The Spectral Zeta Function for P.C.F. Self-similar Fractals

Simon Waddington

Mathematisches Institut, Bunsenstr. 3-5, D-37073 Göttingen.

June 2, 1996

Abstract

We study the spectral zeta function for the Laplacian, with Dirichlet boundary conditions, on p.c.f self similar fractals. We use this zeta function to give an alternative proof of an analogue of Weyl’s formula, obtained by Kigami and Lapidus [6], and additionally obtain precise error terms. Further, we derive an analogue of the Selberg Trace Formula, relating the eigenvalues of the Laplacian to lengths of closed orbits of a semiflow whose cross section is the fractal set.

The main results of this paper provide solutions to conjectures in [7] and [8].

0 Introduction

During the past several years, there has been increasing interest in studying analysis on fractal sets. Diffusion processes were first constructed on certain fractals as limits of random walks on finite graphs, approximating the fractal [2], [9]. This approach leads to a natural definition of the Laplacian as the infinitessimal generator of Brownian motion.

Later, Kigami [5] introduced an “analytical” approach, defining the Laplacian as a limit of a sequence of finite difference operators. This allowed the development of a general theory of harmonic analysis, modelled on classical harmonic analysis on Euclidean spaces. In particular, a version of the Dirichlet problem can be formulated and solved, and there are also analogues of Green’s functions and harmonic functions. So far this theory has been restricted to a special class of fractals, termed “post critically finite”. The canonical example in this class is the Sierpinski gasket.

One particularly interesting result is an analogue of Weyl’s asymptotic formula for the growth rate of the eigenvalues of the Laplacian, which was proved in [6]. The proof is based on an application of the Renewal Theorem from probability theory. In this paper, we adopt a more classical approach. We define an analogue of the Minakshisundaram-Pleijel zeta function for compact Riemannian surfaces [12]. An analysis of the analytic behaviour of this zeta function near its critical line of convergence yields the analogue of Weyl’s formula, together with explicit error terms. These error terms cannot easily be obtained using the original method.

Further, we prove an analogue of Weyl’s asymptotic estimate [11], and a result concerning the distribution of eigenvalues in remote intervals. Finally, we compute an explicit expression for the heat kernel of the Laplacian, as well as a general trace formula. This result can be regarded as an analogue of Selberg’s Trace Formula, which relates the eigenvalues of the Laplacian on a compact Riemannian surface to the geodesic flow on the unit tangent bundle of the manifold. Our formula relates the spectrum of the Laplacian to the lengths of closed orbits of a semiflow, whose cross section is the fractal set. The semiflow itself depends on the harmonic structure we define on the fractal.

The main results of this paper provide solutions to the conjectures proposed in section six of [7], as well as conjecture Q_{6} in [8].

1 The Laplacian on p.c.f. fractals

In this section, we give some background definitions and results which will be used later in the paper. For a more detailed expositions, together with proofs, see [5].

Let K be a compact, metrizable topological space and let T_{i} : K → K for i = 1, 2, …, N be continuous mappings, for some N ≥ 2.

Let Σ = {1, 2, …, N}^{IN∪{0}} be a space of one-sided infinite sequences of the symbols {1, 2, …, N}. The space Σ is a metric space, when endowed with the metric d, given by

d((x_{n}), (y_{n})) = ∑_{n=0}^{∞} \frac{δ(x_{n}, y_{n})}{2^{n}}

where

δ(x_{n}, y_{n}) = \{ \begin{matrix}0 , & if x_{n} = y_{n} \\ 1 , & otherwise. \end{matrix}

The shift map σ : Σ → Σ given by (σx)_{n} = x_{n+1} is a continuous surjective N-to-1 map. The inverse branches of σ are the maps σ_{i} : Σ → Σ for i = 1, 2, …, N given by σ_{i}x = ix. For each m ≥ 1, let Σ_{m} = {1, 2, …, N}^{m} be the collection of words of length m. For w = w_{0}w_{1} … w_{m−1} ∈ Σ_{m}, we define T_{w} : K → K by T_{w} = T_{w_{0}} ∘T_{w_{1}} ∘⋯∘T_{w_{m−1}} and K_{w} = T_{w}(K).

Definition 1 The pair (K, (T_{i})_{i=1}^{N}) is called a self-similar structure if the following two conditions hold:

(i) Each map T_{i} : K → K for i = 1, 2, …, N is injective.

(ii) There exists a continuous surjective map π : Σ → K such that π ∘ σ_{i} = T_{i} ∘ π for i = 1, 2, …, N.

For a self-similar structure (K, (T_{i})_{i=1}^{N}), define subsets of Σ, called the critical set C and the post-critical set P, by

C = π^{−1}( ∪_{i≠j} T_{i}(K) ∩ T_{j}(K))

and

P = ∪_{n≥1}σ^{n}(C)

respectively.

Definition 2 A self-similar structure (K, (T_{i})_{i=1}^{N}) is called post critically finite (p.c.f) if |P| < ∞.

We now define a sequence of finite sets approximating K. Let V_{0} = π(P), and for each m ≥ 1, define

V_{m} = ∪_{w∈Σ_{m}}T_{w}(V_{0})

and also let V_{∗} = ∪_{m≥0}V_{m}. It is easy to see that V_{m} ⊂ V_{m+1} for all m ≥ 0 and \overline{V_{∗}} = K. The set V_{0} plays the role of the potential-theoretic boundary of K, and will be denoted by ∂K. In the same spirit, we also define V_{m}^{0} = V_{m} \ V_{0} and K^{0} = K \ ∂K.

We now introduce two items of notation. Let U, V be sets.

(i) Let l(V) = {f : V → IR} and let L(U, V) denote the space of all linear operators l(U) → l(V). In the case U = V, we denote L(U, U) by L(U).

(ii) Let C(K) = {f ∈ l(K) : f is continuous on K}, and endow the space with the supremum norm ∥.∥_{∞}. Note that C(K) can be viewed as a subspace of l(V_{∗}).

Next we define finite difference operators on V_{0}.

Definition 3 We say that D ∈ H(V_{0}) if D ∈ L(V_{0}) and D satisfies the following properties:

(i) D^{t} = D

(ii) D is irreducible (i.e. for all (p, q) ∈ V_{0} × V_{0}, there exists (p_{n})_{n=1}^{k} with p_{n} ∈ V_{0} for n = 1, 2, …, k and p_{1} = p, p_{k} = q such that D(p_{n}, p_{n+1}) ≠ 0 for all n = 1, 2, …, k − 1.)

(ii) D(p, p) < 0 and ∑_{q∈V_{0}} D(p, q) = 0 for all p ∈ V_{0}.

(iii) D(p, q) ≥ 0 for all p ≠ q.

