ ) = λ | Λ | Z Λ dxf ω Λ ( x )Π ωx , Λ ( β, λ ) , (3 . pressure which is an useful expression for the computations developed ahead. We conclude this sectionby proving the following inequality concerning the absolute value of the coefficients c ωn ( x , β, Λ)definied above. Proposition 3.1 For any x ∈ Λ and any ω ∈ Ω ∗ , it holds that | c ωn ( x , β, Λ) | ≤ e βκρ ω n ( n + 1) n − ( n + 1)! e β B v ( n +1) [ C v ( β )] n . (3 . cogen with C v ( β ) defined in (2.11). Proof . We first recall that the potential v is stable with stability constant B v , therefore we canuse the bound proved in [22] (see there Proposition 1), namely, | Φ T ( x , x , . . . , x n ) | ≤ e β B v ( n +1) X τ ∈ T n Y { i,j }∈ E τ (1 − e − β | V ( x i − x j ) | ) , (3 . P Y where T n is the set of trees with vertex set { , , , . . . , n } . Moreover, for any x ∈ Λ, by Remark3.1 we have that f ω Λ ( x ) ≤ e κβρ ω . Thus | c ωn ( x , β, Λ) | ≤ n + 1)! Z Λ dx . . . Z Λ dx n | Φ T ( x , x , . . . , x n ) | f ω Λ ( x ) . . . f ω Λ ( x n ) ≤ e βκρ ω n n + 1)! Z Λ dx . . . Z Λ dx n e β B v ( n +1) X τ ∈ T n Y { i,j }∈ E τ (1 − e − β | V ( x i − x j ) | ) ≤ e βκρ ω n e β B v ( n +1) ( n + 1)! X τ ∈ T n Z Λ dx . . . Z Λ dx n Y { i,j }∈ E τ (1 − e − β | V ( x i − x j ) | ) . Now, for any n ∈ N and τ ∈ T n we have (see e.g. Lemma 3 in [22]) Z Λ dx . . . Z Λ dx n Y { i,j }∈ E τ (1 − e − β | V ( x i − x j ) | ) ≤ [ C v ( β )] n . (3 . Therefore, | c ωn ( x , β, Λ) | ≤ e βκρ ω n e β B v ( n +1) ( n + 1)! [ C v ( β )] n X τ ∈ T n ≤ e βκρ ω n e β B v ( n +1) ( n + 1)! [ C v ( β )] n ( n + 1) n − where in the last line we have used the Cayley formula (see [5] ), i.e. | T n | = ( n + 1) n − . (cid:3) We stress that bound (3.11) is very crude and it may be quite strongly improved dependingon the distance of the point x from the border of Λ. We will analyze in some more detail thebehaviour of the coefficients c ωn ( x , β, Λ) in the next section.9 .2 On the behavior of c ωn ( x , β, Λ) Recall that we are supposing Λ to be a cube of size 2 L centered at the origin. Let us choose amonotonic increasing continuous function h ( L ) such that lim L →∞ h ( L ) = ∞ , lim L →∞ h ( L ) /L = 0and define Λ h = { x ∈ Λ : d Λ x > h ( L ) } and Λ ∗ h = Λ \ Λ h so that Λ h is a cube centered at the origin with size 2( L − h ( L )) fully contained in Λ and clearlylim Λ →∞ | Λ h || Λ | = 1 (3 . uno and lim Λ →∞ | Λ ∗ h || Λ | = 0 . (3 . due Remark 3.2 Observe that √ L is an example of a function that satisfies the proprieties describedabove for the function h ( L ) . However, while the properties (3.14) and (3.15) are essential for