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Transfer operators and dynamical zeta functions for a class of lattice spin models

We investigate the location of zeros and poles of a dynamical zeta function arizing in a class of lattice spin models introduced in the 60-ties by M. Kac. The transfer operator method allows us to prove the xistence of infinitely nontrivial zeros of this function on the real line. For certain parameter values there exist also infinitely many trivial equally spaced zeros on a line parallel to the imaginary axis. Hence also for this kind of dynamical zeta function some kind of Riemann hypothesis seems to hold.


Introduction
The transfer matrix method has played an important role in statistical mechanics ever since E. Ising for the first time used this method to solve his 1-dimensional lattice spin model with nearest neighbour interaction. The method was extended later to treat higher dimensional models with arbitrary finite range interactions. The most satisfying theory for this method from the mathematical point of view goes back to D. Ruelle who introduced the so called transfer operator for 1-dimensional lattice spin systems with arbitrary long range interactions (see [Ru68]). Continued interest in such systems is related to the fact that these systems show up in a rather natural way within the so called thermodynamic formalism for dynamical systems (see [Ru78]). Thereby the transfer operator can be used for instance to construct invariant measures for such systems and to characterize their ergodic properties (see [Ba00]).
Another nice application of this method is to the theory of dynamical zeta functions (see [Ru92], [Ru94]). These functions can be interpreted as generating functions for the partition functions of the system constructed in complete analogy to the partition functions of lattice spin systems. It turned out that the transfer operator method had indeed been used already some time ago in the p-adic setup of zeta functions by B. Dwork (see [Dw60] or [Ro86]) who constructed such an operator to show rationality of the Artin-Weil zeta function for projective algebraic varieties over finite fields and proved in this way part of the Weil conjectures (see [We49]). More recently the method also yielded a completely new approach to Selberg's zeta function, which can also be viewed as a dynamical zeta function for the geodesic flow on surfaces of constant negative curvature (see [Ma91]). Indeed, the aforementioned Artin-Weil zeta function is nothing but the dynamical Artin-Mazur zeta function (see [ArMa65]) for the Frobenius map of the algebraic variety. Typically, such dynamical zeta functions can be expressed in terms of some kind of Fredholm determinant of the transfer operator, which therefore allows a spectral interpretation of the zeros and poles of these functions. The existence of such an interpretation is one of the challenging open problems for all arithmetic zeta functions of number theory and algebraic geometry (see [Be86], [Co96], [De99]). Obviously such a spectral interpretation is also closely related to another famous open problem for these zeta and more general L-functions, namely the general Riemann hypothesis: one expects that the zeros and poles of such functions are located on critical lines in the complex plane as one does in the special case of the well known Riemann zeta function. Presently it is not known whether such a conjecture makes sense also for general dynamical zeta functions which are, unlike the Selberg-or the Artin-Weil zeta functions, not related to arithmetics.
In the present paper we address this problem for the Ruelle zeta function of a certain subshift of finite type which in the physics literature has become known as the Kac-Baker model (see [Ka66]). M. Kac got interested in this model while trying to understand the mathematics behind the phenomenon of phase transitions in systems of statistical mechanics with weak long-range interactions like in the van der Waals gas (see [Ba61]). The model he considered is an Ising spin system on a 1-dimensional lattice with a 2-body interaction given by a finite superposition of terms decaying exponentially fast with the distance between the different spins. His real interest was certainly in a system with a continuous superposition of such exponentially decaying terms to model also interactions decaying only polynomially fast as is the case in the van der Waals gas. However his method did not allow him to treat this limiting case in a rigorous way.
We have chosen the Kac model since its dynamical zeta function can be understood rather well by the transfer operator method. On the other hand there seems to be no obvious connection of this zeta function to any arithmetic zeta functions for which a general Riemann hypothesis is known to hold.
There exist two rather different transfer operators for this model which allow to express its zeta function as Fredholm determinants of these operators (see [Ma80], [ViMa77], [Ka66]). Up to now, however, it was not known how these two operators are related to each other. Our investigations show that the Ruelle operator is basically equivalent to Kac's original transfer operator through a Segal-Bargmann transformation establishing a unitary map between the Hilbert space of square integrable functions on the real line, where the Kac operator acts, and the Fock space of entire functions on the complex plane square integrable with respect to a certain weight function, to which the Ruelle operator can be restricted. Since the Kac operator K β is a symmetric, positive definite trace class operator for positive β and has real spectrum also for negative β, and the Ruelle operator L β defines a family of trace class operators holomorphic in the variable β we can show that the zeta functions of a whole class of Kac models extend meromorphically to the entire complex β-plane and have infinitely many nontrivial zeros on the real line. For a special case of the parameters we can also show the existence of infinitely many "trivial" zeros of this dynamical zeta function located on a line parallel to the imaginary axis in the complex β-plane. Thus for this function an analogue of the Riemann hypothesis seems plausible. The present paper generalizes analogous results in [HiMa01] for the case of an interaction consisting of a single exponentially decaying term.