The operators in H(V_{0}) induce finite difference operators H_{m} ∈ L(V_{m}) as follows. Let r = (r_{1}, r_{2}, …, r_{N}) ∈ (0, ∞)^{N} and define a linear operator H_{m} ∈ L(V_{m}), for each m ≥ 1, by

H_{m} = ∑_{w∈Σ_{m}} \frac{1}{r_{w}} R_{w}^{t}DR_{w}

where R_{w} ∈ L(V_{m}, V_{0}) is defined by R_{w} = f ∘ T_{w}, and r_{w} = r_{w_{0}}r_{w_{1}} … r_{w_{m−1}} for w = w_{0}w_{1} … w_{m−1} ∈ Σ_{m}.

Definition 4 A function f ∈ C(K) is called harmonic if (H_{m}f)(p) = 0 for all p ∈ V_{m}^{0} and m ≥ 1.

Thus harmonic functions are continuous functions which satisfy a mean value property. In order to guarentee the existence of non-constant harmonic functions, we require a further restriction on the pair (D, r), which we will describe in Definition 1. Since H_{1} is a symmetric matrix, we may write

H_{1} = ( \begin{matrix}T & J^{t} \\ J & X \end{matrix} )  (1.1)

where T ∈ L(V_{0}), J ∈ L(V_{0}, V_{1}^{0}) and X ∈ L(V_{1}^{0}). It is also possible to show that the matrix X is invertible (see [5], Lemma 2.7).

Definition 5 The pair (D, r) ∈ H(V_{0}) × (0, ∞) is called a harmonic structure if there exists λ > 0 such that

D = λ(T − J^{t}X^{−1}J).

Further, a harmonic structure is called regular if r_{i} < λ for all i = 1, 2, …, N.

Henceforth, we shall assume that (D, r) is a regular harmonic structure. The natural discrete Dirichlet form associated to H_{m} is given by

E^{(m)}(f, g) = λ^{m}f^{t}H_{m}g

for f, g ∈ l(V_{m}).

By Corollary 6.14 in [5], for each f ∈ l(V_{∗}), the sequence (E^{(m)}(f|_{V_{m}}, f|_{V_{m}}))_{m=1}^{∞} is monotonic non-decreasing. Thus lim_{m→∞} E^{(m)}(f|_{V_{m}}, f|_{V_{m}}) exists provided we allow the value of the limit to be infinity.

Definition 6 A subspace F ⊂ l(V_{∗}) is defined by

F = {f ∈ l(V_{∗}) : lim_{m→∞} E^{(m)}(f|_{V_{m}}, f|_{V_{m}}) < ∞}

and a symmetric form E on F is given by

E(f, g) = E^{(m)}(f|_{V_{m}}, g|_{V_{m}}).

For k ≥ 0, let F_{k} be the subspace of F given by

F_{k} = {f ∈ F : f|_{V_{k}} = 0}.

Let E_{k} = E|_{F×F} denote the restriction of the symmetric form E to F_{k} ×F_{k}. (Frequently, we shall also refer to the restricted Dirichlet form as E.)

Now we introduce the class of Bernoulli probability measures on K. Let (μ_{1}, μ_{2}, …, μ_{N}) ∈ (0, 1)^{N} satisfy ∑_{i=1}^{N} μ_{i} = 1, and define a Borel probability measure μ on K by

μ(K_{w}) = μ_{w_{0}}μ_{w_{1}} … μ_{w_{m−1}}

for each w = w_{0}w_{1} … w_{m−1} ∈ Σ_{m} and m ≥ 1. (In fact it is possible to consider a more general class of measures [6].)

Proposition 1 The bilinear symmetric forms (E, F) and (E_{k}, F_{k}), for k ≥ 0 are local regular Dirichlet forms on L^{2}(K, μ). In particular, F_{k} ⊂ C(K) for all k ≥ 0.

Associated to the Dirichlet form E_{0} is a self adjoint operator called the Laplacian. In particular, there is a dense linear subspace D_{μ} of C(K) and an operator Δ_{μ} : D_{μ} → C(K) satisfying

E_{0}(f, g) = − ∫ fΔgdμ

for all f ∈ F_{0}. It is important to note that the definition of the Laplacian depends on μ, whereas the definition of E does not.

A function h ∈ F_{k} is called an eigenfunction of E_{k} with eigenvalue θ ∈ IR if

E_{k}(h, g) = − ∫ hgdμ

for all g ∈ F_{k}. Furthermore, h ∈ F_{k} is an eigenvalue of E_{k} with eigenvalue θ if and only if h ∈ D_{μ}, h|_{V_{k}} = 0 and h is an eigenfunction of −Δ with eigenvalue θ, (see [6] Proposition 5.2.)

By remarking that for each k ≥ 0, the natural inclusion map i_{k} : F_{k} → L^{2}(K, μ) is a compact operator (see [6], Lemma 5.4), it follows from Theorem 6.2 in [15] that the eigenvalues of E_{k} are countable, non-negative and of finite multiplicity. Further, there is a single accumulation point at infinity.

Let (θ_{n}^{(k)})_{n=1}^{∞} denote the eigenvalues of (E_{k}, F_{k}) ordered so that

0 ≤ θ_{1}^{(k)} ≤ θ_{2}^{(k)} ≤ ⋯ ↑ ∞

for all k ≥ 0. Define the eigenvalue counting function ρ_{k} for E_{k}, for each k ≥ 0, by

ρ_{k}(t) = ♯{n : θ_{n}^{(k)} ≤ t}

for any t ∈ IR.

The following lemma follows by applying of Corollary 4.7 in [6].

Lemma 1 For any k ≥ 0 and t ∈ IR,

ρ_{k+1}(t) ≤ ρ_{k}(t) ≤ ρ_{k+1}(t) + ♯(V_{0}).

Now define numbers γ_{i} ∈ (0, 1) for i = 1, 2, …, N by

γ_{i} = (\frac{r_{i}μ_{i}}{λ} )^{\frac{1}{2}}

where r = (r_{1}, r_{2}, …, r_{N}), λ > 0 are given by the harmonic structure and the μ_{i} are the weights associated to the measure μ.

The unique positive real number d, called the spectral dimension is defined by

∑_{j=1}^{N} γ_{j}^{d} = 1.

Define a function R : IR → IR by

R(t) = ρ_{0}(t) − ∑_{j=1}^{N} ρ_{0}(γ_{j}^{2}t).  (1.2)

We recall that any discrete subgroup of IR can be written in the form T ZZ, where T > 0 is the least positive element, and is called the generator.

The following analogue of Weyl’s asymptotic formula for the eigenvalues of the Laplacian was derived in [6] (Theorem 2.4).

Theorem 1 [6] The function R is bounded and continuous from the right. Further, the following statements hold.