ourtask, the function rule of h ( L ) does not play an important role in the calculations ahead. Let us now choose L large enough in such a way that h ( L ) > R , so that n h ( L ) . = (cid:22) h ( L ) R − (cid:23) (3 . nl is greater that or equal to one. Observe thatlim Λ →∞ n h ( L ) = + ∞ . (3 . limn Theorem 3.1 Let x ∈ Λ h and let ω ∈ Ω ∗ . Then, for all n ≤ n h ( L ) , we have that c ωn ( x , β, Λ) = c ∅ n ( x , β, Λ) . (3 . inside Moreover, for all n ∈ N ∪ { } the following bound holds | c ∅ n ( x , β, Λ) | ≤ ( n + 1) n − ( n + 1)! e β B v ( n +1) [ C v ( β )] n . (3 . c n Proof . Let us start by proving identity (3.18). We recall the definition (3.9) of Φ T ( x , x , . . . , x n ).If g is any connected graph with vertex set [ n ] , as v is finite range with v ( x ) = 0 if k x k ≥ R , wehave that Y { i,j }∈ E g h e − βv ( x i − x j ) − i = 0 (3 . g i ∈ [ n ], k x − x i k ≥ nR . Indeed, given g ∈ G n , suppose that there exists a vertex i ∈ [ n ] of g such that k x − x i k ≥ nR . Then there exists a path p i, = { , i , i , . . . , i k − , i k ≡ i } contained in g connecting 0 to i , once g is connected. Let E p i, be the edge set of such a path.Therefore Y { i,j }∈ E g h e − βv ( x i − x j ) − i = Y { i,j }∈ E g \ E pi, h e − βv ( x i − x j ) − i k Y s =1 h e − βv ( x s − − x s ) − i . Observe that in any case k ≤ n , the hypothesis that k x − x i k ≥ nR implies that at least for one s ∈ [ k ] we have that k x s − − x s k ≥ R and thus v ( x s − − x s ) = 0, so that e − βv ( x s − − x s ) − k x − x i k ≥ nR , then k Y s =1 h e − βv ( x s − − x s ) − i = 0and thus (3.20) follows.The discussion above immediately implies that Φ T ( x , x , . . . , x n ) = 0 if there exists i ∈ [ n ] suchthat k x i − x k > nR and thus we can rewrite c ωn ( x , β, Λ) as follows c ωn ( x , β, Λ) = 1( n + 1)! Z x ∈ Λ k x − x k≤ nR dx . . . Z xn ∈ Λ k x − xn k≤ nR dx n Φ T ( x , x , . . . , x n ) f ω Λ ( x ) . . . f ω Λ ( x n ) . (3 . cexb Let us now suppose that n ≤ n h ( L ) , i.e., n ≤ h ( L ) R − , whence, as by hypotheses x ∈ Λ h , then d Λ x ≥ h ( L ) and we have d Λ x ≥ ( n + 1) R. (3 . dist Moreover, by the triangular inequality, d Λ x i ≥ d Λ x − k x i − x k , hence d Λ x i ≥ ( n + 1) R − k x i − x k ≥ ( n + 1) R − nR ≥ R where in the intermediate inequality we used (3.22) and in the last inequality we used that any n -uple ( x , . . . , x n ) contributing to the integral of the r.h.s. of (3.21) is such that, for any i ∈ [ n ], k x i − x k ≤ nR .In conclusion we have shown that if n ≤ n h ( L ) , then for any n -uple ( x , . . . , x n ) ∈ Λ n such that k x − x i k ≤ nR for all