In detail the present paper is organized as follows: in Section 2 we recall the definition of the Kac-Baker models and derive their Ruelle transfer operators. Further, we show how the dynamical zeta function of these models can be expressed through Fredholm determinants of the Ruelle operators. In Section 3 we derive, basically following Gutzwiller (see [Gu82]), the Kac operator appropriate to our problem and show that the zeta function can be expressed for positive β also in terms of Fredholm determinants of this Kac operator. In Section 4 we show how the kernel of the Kac operator is related to a certain form of Mehler's formula for the Hermite functions which allows us to diagonalize an integral operator closely related to the Kac operator. In Section 5 we introduce the Fock spaces and the Segal-Bargmann transform and show how the Ruelle and Kac operators can be directly related to each other. There we show that for real β the two operators have the same spectrum and give explicit expressions relating the eigenfunctions of the two operators for nonvanishing eigenvalues. In Section 6 we derive the asymptotic behavior of the eigenvalues of the Ruelle operator for large positive and negative values of β and apply it to get the results on the location of poles and zeros of the zeta function on the real line. In Section 7 we give explicit expressions for the matrix elements of a modified Kac-Gutzwiller operator in the Hilbert space basis given by the Hermite functions which seem best suited for future numerical calculations.
restricted to the finite sublattice Z [n−1] = {0, 1, . . . , n − 1} is then given as When inserting the explicit form of φ one gets For β ∈ C the partition functions Z n (β) for the finite sublattices Z [n−1] with periodic boundary conditions are defined as where Per n denotes the set of configurations ξ ∈ F Z+ which are periodic with period n. That means ξ ∈ Per n if and only if ξ i+n = ξ i for all i ∈ Z + . Defining the shift τ : one has Per n = Fix τ n = ξ ∈ F Z+ : τ n ξ = ξ .
To the dynamical system F Z+ , τ one can associate the Ruelle zeta function Note that |U n (ξ)| ≤ n m l=1 λ l J l 1−λ l =: nc so that |Z n (β)| ≤ (2e |β|c ) n . Therefore the series defining ζ R converges in a neighborhood of (0, 0) in C 2 . We will show in Proposition 2.4 that ζ R can in fact be extended to a meromorphic function on C 2 .
To determine the analytic properties of this function one makes use of the transfer operator technique. Note first that the configuration space F Z+ is compact and metrizable with respect to the product topology. For each β ∈ C one can define the Ruelle transfer operator L β which will act on the space of observables of the lattice spin system, i.e. on C(F Z+ ), the space of continuous functions on F Z+ . In this framework one sets Inserting the explicit expression for U 1 (η) one finds where (σ, ξ) := η with η 0 = σ and η j = ξ j−1 for all j ∈ N. Generalizing the arguments for the case m = 1 in [Ma80] one introduces the map z = (z l ) l=1,..,m : F Z+ → R m defined by Since z l (σ, ξ) = σλ l + λ l z l (ξ) for all l, the operator L β leaves the space of functions z * (ϕ) := ϕ • z with ϕ ∈ C(R m ) invariant. Thus we obtain a factorization , which we will denote again by L β and which is of the form where Λ is the diagonal matrix with diagonal elements λ 1 , . . . , λ m and J = (J 1 , . . . , J m ).
Indeed, by the Ruelle-Perron-Frobenius Theorem (see [Zi00,Chap.4]) the iterates under L β of the constant function f (ξ) = 1 in C(F Z+ ) converge uniformly to the eigenfunction belonging to the leading eigenvalue of L β and hence this eigenfunction belongs to the space C(R m ). Therefore, when restricting the operator to the space C(R m ) one does not lose the leading eigenvalue which is the most important one from the physical point of view. The physically most satisfying operator is obtained in fact by restricting the domain of the operator still further.