(i) If the additive group ∑_{j=1}^{N} ZZ log γ_{j} is a dense subgroup of IR then

ρ_{0}(t) ∼ \frac{∫_{−∞}^{∞} e^{−dt}R(e^{2t})dt}{− ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}}t^{d/2} as t → ∞.

(ii) If ∑_{j=1}^{N} ZZ log γ_{j} is a discrete subgroup of IR with least positive generator T, then

ρ_{0}(t) = (G(\frac{log t}{2}) + o(1))t^{d/2}

where G is a periodic function with period T, given by

G(t) = T \frac{∑_{j=−∞}^{∞} e^{−d(t+jT)}R(e^{2(t+jT)})}{− ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}}.

Moreover, G is continuous from the right and bounded away from zero and infinity.

Remark 1

(i) In the statement of the theorem, we have used two standard pieces of notation. Firstly f(t) ∼ g(t) means that \frac{f(t)}{g(t)} → 1 as t → ∞, and secondly f(t) = o(t) means f(t) → 0 as t → ∞.

(ii) The fact that the function R is bounded and continuous from the right, which ensures that the infinite integral and summation converge.

(iii) One can also the eigenvalue problem with an analogue of the Von Neumann boundary conditions. In this case, identical asymptotic formulae for the eigenvalue counting function can be obtained.

(iv) In [8], a “volume” measure v on K was introduced. In particular, v is a welldefined, positive Radon measure. In case (i),

v(K) = \frac{∫_{−∞}^{∞} e^{−dt}R(e^{2t})dt}{− ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}}

and in case (ii),

v(K) = \frac{1}{T} ∫_{0}^{T} G(t)dt.

Of course, v depends on the associated harmonic structure and the measure μ.

2 Dynamical zeta functions

Let (K, (T_{i})_{i=1}^{N}) be a self-similar structure, and let 0 < γ_{i} < 1 for i = 1, 2, …, N be fixed and satisfy ∑_{j=1}^{N} γ_{j}^{d} = 1. Define a function ϕ : Σ → IR by

ϕ(w) = (γ_{w_{0}})^{2} if w = w_{0}w_{1} … ∈ Σ.

Then define a function ψ : K → IR by ψ(x) = ϕ(l(x)) where

l(x) = max{v ∈ Σ : v ∈ π^{−1}(x)}

and the maximum is with respect to the natural lexicographical ordering of words in Σ. (The ambiguity in the definition occurs at the points of the finite set π(C). Our choice definition of ψ on π(C) is purely arbitrary.)

Now define a space K^{ψ} by

K^{ψ} = {(x, t) ∈ K × IR : 0 ≤ t ≤ ψ(x)}/ ∼

where the equivalence relation ∼ identifies the points (x, ψ(x)) and (T_{j}^{−1}x, 0), where j is chosen so that j = max{k : x ∈ K_{k}}.

Define a semiflow Φ : X → X for t ≥ 0 locally by

Φ_{t}(x, s) = (x, s + t),

respecting the above identification. The semiflow Φ is called (topologically) weak mixing if whenever there exist F ∈ C(K^{ψ}, C^{|}^{|} ) and a ∈ IR such that

F ∘ Φ_{t} = e^{iat}F

for all t ≥ 0, then a = 0 and F is identically constant. The following elementary lemma lists some equivalent criteria.

Lemma 2 The following statements are equivalent:

(i) Φ is not topologically weak mixing.

(ii) The group ∑_{j=1}^{N} ZZ log γ_{j} is a discrete subgroup of IR.

(iii) The least periods of periodic orbits of Φ lie in a discrete subgroup of IR.

Let τ denote a generic closed orbit of Φ and let λ(τ) denote its least period. Since the set π(C) is finite, the set of periodic orbits of Φ which intersect π(C) is also finite. Let p(τ) denote the number of times the orbit τ intersects K × {0}.

Define formally a zeta function Z(s, z) of two coordinates by

Z(s, z) = ∏_{{τ \atop τ∩π(C)=∅}} (1 − e^{−(sd/2)λ(τ)+zp(τ)})^{−1}

where the symbol τ denotes a generic closed orbit of Φ. Formally, we have the identity

Z(s, z) = exp ∑_{{τprime \atop τ∩π(C)=∅}} ∑_{k=1}^{∞} \frac{1}{k}e^{−s(d/2)kλ(τ)+kzp(τ)}

where the summation is restricted to “prime” closed orbits. (A closed orbit τ is called prime if τ ≠ γ^{k} for any closed orbit γ and any k > 1.) Using our symbolic representation of K, we may write

Z(s, z) = exp ∑_{n=1}^{∞} \frac{1}{n} ∑_{{w∈Σn \atop τ_{π(\overline{w})}∩π(C)=∅}} e^{−s(d/2)ϕ^{n}(\overline{w})+nz}

where \overline{w} = www … ∈ Σ is the periodic extension of the finite word w ∈ Σ_{n}. Further,

ϕ^{n}(v) = ∑_{j=0}^{n−1} ϕ(σ^{j}v)

for each n ≥ 1 and v ∈ Σ, and τ_{x} denotes the set

τ_{x} = {x, T_{w_{n−1}}x, T_{w_{n−2}w_{n−1}}x, …, T_{w_{0}w_{1}...w_{n−1}}x}

assuming that T_{w_{0}w_{1}...w_{n−1}}x = x, and T_{w_{0}w_{1}...w_{j}}x ≠ x for all 0 ≤ j < n − 1. We may also define a function

Y (s, z) = exp ∑_{n=1}^{∞} \frac{1}{n} ∑_{w∈Σ_{n}} e^{−s(d/2)ϕ^{n}(\overline{w})+nz}.

The functions Y and Z do not coincide, due to the ambiguity at points of π(C). However, we may write

Y (s, z) = Z(s, z)E(s, z)  (2.1)

where

E(s, z) = exp ∑_{n=1}^{∞} \frac{1}{n} ∑_{{w∈Σn \atop τ_{π(\overline{w})}∩π(C)≠∅}} e^{−s(d/2)ϕ^{n}(\overline{w})+nz}.

In order to interpret E(s, z) as an Euler product, we need to introduce some extra “fictitious” orbits of Φ. By the semiconjugacy relation, we know that if x ∈ π(C) is periodic (i.e. T_{w}x = x for some w ∈ Σ_{n} and n ≥ 1) then each preimage w of x under π is periodic under σ. Thus τ = τ_{x} lifts to finitely many periodic orbits of σ. Let I(τ) denote this set. For β ∈ I(τ) and w ∈ β, define λ(β) = ϕ^{n}(v), where σ^{n}v = v, and n is chosen so that σ^{j}v ≠ v for 0 ≤ j < n. Then we may write

E(s, z) = ∏_{{τ \atop τ∩π(C)≠∅}} ∏_{β∈I(τ)} (1 − e^{−s(d/2)λ(β)+zp(β)})^{−1}

where both products are finite.