i ∈ [ n ], it holds that d Λ x i ≥ R . Recalling the Remark 3.1, we have that f ω Λ ( x i ) = 1 for all i ∈ [ n ] in formula (3.8) when n ≤ n h ( L ) . Therefore, we have the identity c ωn ( x , β, Λ) = 1( n + 1)! Z Λ dx . . . Z Λ dx n Φ T ( x , x , . . . , x n ) = c ∅ n ( x , β, Λ)11or all x ∈ Λ h and for all n ≤ n h ( L ) . Namely, we have proved the statement (3.18).Let us now prove bound (3.19). Recalling that the potential v is stable with stability constant B v , we can once again use the bound (3.12), namely, | Φ T ( x , x , . . . , x n ) | ≤ e β B v ( n +1) X τ ∈ T n Y { i,j }∈ E τ (1 − e − β | V ( x i − x j ) | ) , where T n is the set of trees with vertex set { , , , . . . , n } . Thus Z Λ dx . . . Z Λ dx n | Φ T ( x , x , . . . , x n ) | ≤ e β B v ( n +1) X τ ∈ T n Z Λ dx . . . Z Λ dx n Y { i,j }∈ E τ (1 − e − β | V ( x i − x j ) | ) . (3 . Now, for any n ∈ N and τ ∈ T n we can use again inequality (3.13) and hence we have the upperbound c ∅ n ( x , β, Λ) ≤ n + 1)! e β B v ( n +1) [ C v ( β )] n X τ ∈ T n 1= ( n + 1) n − ( n + 1)! e β B v ( n +1) [ C v ( β )] n where in the last line we have once again used the Cayley formula (see [5]). Note that this boundholds also when n = 0, since 1 ≤ e β B v . This concludes the proof of bound (3.19). (cid:3) Recalling that we chose L large enough in such a way that h ( L ) > R , the definitions of Λ h andΛ ∗ h given at the beginning of Section 3.2 and the Identity (3.10), we can rewrite the finite volumepressure βp ω Λ ( β, λ ) of our system as βp ω Λ ( β, λ ) = λ | Λ | "Z Λ h dxf ω Λ ( x )Π ωx, Λ ( β, λ ) + Z Λ ∗ h dxf ω Λ ( x )Π ωx, Λ ( β, λ ) == λ | Λ | "Z Λ h dx Π ωx, Λ ( β, λ ) + Z Λ ∗ h dxf ω Λ ( x )Π ωx, Λ ( β, λ ) where the last identity follows from Remark 3.1.By Formula (3.18) in Theorem 3.1, we have that for any x ∈ Λ h Π ωx, Λ ( β, λ ) = P ∅ ,n h ( L ) x, Λ ( β, λ ) + Q ω,n h ( L ) x, Λ ( β, λ )where P ∅ ,n h ( L ) x, Λ ( β, λ ) = n h ( L ) X n =0 c ∅ n ( x, β, Λ) λ n (3 . P Q ω,n h ( L ) x, Λ ( β, λ ) = ∞ X n = n h ( L ) +1 c ωn ( x, β, Λ) λ n . (3 . P sio Therefore, posing η ω Λ ( λ, β ) = λ | Λ | Z Λ h dxP ∅ ,n h ( L ) x, Λ ( β, λ ) (3 . eta and ξ ω Λ ( λ, β ) = λ | Λ | "Z Λ h dxQ ω,n h ( L ) x, Λ ( β, λ ) + Z Λ ∗ h dxf ω Λ ( x )Π ωx, Λ ( β, λ ) (3 . xi we have that βp ω Λ ( β, λ ) = η ω Λ ( λ, β ) + ξ ω Λ ( λ, β ) . Note that η ω Λ ( λ, β ) is a polynomial of degree λ n h ( L ) . Let us also define the function Q ∅ ,n h ( L ) x, Λ ( β, λ ) = ∞ X n = n h ( L ) +1 c ∅ n ( x , β, Λ) λ n . (3 . P siob Theorem 3.2 Let D ∅ be the closed disc in the complex plane defined in (2.12) D ∅ = n λ ∈ C : | λ | ≤ e β B v +1 C v ( β ) o . Then the functions Π ∅ x, Λ ( β, λ ) and Q ∅ ,n h ( L ) x, Λ ( β, λ ) are analytic inside the disc D ∅ where, uniformlyin Λ and x , they admit the bounds | Π ∅ x, Λ ( β, λ ) | ≤ (8 / e β B v +1 (3 . bop and | Q ∅ ,n h ( L ) x, Λ ( β, λ ) | ≤ e β B v +1 n / h ( L ) . (3 . boq Proof . We start by proving the analyticity of Π ∅ x , Λ ( β, λ ). We have straightforwardly that | Π ∅ x , Λ ( β, λ ) | ≤ ∞ X n =0 | c ∅ n ( x , β, Λ) || λ | n , then, by Inequality (3.19) in Theorem 3.1, we have | Π ∅ x , Λ ( β, λ ) | ≤ ∞ X n =0 ( n + 1) n − ( n + 1)! e β B v ( n +1) [ | λ | C v ( β )] n = e β B v ∞ X n =0 ( n + 1) n − ( n + 1)! [ e β B v | λ | C v ( β )] n . r ≥ β, r ) = e β B v ∞ X n =0 ( n + 1) n − ( n + 1)! [ e β B v rC v ( β )] n and suppose that there exists r ∗ = max n r ≥ β, r ) < + ∞ o (3 . rstar , then clearly Π ∅ x , Λ ( β, λ ) is analytic for all | λ | ≤ r ∗ and | Π ∅ x , Λ ( β, λ ) | is bounded by Θ( β, r ∗ ) for all λ in the disc | λ | ≤ r ∗ .To show that this is indeed true, let us just recall the Stirling bound, namely √ πnn n e − n ≤ n ! ≤ e √ nn n e − n , (3 . stir for all n ∈ N . So that we may bound e β B v +1 " e ∞ X n =1 [ e β B v +1 rC v ( β )] n ( n + 1) ≤ Θ( β, r ) ≤ e β B v +1 " √ π ∞ X n =1 [ e β B v +1 rC v ( β )] n ( n + 1) . The series P ∞ n =1 ( n + 1) − / [ e β B v +1 rC v ( β )] n converges if e β B v +1 rC v ( β ) ≤ r ∗ defined in (3.31) does exist and it is equal to r ∗ = 1 e β B v +1 C v ( β ) . Moreover, Θ( β, r ∗ ) ≤ e β B v +1 " √ π ∞ X n =1 n + 1) < (8 / e β B v +1 and thus we have proved that Π ∅ x, Λ ( β, λ ) is analytic in the closed disc D ∅ and its modulus is boundedthere by (8 / e β B v +1 .Let us now prove the analyticity and boundedness of the function Q ∅ ,n h ( L ) x, Λ ( β, λ ) defined in (3.28)when λ varies in the complex disc D ∅ . Using once again inequality (3.19) and the Stirling bound(3.32) and assuming that | λ | varies in D ∅ , we have | Q ∅ ,n h ( L ) x, Λ ( β, λ ) | = ∞ X n = n h ( L ) +1 | c ∅ n ( x , β, Λ) || λ | n ≤ ∞ X n = n h ( L ) +1 ( n + 1) n − ( n + 1)! e β B v ( n +1) [ C v ( β )] n | λ | n ≤≤ e β B v +1 √ π ∞ X n = n h ( L ) +1 [ e β B v +1 C v ( β ) | λ | ] n ( n + 1) ≤ e β B v +1 √ π ∞ X n = n h ( L ) +1 n + 1) ≤ e β B v +1 √ π ∞ X n = n h ( L ) +1 n / ≤≤ e β B v +1 √ π Z ∞ n h ( L ) u / du = 2 e β B v +1 √ π n / h ( L ) < e β B v +1 n / h ( L ) . Hence we have proved that | Q ∅ ,n h ( L ) x, Λ ( β, λ ) | is analytic in D ∅ and bounded there according to (3.30).14 heorem 3.3 Given x ∈ Λ and ω ∈ Ω ∗ , let D ω the closed disc in the complex plane defined in(2.13) D ω = n λ ∈ C : | λ | ≤ e κβρ ω e β B v +1 C v ( β ) o . Then the function Π ωx, Λ ( β, λ ) defined in (3.7) is analytic in the disc D ω where, uniformly in Λ , x and ω | Π ωx, Λ ( β, λ ) | ≤ (8 / e β B v +1 . (3 . bp Moreover, let n h (Λ) be the integer defined in (3.16), then the function Q ω,n h ( L ) x, Λ ( β, λ ) defined in(3.25) is analytic in the disc D ω where, uniformly in Λ , x and ω | Q ω,n h ( L ) x, Λ ( β, λ ) | ≤ e β B v +1 n / h ( L ) . (3 . bq Proof . The proof of (3.33) and (3.34) proceeds along the same lines described in the previoustheorem. In order to proof bound (3.33), we can use the bound (3.11) on | c ωn ( x , β, Λ) | given inProposition 3.1 together with the Stirling bound (3.32) to get, for all λ ∈ D ω , | Π ωx, Λ ( β, λ ) | = ∞ X n =0 | c ωn ( x , β, Λ) || λ | n ≤ ∞ X n =0 ( n + 1) n − ( n + 1)! e β B v ( n +1) [ C v ( β ) e κβρ ω ] n | λ | n ≤≤ e β B v +1 " √ π ∞ X n =0 [ e β B v +1 e κβρ ω C v ( β ) | λ | ] n ( n + 1) ≤ e β B v +1 " √ π ∞ X n =0 n + 1) ≤ (8 / e β B v +1 . Concerning now the bound (3.34) we have, | Q ω,n h ( L ) x, Λ ( β, λ ) | = ∞ X n = n h ( L ) +1 | c ωn ( x , β, Λ) || λ | n ≤ ∞ X n = n h ( L ) +1 ( n + 1) n − ( n + 1)! e β B v ( n +1) [ C v ( β ) e κβρ ω ] n | λ | n ≤≤ e β B v +1 √ π ∞ X n = n h ( L ) +1 [ e β B v +1 e κβρ ω C v ( β ) | λ | ] n ( n + 1) ≤ e β B v +1 √ π ∞ X n = n h ( L ) +1 n + 1) ≤≤ e β B v +1 √ π ∞ X n = n h ( L ) +1 n / ≤ e β B v +1 √ π Z ∞ n h ( L ) u / du = 2 e β B v +1 √ π n / h ( L ) < e β B v +1 n / h ( L ) . Proposition 3.2 The function η ω Λ ( λ, β ) defined in (3.26) is analytic in the whole complex planeand its modulus is bounded as | η ω Λ ( λ, β ) | ≤ (8 / e β B v +1 | λ | (3 . bhl as λ varies in the disc D ∅ . Moreover for λ ∈ D ∅ ( β ) it holds that lim Λ →∞ η ω Λ ( λ, β ) = βp ∅ ( β, λ ) (3 . limh roof . The analyticity of η ω Λ ( λ, β ) in the whole complex plane follows trivially from the fact that,by definition (3.26), η ω Λ ( λ, β ) is, as a function of λ , a polynomial of degree n h ( L ) . The fact thatthe modulus | η ω Λ ( λ, β ) | is bounded by the r.h.s. of (3.35) when λ ∈ D ∅ follows trivially from bound(3.29) of Theorem 3.2. Indeed, if λ ∈ D ∅ , recalling the definition (3.24) of P ∅ ,n h ( L ) x, Λ ( β, λ ), it easilyfollows from Theorem 3.2 (see the proof of inequality (3.29)) that | P ∅ ,n h ( L ) x, Λ ( β, λ ) | ≤ n h ( L ) X n =0 | c ∅ n ( x , β, Λ) || λ | n ≤ ∞ X n =0 | c ∅ n ( x , β, Λ) || λ | n ≤ (8 / e β B v +1 . (3 . P bo Therefore, if λ ∈ D ∅ , | η ω Λ ( λ, β ) | ≤ | λ || Λ | Z Λ h dx | P ∅ ,n h ( L ) x, Λ ( β, λ ) | ≤ λ | Λ | Z Λ h (8 / e β B v +1 dx ≤ | λ | (8 / e β B v +1 . In order to prove (3.36), observe that βp ∅ Λ ( β, λ ) = λ | Λ | Z Λ dx Π ∅ x , Λ ( β, λ ) = λ | Λ | Z Λ dx (cid:16) P ∅ ,n h ( L ) x, Λ ( β, λ ) + Q ∅ ,n h ( L ) x, Λ ( β, λ ) (cid:17) , so that η ω Λ ( λ, β ) − βp ∅ Λ ( β, λ ) = λ | Λ | " Z Λ h dxP ∅ ,n h ( L ) x, Λ ( β, λ ) − Z Λ dxP ∅ ,n h ( L ) x, Λ ( β, λ ) − Z Λ dxQ ∅ ,n h ( L ) x, Λ ( β, λ ) , hence | η ω Λ ( λ, β ) − βp ∅ Λ ( β, λ ) | ≤ | λ || Λ | "Z Λ \ Λ h dx | P ∅ ,n h ( L ) x, Λ ( β, λ ) | + Z Λ dx | Q ∅ ,n h ( L ) x, Λ ( β, λ ) | . Now, if λ ∈ D ∅ , by (3.37) and (3.30) we have that | P ∅ ,n h ( L ) x, Λ ( β, λ ) | ≤ (8 / e β B v +1 and | Q ∅ ,n h ( L ) x, Λ ( β, λ ) | ≤ e β B v +1 n / h ( L ) Hence | η ω Λ ( λ, β ) − βp ∅ Λ ( β, λ ) | ≤ | λ | e β B v +1 (8 / | Λ \ Λ h || Λ | + 1 n / h ( L ) = | λ | e β B v +1 (8 / | Λ ∗ h || Λ | + 1 n / h ( L ) . Recalling (3.15), we have that for λ ∈ D ∅ lim Λ →∞ | η ω Λ ( λ, β ) − βp ∅ Λ ( β, λ ) | ≤ lim Λ →∞ | λ | e β B v +1 (8 / | Λ ∗ h || Λ | + 1 n / h ( L ) = 0 (3 . ofco λ ∈ D ∅ , it holds thatlim Λ →∞ η ω Λ ( λ, β ) = lim Λ →∞ βp ∅ Λ ( β, λ ) = βp ∅ ( β, λ ) , which ends the proof. (cid:3) Proposition 3.3 The function ξ ω Λ ( λ, β ) defined in (3.27) is analytic and bounded as far as λ ∈ D ω .Moreover, it holds that lim Λ →∞ ξ ω Λ ( λ, β ) = 0 . (3 . limu Proof. We recall the definition (3.27) of ξ ω Λ ( λ, β ) ξ ω Λ ( λ, β ) = λ | Λ | "Z Λ h dxQ ω,n h ( L ) x, Λ ( β, λ ) + Z Λ ∗ h dxf ω Λ ( x )Π ωx, Λ ( β, λ ) . By Theorem 3.3, both Q ω,n h ( L ) x, Λ ( β, λ ) and Π ωx, Λ ( β, λ ) are analytic and bounded in the closed disc D ω . This immediately implies that ξ ω Λ ( λ, β ) is also analytic (and bounded) in D ω .Concerning the limit (3.39), we can use the bounds (3.33) and (3.34) given in Theorem 3.3 forΠ ωx, Λ ( β, λ ) and Q ω,n h ( L ) x, Λ ( β, λ ) respectively and, recalling Remark 3.1, we have that f ω Λ ( x ) ≤ e βκρ ω and so | ξ ω Λ ( λ, β ) | ≤ | λ || Λ | "Z Λ h dx | Q ω,n h ( L ) x, Λ ( β, λ ) | + Z Λ ∗ h dxf ω Λ ( x ) | Π ωx, Λ ( β, λ ) | ≤ | λ || Λ | Z Λ h dx e β B v +1 n / h ( L ) + Z Λ ∗ h dxe βκρ ω (8 / e β B v +1 ≤ (8 / | λ | e βκρ ω e β B v +1 " | Λ h || Λ | n / h ( L ) + | Λ ∗ h || Λ | . Hence lim Λ →∞ | ξ ω Λ ( λ, β ) | ≤ (8 / | λ | e βκρ ω e β B v +1 lim Λ →∞ " | Λ h || Λ | n / h ( L ) + | Λ ∗ h || Λ | = (8 / | λ | e βκρ ω e β B v +1 " lim Λ →∞ | Λ h || Λ | lim Λ →∞ n / h ( L ) + lim Λ →∞ | Λ ∗ h || Λ | = 0where in the last line, by the definitions given at the beginning of Secion 3.2, we have used thatlim Λ →∞ | Λ h || Λ | = 1 , lim Λ →∞ n h ( L ) = + ∞ , lim Λ →∞ | Λ ∗ h || Λ | = 0 . 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