¿From the form of the operator L β in the space C(R m ) we see that, if g is a holomorphic function on the polycylinder D = {z ∈ C m : |z l | < R l , l = 1, . . . , m} with R l > λ l 1−λ l , then also (L β g) (z) is such a function. In fact, much more is true: Denote by B(D) the Banach space of holomorphic functions on D which extend continuously to the closure D of D. Then, according to [Ma80, Appendix B], we have Proposition 2.1 L β : B(D) → B(D) is with respect to the parameter β ∈ C a holomorphic family of nuclear operators of degree 0 in the sense of Grothendieck (see [Gr55]). In particular all the L β are of trace class.
Proof. The defining equation (1) for the partition function Z n (β) can be rewritten as Using the fact that ξ ∈ Per n implies ξ i+n = ξ i for all i ∈ Z + one gets where σ n+i = σ i for all i. On the other hand the n-th iterate of the transfer operator L β from (4) acting on the Banach space B(D) is given by where λ i := (λ i 1 , . . . , λ i m ). We apply Lemma 2.2 to the maps ψ σ defined by for which the fixed points are given by The result is In view of the identity

But
Together with Proposition 2.1 this proves the following proposition.
Proposition 2.4 The Ruelle zeta function ζ R (z, β) defined in (2) for the Kac-Baker model with decay rates λ = (λ 1 , . . . , λ m ) ∈]0, 1[ m can be extended to a meromorphic family (ζ R (·, β)) β∈C of meromorphic functions on C via the formula Remark 2.5 The analytic properties in the variable z are well known for general lattice spin systems with exponentially fast decaying interactions. For m = 1 Proposition 2.4 was proved in [Ma80].

The Kac-Gutzwiller operator
In [Ka59] M. Kac found another operator whose traces are directly related to the partition functions Z n (β) of the Kac-Baker model. Kac did not work with periodic but open boundary conditions and hence his operator has to be modified a bit to give the partition functions we use here. In the case m = 1 M. Gutzwiller already derived this operator (see [Gu82]) and the case of general m 1 can be handled similarly. Just like Gutzwiller and Kac we start with an identity for Gaussian integrals known already to C. Cramér: For a positive definite (n × n)-matrix A and x ∈ R n we have (see [Cr46] or the appendix of [Fo89]) where B = A −1 .
Any periodic configuration ξ p ∈ Per n can be extended to a periodic configuration on the entire lattice Z with the same period which we again denote by ξ p . Now consider the (n × n)-matrix A (l) given by where we choose the constants γ l so that e −γ l = λ l with the coupling constants λ l of the Kac-Baker model. Using ξ p n = ξ p 0 , . . . , ξ p n−1 one finds (see [Gu82,§6] for this and the following results on matrix calculations) i.e. the left hand side is, up to the constant term given by the sum of the J l , just the interaction energy of the periodic configuration ξ p ∈ Per n for the Kac model.
The matrix B (l) = A (l) −1 has the following form That B (l) is indeed positive definite one can see as follows: for arbitrary x ∈ R n one finds A simple calculation (see also Proposition 4.1) shows that the following identity holds and hence for βJ l > 0 and γ l > 0 one finds x, B (l) x 0. Inserting (6) into formula (5) one calculates The change z (l) =: Performing the summation over all the periodic configurations ξ ∈ Per n then gives Inserting the explicit form of the matrix For β 0 we introduce the kernel function where ξ = (ξ 1 , . . . , ξ m ) ∈ R m and η = (η 1 , . . . , η m ) ∈ R m . We call the associated operator K β on L 2 (R m , dξ) defined by the Kac-Gutzwiller transfer operator. Note that the kernel of K β decreases fast enough to ensure that K β is of trace class with trace K β = R m K β (ξ, ξ)dξ. Calculating the trace of the iterates of K β and comparing the result to the above formula for the partition function Z n (β) after inserting (7) we find To simplify the expression for Z n (β) we introduce the rescaled Kac-Gutzwiller operator G β : In view of which is a simple reformulation of (9) we finally have shown the following proposition.
An argument similar to the one we used for the Ruelle operator L β in the proof of Proposition 2.4 now shows Since the zeta function ζ R (z, β) is meromorphic and its divisor for fixed β uniquely determined, it is not too difficult to see that at least for generic values of the parameters β > 0 and λ ∈]0, 1[ m the spectra of the two operators L β and G β have to be identical. This indeed has been shown in the case m = 1 already by B. Moritz in (see [Mo89]). We will show however (see Theorem 5.12) that the spectra of the two operators coincide for any real β and all parameters λ. Hence the Fredholm determinants det(1 − zG β ) and det(1 − zL β ) of the Kac-Gutzwiller operator and the Ruelle operator coincide on the real axis and extend to a holomorphic function in the entire β plane even if the operator G β contrary to the operator L β has itself no such analytic continuation to the entire β-plane.