Let A_{s,z} denote the N × N matrix with elements

A_{s,z}(i, j) = γ_{j}^{ds}e^{z}

for s, z ∈ C^{|}^{|} . Formally, we may write

Y (s, z) = exp ∑_{n=1}^{∞} \frac{1}{n} ∑_{w_{0}w_{1}...w_{n−1}∈Σ_{n}} A_{s,z}(w_{0}, w_{1})A_{s,z}(w_{1}, w_{2}) … A_{s,z}(w_{n−1}, w_{0})

= exp ∑_{n=1}^{∞} \frac{1}{n}tr(A_{s,z})^{n}
= \frac{1}{1 − e^{z} ∑_{j=1}^{N} γ_{j}^{ds}} .

In particular, Y (s, z) is a nowhere vanishing analytic function for Re(s) > 1 and z in a neighbourhood of 0 (depending on s). By using the Perron-Frobenius Theorem for matrices and standard perturbation theory for linear operators, (see [13]), we may analyse the behaviour of Y (s, z) on the critical line Re(s) = 1.

Lemma 3 (i) If Φ is topologically weak mixing, Y (s, z) has a nowhere zero analytic extension to U \ {1}, where U is an open neighbourhood of {s : Re(s) ≥ 1}. Further,

H(s, z) = Y (s, z)(1 − e^{z} ∑_{j=1}^{N} γ_{j}^{ds})

has a nowhere zero analytic extension to s = 1, for |z| sufficiently small, depending on s.

(ii) If Φ is not topologically weak mixing, then Y (s, z) has a nowhere zero analytic extension to U \ \{1 + \frac{2πki}{dT} : k ∈ ZZ\} for |z| sufficiently small (depending on s but not on k). Further, H(s, z) has a nowhere zero analytic extension to the points \{1 + \frac{2πki}{dT} : k ∈ ZZ\}, for |z| small depending on s.

Logarithmic differentiation of Y (s, z) at z = 0 gives

η(s) = \frac{Y′(s, 0)}{Y (s, 0)} = \frac{∑_{j=1}^{N} γ_{j}^{ds}}{1 − ∑_{j=1}^{N} γ_{j}^{ds}} .

So we have

lim_{s→1}(s − 1)η(s) = lim_{s→1} \frac{s − 1}{1 − ∑_{j=1}^{N} γ_{j}^{ds}}
= \frac{1}{−d ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}} .

In the case that Φ is not topologically weak mixing, η(s) is simply periodic with period \frac{2πi}{dT}, that is formally,

η(s + \frac{2πki}{dT}) = η(s)

for all k ∈ ZZ.

We summarise these results in the following proposition.

Proposition 2 The function η(s) is analytic for Re(s) > 1, and

(i) if Φ is topologically weak mixing, then η(s) has an analytic extension to a neighbourhood of Re(s) > 1, except for a simple pole at s = 1, with residue

Res(η, 1) = \frac{1}{−d ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}} ,

(ii) if Φ is not topologically weak mixing, then η is simply periodic with period \frac{2πi}{T d}, and η is analytic for Re(s) > 1 − ε for some ε > 0, except for simple poles at the points 1 + \frac{2kπi}{dT} for k ∈ ZZ, with residue

−(d ∑_{j=1}^{N} γ_{j}^{d} log γ_{j})^{−1}.

By logarithmic differentiation of (2.1), we obtain the identity

η(s) = ∑_{{τprime \atop τ∩π(C)=∅}} ∑_{k=1}^{∞} p(τ)e^{−s(d/2)kλ(τ)} + E(s),  (2.2)

where

E(s) = \frac{∂}{∂z} log E(s, z)|_{z=0}

is an entire function.

3 Self-similarity of Dirichlet forms

In this section, we give a modified version of Proposition 6.2 in [6], overcoming a difficulty in the proof. (In [6], it is claimed that if f is an eigenfunction of (E, F_{1}) with eigenvalue θ, then f ∘ T_{i} is an eigenvalue of (E, F_{0}) with eigenvalue (\frac{r_{i}μ_{i}}{λ})θ. However, it is not shown that f ∘ T_{i} is not identically zero.)

The proof of Proposition 3 is based on the following simple lemma.

Lemma 4 ([6],Lemma 6.1) For f, g ∈ F,

E(f, g) = λ ∑_{i=1}^{N} \frac{1}{r_{i}} E(f ∘ T_{i}, g ∘ T_{i}).

Proof. This follows by a simple calculation. ▷◁

Proposition 3 (i) f is an eigenfunction of (E, F_{0}) with eigenvalue θ if and only if for each i = 1, 2, … N, the function g_{i} ∈ F_{1} defined by

g_{i}(x) = \{ \begin{matrix}f ∘ T_{i}^{−1}(x) , & if x ∈ K_{i} \\ 0  , & if x ∉ K_{i} \end{matrix}  (3.1)

is an eigenfunction of (E, F_{1}) with eigenvalue (\frac{λ}{r_{i}μ_{i}} ) θ.

(ii) Any eigenfunction g of (E, F_{1}) is expressable as a linear combination of the functions g_{i}.

(iii) ρ_{1}(t) = ∑_{i=1}^{N} ρ_{0} (\frac{r_{i}λ_{i}}{λ} t).

Proof. (i) Let f be an eigenfunction of (E, F_{0}) with eigenvalue θ ∈ IR and let g_{i} ∈ F_{1} for i = 1, 2, …, N be as in (3.1). By definition,

E(f, g) = θ ∫ fhdμ

for all f ∈ F_{0}. By Lemma 4, for all k ∈ F_{1}, we have that

E(g_{i}, k) = λ ∑_{j=1}^{N} \frac{1}{r_{j}} E(g_{i} ∘ T_{j}, k ∘ T_{j})
= \frac{λ}{r_{i}} E(f, k ∘ T_{i})
= \frac{λ}{r_{i}} θ ∫ f.k ∘ T_{i}dμ
= (\frac{λ}{r_{i}μ_{i}} ) θ ∑_{j=1}^{N} μ_{j} ∫ g_{i} ∘ T_{j}.k ∘ T_{j}dμ

= (\frac{λ}{r_{i}μ_{i}} ) θ ∫ g_{i}kdμ

and thus

E(g_{i}, k) = (\frac{λ}{r_{i}μ_{i}} ) θ ∫ g_{i}kdμ

for all k ∈ F_{1}. The converse argument is obvious.

(ii) It suffices to check that every eigenfunction g of (E, F_{1}) is expressable as a linear combination of functions of the form (3.1). So let g ∈ F_{1} be an eigenfunction of (E, F_{1}) with eigenvalue θ ∈ IR. Then

E(g, k) = θ ∫ gkdμ  (3.2)

for all k ∈ F_{1}. Define

h_{i}(x) = \{ \begin{matrix}g(x) , & if x ∈ K_{i} \\ 0 , & if x ∉ K_{i} \end{matrix}

for each i = 1, 2, …, N, and note that h_{i} ≢ 0 for at least one such i. Fix such an i, and let

H = ∑_{{j=1 \atop j≠i}}^{N} h_{j}.

Further, for any k ∈ F_{1} define k_{i} ∈ F_{1} by

k_{i}(x) = \{ \begin{matrix}k(x) , & if x ∈ K_{i} \\ 0 , & if x ∉ K_{i} \end{matrix}

Then by (3.2),

E(h_{i} + H, k_{i}) = θ ∫ (h_{i} + H)k_{i}dμ

By the local property of the Dirichlet form, (in particular, since the supports of H and k_{i} have disjoint interiors), E(H, k_{i}) = 0, and thus

E(h_{i}, k_{i}) = θ ∫ h_{i}k_{i}dμ.

Applying the local property again to show that

E(h_{i}, (k − k_{i})) = 0

we have

E(h_{i}, k) = θ ∫ h_{i}kdμ

for all k ∈ F_{1}. Defining f ∈ F_{0} by f = g ∘ T_{i} completes the proof of (ii).

(iii) This follows immediately from parts (i) and (ii). ▷◁

Proposition 4 For all t ∈ IR,

∑_{i=1}^{N} ρ_{0} (\frac{r_{i}μ_{i}}{λ} t) ≤ ρ_{0}(t) ≤ ∑_{i=1}^{N} ρ_{0} (\frac{r_{i}μ_{i}}{λ} t) + ♯(V_{0}).

Proof. The proposition follows by combining Lemma 1 and Proposition 3(iii). ▷◁

4 The spectral zeta function

Let θ_{1}, θ_{2}, θ_{3},… denote the eigenvalues of −Δ|D_{μ} (or equivalently the eigenvalues of (E_{0}, F_{0})), arranged in increasing order and counted according to multiplicity. We also define θ_{0} = 0.

Definition 7 The spectral zeta function ζ(s) is defined formally by

ζ(s) = ∑_{n=1}^{∞} θ_{n}^{−sd/2}.

The function ζ(s) is the analogue of the Minakshisundaram-Pleijel zeta function for the Laplacian on surfaces on constant negative curvature [12]. The zeta function is normalized so that its critical line of convergence is {s : Re(s) = 1}.

By Proposition 4, the function R defined in section one is bounded. Since the spectrum of −Δ is discrete and countable, R is also continuous from the right.

Lemma 5 Formally, the following identity holds:

ζ(s) = (∑_{n=0}^{∞} (R(θ_{n+1}) − R(θ_{n}))θ_{n+1}^{−sd/2}) (1 + η(s)).

where η(s) is given by (2.2).

Proof. First of all, note that

R(t) = ρ_{0}(t) − ρ_{1}(t)
= ♯{n ≥ 1 : θ_{n} ≤ t, θ_{n} is not an eigenvalue of (E, F_{1})}.

Thus in particular, R is a monotonic non-decreasing function. Let b = sup_{t∈R} R(t) and let β_{1}, β_{2}, …, β_{b} be all eigenvalues of (E, F_{0}) which are not eigenvalues of (E, F_{1}). By Proposition 3, the set of eigenvalues of (E, F_{0}) is equal to the set

⋃_{j=1}^{b} (\{β_{j}\} ∪ ⋃_{m≥1} \{\frac{β_{j}}{γ_{w}^{2}} : w ∈ Σ_{m}\})

where

γ_{w} = γ_{w_{0}}γ_{w_{1}} … γ_{w_{m−1}}

for w = w_{0}w_{1} … w_{m−1} ∈ Σ_{m}. Thus we obtain the following expression for ζ(s)

ζ(s) = ∑_{j=1}^{b} β_{j}^{−sd/2}(1 + ∑_{m=1}^{∞} ∑_{w∈Σ_{m}} γ_{w}^{sd})

= ( ∑_{n=0}^{∞} (R(θ_{n+1}) − R(θ_{n}))θ_{n+1}^{−sd/2})(1 + η(s))

as required. ▷◁

Proof of Theorem 1(i) By Proposition 2 and Lemma 5, we have

ζ(s) = (\frac{∑_{n=0}^{∞} (R(θ_{n+1}) − R(θ_{n}))θ_{n+1}^{−d/2}}{−d ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}} ) \frac{1}{s − 1} + J_{1}(s)

for Re(s) > 1, where J_{1}(s) has an analytic extension to an open neighbourhood of {s : Re(s) ≥ 1}. We now require the following well known Tauberian theorem of Wiener and Ikehara [16].

Proposition 5 Let α(t) be a monotonic non-decreasing function which is continuous from the right and satisfies α(0) = 0. Assume that for some A ≠ 0,

∫_{1}^{∞} t^{−s}dα(t) = \frac{A}{s − 1} + J_{2}(s)

for all Re(s) > 1, where J_{2}(s) is analytic in an open neighbourhood of {s : Re(s) ≥ 1}. Then

α(t) ∼ At as t → ∞.

By writing

ζ(s) = ∫_{1}^{∞} t^{−sd/2}dρ_{0}(t)

we may apply Proposition 5 to deduce that

ρ_{0}(t) ∼ (\frac{∑_{n=0}^{∞} (R(θ_{n+1}) − R(θ_{n}))θ_{n+1}^{−d/2}}{−d ∑_{j=1}^{N} γ_{j}^{d} log γ_{j}} ) t^{d/2} as t → ∞.

Finally,

\frac{1}{d} ∑_{n=0}^{∞} (R(θ_{n+1}) − R(θ_{n}))θ_{n+1}^{−d/2} = \frac{1}{d} ∑_{n=1}^{∞} R(θ_{n})(θ_{n}^{−d/2} − θ_{n+1}^{−d/2})
= \frac{1}{2} ∑_{n=1}^{∞} R(θ_{n}) ∫_{θ_{n}}^{θ_{n+1}} u^{−1− \frac{d}{2}} du
= \frac{1}{2} ∫_{0}^{∞} u^{−1− \frac{d}{2}} R(u)du

since R(u) = 0 for u ∈ [0, θ_{1})

= ∫_{−∞}^{∞} e^{−dt}R(e^{2t})dt

after the change of variables u = e^{2t}. This completes the proof the Theorem 1(i). ▷◁

Proof of Theorem 1(ii) First of all, let

Q = −(d ∑_{j=1}^{N} γ_{j}^{d} log γ_{j})^{−1}.