Consider the operators
where m xj denotes the multiplication operator The Hermite functions h α ∈ L 2 (R m , dx) with α = (α 1 , . . . , α m ) ∈ N m 0 are given by (see [Fo89,p.51 The Hermite functions are known to be an orthonormal basis of the Hilbert space L 2 (R m , dx) and the following formula due to Mehler holds (see [Fo89]): where |w| < 1 and ℜ 2 1−w 2 > 0. ¿From the case m = 1 we then get for λ = (λ 1 , . . . , λ m ) with A simple calculation presented for m = 1 in [HiMa01] shows that the following proposition is true.
In Proposition 4.1 (iii) we used only the fact that Then the kernel K β (ξ, η) of the Kac-Gutzwiller transfer operator defined in (8) satisfies Then is an isomorphism of Hilbert spaces. Proof.
If K : L 2 R m , dξ → L 2 R m , dξ is given by an integral kernel K ξ, η then the induced operator as one easily verifies by a straightforward calculation using the transformation formula.
Then (a) the operator K • m √ a is an unbounded operator on L 2 (R m , dx) with kernel K x, y a(y). Proof.
(a) This follows from with integral operators K ′ and K ′′ given by the kernels .
If the kernel |K(x, y)| defines a bounded operator on L 2 (R m , dx), then also the operator K ′′ is bounded.
. . , J l ) and s = s 0 := 2 √ βπJ we choose the function a := cosh s 0 and for c = c 0 := 2 √ π(1, . . . , 1) we obtain Hence for the kernel K ′′ c 0 (x, y) of the operator K ′′ β,c 0 =: K ′′ c 0 , which does not depend on the variable β, one finds From the fact that the Hermite functions h α determine an orthonormal basis of L 2 (R m , dx) one concludes i.e., the h α are the complete set of eigenfunctions of the operator K ′′ c 0 with eigenvalue ρ α := 2λ In particular, K ′′ c 0 is bounded. We will need this result later on. Note that Lemma 4.3 yields the following commutative diagram for the Kac-Gutzwiller operator K β : The functions e πx 2 h α (x) are polynomials of degree |α| in x (see [Fo89,p.52]). In view of the estimate Hence all the h α are contained in L 2 R m , cosh R (ξ)dξ . This together with Mehler's formula in Proposition 4.1 shows immediately that the Kac-Gutzwiller operator K β with kernel K β (ξ, η) in (12) is symmetric and positive definite for β 0. In fact, where R 0,i = √ βJ i , 0 i m so that the eigenvalues of the operator K β are nonnegative for β 0.

Fock Space and Segal-Bargmann Transformation
For t > 0 consider the Hilbert space HL 2 (C m , µ t ) of entire functions F : C m → C with where µ t denotes the weight function The Bargmann transform B t : L 2 (R m , dx) → HL 2 (C m , µ t ) defined via determines a unitary operator in the two Hilbert-spaces (see [Fo89,p.47]).
In the following we are primarily interested in the case t = 1. In this case we denote the space HL 2 (C m , µ t ) simply by F m and call it the Fock space over C m . The transform B 1 is then denoted by B and hence where z * l = z l simply is the complex conjugate of z l . An orthonormal basis of Fock space F m is given by the functions where z α = Proof. Consider first F (z) = ζ α (z). In view of (13) and (14) we have But ζ α (z) is homogeneous of degree α i in z i and hence Therefore the claim is true for the basis elements ζ α of F m and hence also for any F ∈ F m .
For r ∈ R m define the translation operator τ r : L 2 (R m , dx) → L 2 (R m , dx) by and for s ∈ R \ {0} define the multiplication operator µ s : Since Z * l and µ s commute, it makes sense to write Proposition 5.2 For any α ∈ N m 0 we have Proof. For f ∈ C 1 (R m ) and 1 ≤ j ≤ m we calculate From this it follows immediately that Z * α • τ r = τ r • Z * + µ r α .
For the Hermite function h 0 (x) one now gets where we used the notation For s ∈ R m denote by exp s : R m → R the function defined by exp s (x) := e s·x . Then one has Proposition 5.3 For α ∈ N m 0 and s ∈ R m the following identities hold Iterating this calculation proves the first identity of the proposition. The second identity is proved in the same way.
where we have denoted the function e r·z also by exp r .
Proof. Using (15) we calculate Consider next the operator in F m induced from the multiplication operator m exp s in L 2 (R m , dx). Since the ζ α form a basis in the space F m the claim of the proposition is true.