Then since η(s) is simply periodic with period \frac{2πi}{dT} and has simple poles at the points s = 1 + \frac{2πi}{dT}k for k ∈ ZZ, with residue Q, we may express η(s) as

η(s) = \frac{dT Q}{1 − e^{−dT (s−1)}} + J_{3}(s)

where J_{3}(s) is analytic for Re(s) > 1 − ε, for some ε > 0. Thus we may write

ζ(s) = ∑_{k=1}^{∞} (R(θ_{k+1}) − R(θ_{k}))θ_{k+1}^{−sd/2}(1 + dT Q ∑_{n=0}^{∞} e^{−dT ns}e^{T dn}) + J_{4}(s)

By carrying out a similar rearrangement as in part (i),

ζ(s) = dT Q ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k}))θ_{k+1}^{−sd/2} ∑_{n=1}^{∞} e^{−dT ns}e^{T dn} + J_{5}(s)
= dT Q ∑_{n=1}^{∞} e^{T dn} ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k}))(e^{2T n}θ_{k+1})^{−sd/2} + J_{5}(s)
= dT Q ∫_{1}^{∞} t^{−sd/2}dξ(t) + J_{5}(s)

where

ξ(t) = ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k})) ∑_{{n:e^{2T N} θ_{k+1}≤t}} e^{T dn}

which is monotonic increasing, continuous from the right and satisfies ξ(0) = 0. So we have

∫_{1}^{∞} t^{−sd/2}dρ_{0}(t) = T Qd ∫_{1}^{∞} t^{−sd/2}dξ(t) + J_{5}(s)

where J_{5}(s) is analytic for Re(s) > 1 − ε, for some ε > 0. We conclude that

\frac{ρ_{0}(t)}{t^{d/2}} = T Qd ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k})) ∑_{\{n:e^{2T n}θ_{k+1}≤t\}} e^{T dn} + O(t^{−ε})

= T Qd ∑_{k=1}^{∞} R(θ_{k}) ∑_{\{n: \frac{t}{θ_{k+1}} <e^{2T n}≤ \frac{t}{θ_{k}} \}} e^{T dn} + O(t^{−ε})

= T Qd ∑_{k=1}^{∞} R(θ_{k}) ∑_{{n:θ_{k}≤te^{−2T n}<θ_{k+1}}} e^{T dn} + O(t^{−ε})

= T Qd ∑_{k=1}^{∞} e^{T dk}R(te^{−2kT}) + O(t^{−ε})
= T Qd ∑_{k=−∞}^{∞} e^{T dk}R(te^{−2kT}) + O(t^{−ε})
= T Qd ∑_{k=−∞}^{∞} e^{−T dk}R(te^{2kT}) + O(t^{−ε})

from which the result follows by a simple rearrangement. ▷◁

We now derive an analogue of Weyl’s asymptotic estimate for the heat kernel (see [11], page 234). This result will be required in section six. Let Γ(s) denote the Gamma function.

Corollary 1 (i) If the additive group ∑_{j=1}^{N} ZZ log γ_{j} is a dense subgroup of IR then

lim_{t→0^{+}} t^{d/2}( ∑_{n=1}^{∞} e^{−θ_{n}t}) = Γ(\frac{d}{2} + 1)v(K).

(ii) If the additive group ∑_{j=1}^{N} ZZ log γ_{j} is a discrete subgroup of IR, with generator T, then

t^{d/2}( ∑_{n=1}^{∞} e^{−θ_{n}t}) = Γ(\frac{d}{2} + 1)(G(\frac{log t}{2}) + o(1)) as t → 0^{+}.

Proof. (i) For all t > 0, define

ϑ(t) = ∑_{n=1}^{∞} e^{−tθ_{n}}.

Applying the Mellin transform to ϑ gives formally

∫_{0}^{∞} t^{s−1}ϑ(t)dt = ∑_{n=1}^{∞} ( ∫_{0}^{∞} t^{s−1}e^{−tθ_{n}}dt)
= ( ∫_{0}^{∞} u^{s−1}e^{−u}du) ∑_{n=1}^{∞} θ_{n}^{−s}
= Γ(s)ζ(\frac{2s}{d}).

But

Γ(s)ζ(\frac{2s}{d}) = \frac{Γ(\frac{d}{2})v(K)}{s − \frac{d}{2}} + ψ_{1}(s)

where ψ_{1} is analytic in an open neighbourhood of \{s : Re(s) ≥ \frac{d}{2}\}. Further,

s ∫_{0}^{∞} t^{s−1}ϑ(t)dt = s ∫_{0}^{1} t^{s−1}ϑ(t)dt + ψ_{2}(s)
= ∫_{1}^{∞} u^{−s}dϑ(\frac{1}{u}) + ψ_{2}(s)

where ψ_{2}(s) is analytic for Re(s) > 0. Thus

∫_{1}^{∞} u^{−s}dϑ(\frac{1}{u}) = \frac{Γ(\frac{d}{2} + 1)v(K)}{s − \frac{d}{2}} + ψ_{3}(s)

where ψ_{3}(s) is analytic in an open neighbourhood of \{s : Re(s) ≥ \frac{d}{2}\}. Since u ↦ ϑ(\frac{1}{u}) is monotonic increasing and continuous, and

lim_{u→0^{+}} ϑ(\frac{1}{u}) = 0,

we can apply the Wiener-Ikehara Tauberian Theorem again to deduce the result.

(ii) By duplicating the argument for part (i), we have

∫_{1}^{∞} u^{−s}dϑ(\frac{1}{u})

= Γ(\frac{d}{2} + 1) ∑_{k=1}^{∞} (R(θ_{k+1}) − R(θ_{k}))θ_{k+1}^{−s}(1 + dT Q ∑_{n=0}^{∞} e^{−2T ns}e^{T dn}) + ψ_{5}(s)

where ψ_{5}(s) is analytic in an open neighbourhood of \{s : Re(s) ≥ \frac{d}{2}\}. The result then follows by the argument above for Theorem 1(ii). ▷◁

We now derive a result which describes the distribution of the eigenvalues of −Δ in remote intervals. The proof is a simple adaptation of the argument in pages 110-111 of [13].

Corollary 2 Suppose that ∑_{j=1}^{N} ZZ log γ_{j} is not a discrete subgroup of IR. The as k → ∞, \{θ_{n} : k < θ_{n} ≤ k + 1\} is distributed according to the probability density \frac{de^{dt/2}}{2(e^{d/2}−1)}.

Proof. Using Theorem 1(i), we have

∑_{θ_{n}≤t} θ_{n}^{ia} = ∫_{1}^{t} u^{ia}dρ_{0}(u)

and hence

∑_{θ_{n}≤t} θ_{n}^{ia} ∼ \frac{dt^{ia}}{d + 2ia}ρ_{0}(t).