Remark 5.7 According to Proposition 5.6 we can view B•m exp s •B −1 as an unbounded operator on F m which is defined on a dense linear subspace, namely the space of polynomial functions.
For the following we need the densely defined unbounded operator C s : F m → F m defined as ¿From Proposition 5.6 we deduce for polynomial F , so indeed C s is densely defined. Composing this unbounded operator with the bounded operator M λ of Proposition 5.1 we actually get a bounded operator on F m as the following proposition shows.
But we have M λ F (z) = 2λ 1 2 F (Λz) and hence the claim follows.
Then consider the induced operator Inserting the expressions for ν α and C s • M λ one gets Choose next the parameters s = s 0 and α = α 0 with s 0,i = 2 πβJ i and α 0,i = π βJ i , 1 ≤ i ≤ m.
We then get But this operator has up to the multiplicative factor Proof. Inserting the definitions of the operator C s and M λ we find Bf is an eigenfunction of the operator C s • M λ for the same eigenvalue ̺. For s = s 0 = 2(πβJ) 1 2 , however, the operator Hence one concludes that f ∈ L 2 (R m , dx) is an eigenfunction of the operator On the other hand given an eigenfunction F = F (z) of the operator L β : F Inserting the explicit expression for α 0 = 1 √ J1 , . . . , 1 √ Jm π β we therefore get Obviously the integral operator G β,c 0 depends on β holomorphically and hence defines a holomorphic family of trace class operators in the Hilbert space L 2 (R m , dx). Its Fredholm determinant det(1 − z G β ) hence is an entire function in the entire β plane coinciding with the Fredholm determinant of the Ruelle operator L β . To relate finally the operator m cosh s 0 • K ′′ c 0 and its eigenfunctions to those of the Kac-Gutzwiller operator K β with kernel K β (ξ, η) we use Proposition 5.11 For real β consider the two operators m cosh s 0 • K ′′ c 0 and m √ coshs 0 acting on the space L 2 (R m , dx). Then the following two statements hold: (ii) Conversely, if g ∈ L 2 (R m , dx) is an eigenfunction of the operator m √ coshs 0 with eigenvalue ρ = 0, then the function f = cosh s 0 · g is an eigenfunction of the operator m cosh s 0 • K ′′ c 0 in L 2 (R m , dx) for the same eigenvalue. Proof.
since cosh s 0 ≥ 1 and a simple calculation shows that this function is an eigenfunction of the operator m √ coshs 0 for the same eigenvalue.
If now g ∈ L 2 (R m , dx) is an eigenfunction of the operator m √ coshs 0 for the nonzero eigenvalue ρ, then a simple calculation shows that the function f := g cosh s ∈ L 2 (R m , 1 coshs dx) is an eigenfunction of the operator m cosh s 0 • K ′′ c 0 for the eigenvalue ρ = 0 and the above argument applied to φ = ρ(4π) − m 2 f shows that f ∈ L 2 (R m , dx).
This leads us to the main result of this section: Proof. We have seen already in Proposition 5.10 that the operators m l=1 (λ l exp βJ l ) − 1 2 m cosh s 0 • K ′′ c 0 and L β are conjugate via the map ν α −1 0 • B. In fact, we have the commutative diagram with c 0 = 2 √ π(1, . . . , 1) and s 0 = 2 √ βπJ . Furthermore Proposition 5.11 shows that for real β the operators m cosh s 0 •K ′′ c 0 and m √ coshs 0 have the same nonvanishing eigenvalues. But the last operator is conjugate to the operator K β with kernel K β (ξ, η) through the map R c 0 . Hence if f ∈ L 2 R m , dξ is an eigenfunction of the integral operator G β , then R c 0 f (x) = c is an eigenfunction of the integral operator m l=1 λ l e βJ l − 1 2 m √ coshs 0 for the same eigenvalue. But then is an eigenfunction of the operator 1 m l=1 (λle βJ l ) 1 2 · m cosh s 0 • K ′′ c 0 again for the same eigenvalue. Therefore m for yet again the same eigenvalue.
Inserting c 0 = (2 √ π, . . . , 2 √ π) one therefore finds for F (z) On the other hand, given an eigenfunction F ∈ F To discuss the zeros and poles of the Ruelle zeta function for the Kac-Baker models we need the following proposition.
Proposition 5.13 For β real the operators G β and L β have real spectrum.
Proof. For β real and f ∈ L 2 R m , dξ an eigenfunction of the operator G β with eigenvalue ̺ consider the scalar product On the other hand f coshR 0 , f = 0, since according to Mehlers formula in Proposition 4.1 for f an eigenfunction of G β . But then it is clear that ̺ must be identical to ̺ and hence real.