Thus

∑_{t<log θ_{n}≤t+1} θ_{n}^{ia} ∼ v(K)e^{t( \frac{d}{2} +ia)}d(\frac{e^{\frac{d}{2} +ia} − 1}{d + 2ia})

and

∑_{t<log θ_{n}≤t+1} 1 ∼ v(K)e^{td/2}(e^{d/2} − 1)

as n → ∞. Thus

\frac{∑_{t<log θ_{n}≤t+1} θ_{n}^{ia}}{∑_{t<log θ_{n}≤t+1} 1} ∼ \frac{e^{tia}d(e^{\frac{d}{2} +ia} − 1)}{(e^{\frac{d}{2}} − 1)(d + 2ia)}.

Choose a = 2πk for k ∈ ZZ, so that

\frac{∑_{t<log θ_{n}≤t+1} θ_{n}^{2πki}}{∑_{t<log θ_{n}≤t+1} 1} ∼ \frac{de^{2πikt}}{d + 4πik}

or equivalently

\frac{∑_{t<θ_{n}≤t+1} θ_{n}^{2πki}}{∑_{t<θ_{n}≤t+1} 1} ∼ \frac{dt^{2πik}}{d + 4πik}

as n → ∞. In particular, the latter is the Fourier transform of \frac{de^{dt/2}}{2(e^{d/2}−1)} translated through an angle 2π log t. ▷◁

5 Error terms in Weyl’s formula

In this section, we will prove the following estimate on the rate of convergence in Theorem 1.

Theorem 2 (i) Suppose that for some 1 ≤ j ≤ N, the numbers {log γ_{i}/ log γ_{j} : i ≠ j} are rationally independent algebraic integers. Then there exists a real number α > 0 such that

ρ_{0}(t) = v(K)t^{d/2}(1 + O(\frac{1}{(log t)^{α}} )) as t → ∞.

(ii) Suppose that ∑_{j=1}^{N} ZZ log γ_{j} is a discrete subgroup of IR. Then there exists β > 0 such that

ρ_{0}(t) = t^{d/2}(G(\frac{log t}{2}) + O(t^{−β})) as t → ∞.

Part (ii) of Theorem 2 follows immediately from the calculation in section four, so it only remains to prove part (i).

Proof of Theorem 1(i). We first introduce the standard unweighted dynamical zeta function for the semiflow Φ, namely

χ(s) = Z(s, 0) = ∏_{τ} (1 − e^{−sdλ(τ)/2})^{−1}

normalised so that χ(s) has {s : Re(s) = 1} as its critical line of convergence. We may rewrite χ(s) as

χ(s) = \frac{1}{1 − ∑_{j=1}^{N} γ_{j}^{ds}}

by using the arguments of section two. It then follows that

η(s) = \frac{χ′(s)}{χ(s)}β(s)  (5.1)

where

β(s) = \frac{∑_{j=1}^{N} γ_{j}^{ds}}{−d ∑_{j=1}^{N}(log γ_{j})γ_{j}^{ds}} .  (5.2)

Under the hypotheses of the theorem, it is shown in [14] that χ′(s)/χ(s) has a pole free region of the form

\{s : Re(s) ≥ 1 − \frac{1}{(1 + Im(s))^{α}} , s ≠ 1\}.  (5.3)

Further, this then implies that

A(t) = t^{d/2}(1 + O(\frac{1}{(log t)^{β})}))  (5.4)

for some β > 0, where

\frac{χ′(s)}{χ(s)} = ∫_{1}^{∞} t^{−sd/2}dA(t).

By (5.1) and (5.2), the poles of β(s) occur precisely at the zeros of χ′(s). Thus η(s) has a pole-free region of the form (5.3). Finally, by Lemma 5, we also deduce that ζ(s) has a pole-free region of the form (5.3). Thus we may replace χ′(s)/χ(s) by ζ(s)/v(K) in the argument in [14] to deduce that

ρ(t) = v(K)t^{d/2}(1 + O(\frac{1}{(log t)^{β})}))

as required. ▷◁

6 A trace formula for the Laplacian

In this section, we prove a trace formula for the heat kernel of the Laplacian (Theorem 3) and deduce a more general trace formula (Theorem 4). These formulae relate the eigenvalues of the Laplacian to the closed orbits of the semiflow Φ, introduced in section two.

Given a function f : IR → IR, we define formally the Fourier transform f̂: IR → IR by

f̂(u) = ∫_{−∞}^{∞} f(t)e^{−itu}dt.

In accordance with common practice, given a periodic orbit τ we define N(τ) = e^{λ(τ)}. Define a set P by

P = \{τ : τ ∩ π(C) = ∅\} ∪ ⋃_{{τprime \atop τ∩π(C)≠∅}} I(τ).

So P is the set of all true and fictitious prime periodic orbits of Φ. The function R : IR → IR is defined in equation (1.2).

Theorem 3 For all t > 0,

∑_{n=1}^{∞} e^{−θ_{n}t} = t(R̂(−it) + ∑_{n=1}^{∞} ∑_{τ∈P} p(τ)N(τ)^{−2n}R̂( − itN(τ)^{−2n}))

Proof. Using the argument in the proof Lemma 5,

∑_{n=1}^{∞} e^{−θ_{n}t}

= ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k}))(e^{−tθ_{k+1}} + ∑_{n=1}^{∞} ∑_{w_{0}w_{1}...w_{n−1}∈Σ_{n}} exp { − tθ_{k+1}(γ_{w_{0}}γ_{w_{1}} … γ_{w_{n−1}})^{2}})

= ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k}))(e^{−tθ_{k+1}} + ∑_{n=1}^{∞} ∑_{{τ \atop τ∩π(C)=∅}} p(τ) exp \{ − tθ_{k+1}e^{−2nλ(τ)}\}

+ ∑_{n=1}^{∞} ∑_{{τ \atop τ∩π(C)≠∅}} ∑_{β∈I(τ)} p(β) exp \{ − tθ_{k+1}e^{−2nλ(β)}\})

= ∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k}))(e^{−tθ_{k+1}} + ∑_{n=1}^{∞} ∑_{τ∈P} p(τ) exp { − tθ_{k+1}N(τ)^{−2N}})  (6.1)

We shall analyse the two terms in (6.1) separately. Firstly,

∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k}))e^{−tθ_{k+1}} = ∑_{k=1}^{∞} R(θ_{k})(e^{−tθ_{k}} − e^{−tθ_{k+1}})
= ∑_{k=1}^{∞} R(θ_{k})( ∫_{θ_{k}}^{θ_{k+1}} te^{−tu}du)
= ∫_{0}^{∞} R(u)te^{−tu}du
= t ∫_{−∞}^{∞} R(u)e^{−tu}du

since R(u) = 0 for u ≤ 0,

= tR̂(−it).

Similarly, for the second term in (6.1), we have

∑_{k=0}^{∞} (R(θ_{k+1}) − R(θ_{k})) ∑_{n=1}^{∞} ∑_{τ∈P} p(τ) exp { − tθ_{k+1}N(τ)^{−2n}}

= ∑_{n=1}^{∞} ∑_{τ∈P} p(τ) ∑_{k=1}^{∞} R(θ_{k})( exp { − tθ_{k}N(τ)^{−2n}} − exp { − tθ_{k+1}N(τ)^{−2n}})
= ∑_{n=1}^{∞} ∑_{τ∈P} p(τ) ∑_{k=1}^{∞} R(θ_{k})( ∫_{θ_{k}}^{θ_{k+1}} exp { − tuN(τ)^{−2n}}du)tN(τ)^{−2n}
= t ∑_{n=1}^{∞} ∑_{τ∈P} p(τ)N(τ)^{−2n}( ∫_{0}^{∞} R(u) exp { − tuN(τ)^{−2n}}du)
= t ∑_{n=1}^{∞} ∑_{τ∈P} p(τ)N(τ)^{−2n}( ∫_{−∞}^{∞} R(u) exp { − tuN(τ)^{−2n}}du)
= t ∑_{n=1}^{∞} ∑_{τ∈P} p(τ)N(τ)^{−2n}R̂( − itN(τ)^{−2n})

which completes the proof of Theorem 1. ▷◁ Let β > 1 and δ > \frac{d}{2} + 1, and define a special class of C^{∞} functions on [0, ∞) by

Δ_{β,δ} = \{h ∈ C^{∞}([0, ∞)) : sup_{t∈[1,∞)} t^{β}|h(t)| < ∞, sup_{t∈(0,1)} \frac{|h(t)|}{t^{δ}} < ∞\}.

Introduce the Laplace transform Lh of h ∈ Δ_{β,δ} by

(Lh)(u) = ∫_{0}^{∞} h(t)e^{−ut}dt.

Then Lh is well defined and lies in C^{1}([0, ∞)). Theorem 4 Let h ∈ Δ_{β,δ} and g = Lh. Then

∑_{n=1}^{∞} g(θ_{n}) = − ∫_{−∞}^{∞} R(t)g′(t)dt − ∑_{n=1}^{∞} ∑_{τ∈P} N(τ)^{−2n} ∫_{−∞}^{∞} R(u)g′(uN(τ)^{−2n})dt.

Proof. By our assumption on h ∈ Δ_{β,δ} and Corollary 1, we may write

∑_{n=1}^{∞} g(θ_{n}) = ∫_{0}^{∞} h(t)( ∑_{n=1}^{∞} e^{−θ_{n}t})dt  (6.2)

where both sides converge uniformly. By applying Theorem 3, we obtain

∑_{n=1}^{∞} g(θ_{n}) = ∫_{−∞}^{∞} R(u)( ∫_{0}^{∞} th(t)e^{−tu}dt)du

+ ∑_{n=1}^{∞} ∑_{τ∈P} p(τ)N(τ)^{−2n} ∫_{−∞}^{∞} R(u)( ∫_{0}^{∞} th(t) exp { − tuN(τ)^{−2n}}dt)du

= − ∫_{−∞}^{∞} R(u)g′(u)du − ∑_{n=1}^{∞} ∑_{τ∈P} p(τ)N(τ)^{−2n}( ∫_{−∞}^{∞} R(u)g′(uN(τ)^{−2n})du.)

As R has compact support, both integrals converge absolutely. ▷◁

Finally, we can interpret the summation ∑_{n=1}^{∞} g(θ_{n}) as a trace in the following. Let H be a Hilbert space and let K be the ideal of compact linear operators on H. Given B ∈ K, let χ_{n} = χ_{n}(B) denote the characteristic values of \sqrt{B^{∗}B}, written in non-increasing order, and according to multiplicity.

Consider the ideal I defined by

I = {B ∈ K : ∑_{n=1}^{∞} χ_{n}(B) < ∞}.

which is just the ideal of trace class operators on H. Let

−Δ = ∫ θdE_{θ}

denote the spectral decomposition of −Δ. Then we may define an operator g(−Δ) by

g(−Δ) = ∫ g(θ)dE_{θ}.

By (6.2) and our assumptions on g,

| ∑_{n=1}^{∞} g(θ_{n})| < ∞

so we have that g(−Δ) ∈ I.

We summarise this discussion in the following proposition.

Proposition 6 For g ∈ L(Δ_{β,δ}), tr g(−Δ) is well defined, and

tr g(−Δ) = ∑_{n=1}^{∞} g(θ_{n}).

References

[1] M. T. Barlow, Harmonic analysis on fractal spaces, Asterisque 206 (1992), 345-368.

[2] M. T. Barlow, E. A. Perkins, Brownian motion on the Sierpinski gasket, Prob. Th. and Rel. Fields 79 (1988),542-642.

[3] M. Fukushima, Dirichlet Forms and Markov Processes, Kodansha and North-Holland, 1980.

[4] M. Fukushima, T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Analysis 1 (1992), 1-35.

[5] J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 355 No. 2 (1993), 721-755.

[6] J. Kigami, M. L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar sets, Comm. Math. Phys. 158 (1993), 93-125.

[7] M. L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture. In: Ordinary and Partial Differential Equations, eds. B. D. Sleeman, R. J. Jarvis, Pittman Research Notes in Mathematics vol. 289, Longman, London, 1993, pp 126-209.

[8] M. L. Lapidus, Analysis on fractals, Laplacians on self-similar sets, noncommutative geometry and spectral dimensions, Topological Methods in Nonlinear Analysis, Journal of the Julius Schauder Center 4 (1995), 137-195.

[9] T. Lindstrom, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 420 (1990).

[10] L. Malozemov, The integrated density states for the difference Laplacian on the modified Koch graph, Comm. Math. Phys. 156 (1993), 387-397.

[11] H. P. McKean, Selberg’s Trace Formula as applied to a compact Riemannian surface, Comm. Pure and Appl. Math. 14 (1972), 225-246.

[12] S. Minakshisundaram, A. Pleijel, Some properties of the eigenfunctions of the Laplace operator on Riemann manifolds, Can. J. Math. Vol. 1 (1949), 242-256.

[13] W. Parry, M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Asterisque 187-188 (1990).

[14] M. Pollicott, Error terms in “Prime Orbit Theorems” for locally constant suspended flows, Quart. J. Math Oxford (2) 41 (1990), 313-323.

[15] M. Reed, B. Simon, Methods of Modern Mathematical Physics, Vol. IV, Analysis for operators, New York, Academic Press, 1978.

[16] N. Wiener, The Fourier integral and certain of its applications, Cambridge University Press, Cambridge, 1967.