The Ruelle zeta function for the Kac-Baker model
Recall from Proposition 2.4 that the Ruelle zeta function ζ R (z, β) = exp ∞ n=1 z n n Z n (β) with Z n (β) the partition function for the Kac-Baker model can be written as with the operator L β : B(D) → B(D) given by where Λ is the m × m-diagonal matrix with the λ j , j = 1, . . . , m, as diagonal elements.
Remark 6.1 For β = 0 one then finds for the spectrum σ (L 0 ) of L 0 In fact, by induction over |α| one finds a polynomial eigenfunction for each 2λ α . In the case m = 1 one obtains 1, z, z 2 + λ 2 λ 2 −1 , z 3 + 3λ 2 λ 2 −1 z and so on. On the other hand, one can show that all eigenfunctions are polynomial, since the derivative of an eigenfunction for the eigenvalue ρ is again an eigenfunction for the eigenvalue ρλ −1 (in the case m = 1) so that infinitely non-vanishing derivatives of an eigenfunction would contradict the compactness of the operator. For m > 1 the argument is similar.
Hence the Ruelle function ζ R (z, β) for β = 0 has the form By induction on m one shows Indeed for m = 1 one has For m = n + 1, on the other hand, one calculates This result does not come as a surprise since for β = 0 the Ruelle zeta function for the Kac-Baker model is just the Artin-Mazur zeta function for the full subshift over two symbols determined by Bowen and Lanford in [BoLa75].
To determine the zeros and poles of the Ruelle function ζ R (β) := ζ R (1, β) on the real β-axis one has to investigate the zeros of the Fredholm determinants det (1 − λ α L β ). Obviously this function takes the special value −1 at the point β = 0. We know already from our discussion of the Kac-Gutzwiller operator G β that all its eigenvalues and hence also those of the Ruelle operator L β are real for real β and nonnegative for β 0. The poles and zeros of the Ruelle zeta function ζ R (1, β) can be located only at those values of β where one of the numbers (λ α ) −1 , α ∈ {0, 1} m belongs to the spectrum σ (L β ) of L β . For β = 0 we have seen that σ can be true only for finitely many β ∈ N m 0 since α ∈ {0, 1} m can take only finitely many values.
Furthermore, for infinitely many β ∈ N m 0 the eigenvalue 2λ β is strictly smaller than (λ α ) −1 for all α ∈ {0, 1} m . Hence, if we can show that infinitely many eigenvalues ̺(β) of L β tend to +∞ for |β| → ∞, then the Ruelle zeta function will have infinitely many "nontrivial" zeros and poles on the real β-axis at least for generic values of λ for which possible cancellations of zeros in the quotients of the different Fredholm determinants do not occur.
To derive the asymptotic behavior of the eigenvalues of the transfer operator for |β| → ∞ we remark first that the eigenspace of any eigenvalue ̺ of L β has a basis consisting of eigenfunctions which are also eigenfunctions of the operator P : B(D) → B(D) defined as We call the eigenfunctions with P f = f even and those with P f = −f odd. Consider then the two operators L ± β : B(D) → B(D) defined via respectively, The eigenfunctions of L + β and L − β are just the even, respectively odd, eigenfunctions of L β . We call the corresponding eigenvalues even, respectively odd. Since L β and P commute one therefore finds for the spectrum σ(L β ) of L β : Extending a result of B. Moritz (see [Mo89]) for m = 1 to the general case m ∈ N we find for the restricted range 0 λ i 1 2 for 1 i m of the parameters λ and arbitrary N 1: Theorem 6.2 (i) For β → ±∞ the N leading even eigenvalues ρ α of L β behave like (ii) For β → ±∞ the N leading odd eigenvalues ρ α of L β behave like Proof. Consider first the even eigenvalues and their asymptotic behavior for β → +∞ . Then Writing g(z) = exp βJ · (1 − Λ) −1 z u(z) we get for u the equation and hence Replacing z by λ + z and introducing the function h(z) := u(λ + z) one arrives at the equation Define an operator T + β;λ : Then T + β,λ is nuclear and its eigenvalues are just the numbers ̺ = ̺ exp −βJ · (1 − Λ) −1 λ , where ̺ is an eigenvalue of the operator L + β . If T λ : B D R → B D R denotes then the composition operator one finds for λ = (λ 1 , . . . , λ m ) with 0 < λ i < 1 2 for all 1 ≤ i ≤ m: since for these values of λ i one can find R i such that for all 1 i m But the eigenvalues of the operator T λ can be determined explicitly: they are given by the numbers λ α with α ∈ N m 0 . ¿From this the asymptotic behavior of the leading eigenvalues ̺ α of L + β and hence the asymptotic behavior of the leading even eigenvalues of L β follows immediately.
The proof of the behavior of the odd eigenvalues for β → +∞ follows the same line of arguments applied to the operator L − β instead of L + β .
For the asymptotic behavior of the even eigenvalues for β → −∞ consider the operator and the behavior of its eigenvalues for β → +∞. In this case we write an eigenfunction g as and get for u the equation Hence ̺ is an eigenvalue of the operator T − β;λ : is nuclear of order zero on the Banach space B D R . The spectrum of T λ , however, is given by the numbers (−λ) α with α ∈ N m 0 and hence the leading even eigenvalues ̺ α of the operator L β behave for β → −∞ like (−1) |α| λ α exp −βJ · (1 + Λ) −1 λ .
In exactly the same way one shows that the leading odd eigenvalues ̺ α of L β behave for β → −∞ Theorem 6.2 shows that both for β → ±∞ infinitely many eigenvalues of L β tend to +∞. Therefore the determinants det (1 − λ α L β ) with α ∈ {0, 1} m have infinitely many zeros in the real variable β. Therefore for generic λ with 0 < λ i < 1 2 for 1 i m the Ruelle zeta function has infinitely many poles and zeros on the real axis.
For special values of the parameters λ some eigenfunctions and their eigenvalues for the Ruelle operator L β are explicitly known. If λ = λ 0 with λ 0,i = λ 0 = 1 2 for all 1 i m, then the identity sinh 2βJ · 1 2 (z ± 1) = 1 2 e βJ·z±βJ·1 − e −βJ·z∓βJ·1 and the calculation e βJ·z sinh 2βJ · 1 2 (z + 1) + e −βJ·z sinh 2βJ · 1 2 (z − 1) = shows that the functions with P n (z) a polynomial homogeneous of degree n in all the variables z l which is invariant under all translations of the form z l → z l + c for real c, are indeed eigenfunctions of L β with eigenvalue The dimension of the space B(D) n of eigenfunctions defined by (19) for fixed n is the dimension of the above space of polynomials and will be calculated in the following proposition. Proof. Let f ∈ V m,n,c . For fixed z consider first the polynomial t → f (z − tc) − f (z). It is of degree less or equal n and has infinitely many zeros (for t ∈ Z), so it is zero. Thus we have The Ruelle zeta function ζ R (β) for our special choice of the parameter λ = λ 0 = ( 1 2 , . . . , 1 2 has the form as one easily checks. According to Proposition 6.3 the contribution of the eigenspaces B(D) n for the eigenvalues ̺ 1,n to the Ruelle function ζ R (β) is given by with k + r = n.
Remark 6.4 Note that for any β the B(D) n represent the entire eigenspaces for the eigenvalues ̺ 1,n . This is true for β = 0 by Remark 6.1 and Proposition 6.3. Indeed the eigenspace of the eigenvalue ̺ = 2λ r 0 of the operator L 0 has dimension m+r−1 m−1 as one can easily check. On the other hand we have is just the dimension of the eigenspace of the eigenvalue ̺ j,k (β) = ̺ j (β)λ k 0 . But for β = 0 the eigenvalues ̺ j (β) are given by 2λ j 0 . Since the eigenvalue ̺ 1,k (β) is holomorphic in the entire β-plane the dimension of its eigenspace does not depend on β (see [Kt66], p.68) and is given by m+k−2 m−2 .
We will check below that (20) can be reduced to the expression To see this one needs the following result on binomial coefficients.
Proposition 6.5 For all m 2, all r 0 and all l m − 1 the following identity holds where ⌊r⌋ is the largest integer less or equal than r.
The contribution (20) of the eigenspace B(D) n can be rewritten first as .
For the cases n = 0 and n = 1 one finds the factors 1 1−̺1 and 1 − λ 0 ̺ 1 . For n 2, however, Proposition 6.5 shows that all the other eigenvalues ̺ 1,n contribute the trivial factor 1 to the zeta function. This proves i.e. the reduction of (20) to (21). This factor leads to "trivial" zeros β n of the Ruelle zeta function with β n = log 2+2πin J where J = m l=1 J l . This conclusion can be drawn only if there are no cancellations with other eigenvalues, which at the moment we cannot exclude.
For the Kac-Baker model with this special choice of parameters λ = λ 0 there exists another Ruelle transfer operator which in a certain sense is simpler than L β . These parameters namely describe a Kac-Baker model with an interaction consisting of one exponentially decreasing term only and whose strength is just J = m l=1 J l . The Ruelle transfer operator for this model is L β g(z) = e βJz g( 1 2 + 1 2 z) + e −βJz g(− 1 2 + 1 2 z), where g is now holomorphic in a disc D in the complex plane C. The Ruelle zeta function in terms of the Fredholm determinants of this operator has the form ζ R (β) = det(1 − 1 2 L β ) det(1 − L β ) .
It is easy to see that the function g 1 (z) = sinh (2βJz) is an eigenfunction of the operator L β with eigenvalue ̺ 1 = exp(βJ). This eigenvalue leads exactly to the zero of the Ruelle function it is easy to see that the function f k is an eigenfunction of the operator L β with eigenvalue ̺ k . But with f k also the functions f k,n (z) := P n (z)f k (z), n = 0, 1, ... with P n a polynomial homogeneous of degree n in m variables and invariant under translations z l → z l + c for c ∈ R are eigenfunctions of L β with eigenvalue ̺ k,n = ̺ k λ n 0 as we have seen already for the eigenfunction f 1,n . Their degree of degeneracy is again given by m+n−2 n . The eigenfunctions g 1 (z) and f 1,n (z) are only special cases of this quite general connection between the eigenfunctions and eigenvalues of the two operators L β and L β . An argument similar to the case k = 1 shows that among all the eigenvalues ̺ k,n = ̺ k λ n 0 only ̺ k,n with n ∈ {0, 1} give nontrivial contributions to the Ruelle zeta function, which coincide exactly with the contributions of the eigenvalues ̺ k of the operator L β to the Ruelle zeta function for the model with λ = λ 0 .
Our discussion shows that there exist infinitely many trivial zeros of the Ruelle zeta function for the Kac model at least for this special parameter λ = λ 0 as long as there are not an infinite number of cancellations occuring among the eigenvalues, which one would certainly not expect. The structure of the zeros of the dynamical zeta functions of this family of Kac-Baker models of statistical mechanics hence seems to be very similar to the one well known for arithmetic zeta functions like the Riemann function. It would be interesting to determine their nontrivial zeros and poles and their distribution at least numerically.

Matrix elements of the Kac-Gutzwiller operator G β
For the numerical determination of the zeros of the Ruelle zeta function ζ R (β) = ζ R (1, β) the modified Kac-Gutzwiller operator G β seems to be best suited. This was realized already in the case m = 1 by Gutzwiller in (see [Gu82,§7]). For a numerical study of the eigenvalues and eigenfunctions of a closely related Ruelle operator see [Thr94]. It turns out that also for general m the matrix elements G α,δ of the operator G β can be determined explicitly in the basis of L 2 (R m , dξ) given by the Hermite functions h α . We start with the identity (see [Gu82,(41) Expanding the exponentials on the right hand side in σ i and ρ i leads to It is not too difficult to see that where Φ(·, ·; ·) denotes the confluent hypergeometric function. Therefore the matrix elements A α,δ = m l=1 A α l ,δ l are given by where Φ µ − M , µ + 1; −R 2 0 := m i=1 Φ µ i − M i , µ i + 1; −R 2 0,i . Finally we then find for the matrix elements G α,δ of the modified Kac-Gutzwiller operator G β This shows that G α,δ = 0 only iff |α + δ| = 0 mod 2 which generalizes the result for m = 1 by Gutzwiller to arbitrary m. Since G β h α , h δ = 0 if |α| mod 2 = |δ| mod 2 also for general m the Hilbert space L 2 (R m , dξ) can be decomposed into two subspaces invariant under the operator G β spanned by the Hermite functions h α with |α| = 0 mod 2 respectively |α| = 1 mod 2. Obviously this property of the operator G β corresponds to the fact that the Ruelle operator L β leaves invariant the subspaces of the Banach space B(D) spanned by the functions F = F (z) which are even, respectively odd, under the transformation z → −z. This follows from the fact that the Segal-Bargmann transform B maps the Hermite functions h α to the functions ζ α in (16) which under the above transformation z → −z have parity |α| mod 2 as one checks easily. One can use the representation of the operator G β in terms of the matrix G β to calculate its traces. For instance one finds Because the confluent hypergeometric function Φ(−n, 1; x) is identical to the Laguerre polynomial L n (x) = L 0 n (x) we get by using the generating function for these polynomials the result which coincides with trace L β .