On the analytic continuation of rank one Eisenstein series

by

WERNER MÜLLER ^{∗}

0. Introduction Let X be a hyperbolic surface of finite area and let Δ : C^{∞}(X) → C^{∞}(X) be the Laplacian of X. Regarded as linear operator in L^{2}(X) with domain C_{0}^{∞}(X), Δ is essentially selfadjoint. Let \overline{Δ} be the unique self-adjoint extension of Δ. The continuous spectrum of \overline{Δ} can be described explicitly by Eisenstein series (cf. [S], [H]). Let Γ ⊂ PSL(2, ℝ) be a discrete cofinite subgroup such that X = Γ\H where H denotes the upper half–plane. The Eisenstein series are attached to the parabolic fixed points of Γ. If a ∈ ℝ ∪ {∞} is a given parabolic fixed point of Γ, denote by Γ_{a} the stabilizer of a in Γ. Pick σ ∈ PSL(2, ℝ) such

that σa = ∞ and σΓ_{a}σ^{−1} = \{( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) | n ∈ ℤ\}. Then the Eisenstein series attached to

a is defined by

E_{a}(z, s) = ∑_{γ∈Γ_{a}\Γ} Im(σγz)^{s}, Re(s) > 1.

This series is absolutely convergent in the half–plane Re(s) > 1 and in this half-plane, it satisfies

(1)  ΔE_{a}(z, s) = s(1 − s)E_{a}(z, s).

One of the main features of the Eisenstein series is the fact that as functions of s, they admit meromorphic continuations to ℂ and satisfy a system of functional equations. Furthermore, the functions obtained by analytic continuation also satisfy (1) and, by restriction of the spectral parameter to the line Re(s) = 1/2, one obtains generalized eigenfunctions of the Laplacian. There exist various methods to achieve the analytic continuation of Eisenstein series [S], [F], [C]. The purpose of this paper is to present a new method which may also be used in other cases to construct generalized eigenfunctions.

We shall work in the setting of complete surfaces X of finite area with finitely many cusps. For simplicity, we suppose that X has a single cusp. Thus

X = X_{0} ∪ Z

where X_{0} is a compact surface with smooth boundary ∂X_{0} = S^{1} and

Z ≅ [1, ∞) × S^{1}, equipped with the metric ds^{2} = (dy^{2} + dθ^{2})/y^{2}.

* Universität Bonn, Mathematisches Institut, Beringstrasse 1, D-53115 Bonn, Germany. E-mail: mueller@rhein.iam.uni-bonn.de As above, the Laplacian Δ of X is essentially self-adjoint. The starting point for our method is the same as in [C]. As observed by Colin de Verdiere [C], for Re(s) > 1/2, s ∉ (1/2, 1], there is a unique solution of the equation ΔE(z, s) = s(1 − s)E(z, s) such that y^{s} − E((y, θ), s) is square integrable on Z. This solution can be constructed as follows. Pick f ∈ C^{∞}(ℝ) such that f(y) = 0 for y ≤ 1 and f(y) = 1 for y ≥ 2. Set h(s) = (Δ − s(1 − s))(f(y)y^{s}). Then we have h(s) ∈ C_{0}^{∞}(X) and

(2)  E(z, s) = f(y)y^{s} − (\overline{Δ} − s(1 − s))^{−1}(h(s)), Re(s) > 1/2, s ∉ (1/2, 1].

Since the continuous spectrum of \overline{Δ} equals [1/4, ∞) (see e.g. [Mü1], p.253), (2) is a meromorphic function in the half–plane Re(s) > 1/2. If X is hyperbolic, then E(z, s) coincides in Re(s) > 1 with the Eisenstein series attached to the unique cusp of X. In particular, (2) provides the analytic continuation of the Eisenstein series up to the critical line Re(s) = 1/2. To continue E(z, s) across the line Re(s) = 1/2, it is sufficient to continue the resolvent (\overline{Δ} − s(1 − s))^{−1} to a meromorphic function on ℂ. This can not be achieved in L^{2}(X) and, therefore, we replace L^{2}(X) by certain weighted L^{2} spaces L_{δ}^{2}(X), δ ∈ ℝ. For each δ > 0, we may regard (\overline{Δ} − s(1 − s))^{−1} as operator from L_{δ}^{2}(X) into L_{−δ}^{2}(X) and as operator in these spaces, the resolvent admits a meromorphic continuation to ℂ.

We shall now briefly describe the content of the paper. In §1 we construct an appropriate parametrix Q(s) for the resolvent in the half–plane Re(s) > 1/2 and then we continue analytically Q(s) to a meromorphic function with values in the bounded operators from L_{δ}^{2}(X) into L_{−δ}^{2}(X). In §2 we use the parametrix to continue the resolvent to a meromorphic operator valued function on ℂ. In §3 we construct and study the corresponding generalized eigenfunctions. In §4 we discuss ℚ–rank one locally symmetric spaces of finite volume and we show how our method can be extended to this case.

1. The parametrix In this § we shall construct a parametrix for (\overline{Δ} − λ)^{−1}. Let X_{1} = X_{0} ∪ ([1, 2] × S^{1}) and let M be a closed Riemannian surface such that X_{1} is isometrically embedded into M. Let Δ^{M} be the Laplacian of M. Since M is closed, the resolvent (Δ^{M} −λ)^{−1} is a meromorphic function of λ ∈ ℂ with simple poles at the eigenvalues of Δ^{M}. Let Q_{1}(x, x′, s) be the restriction of the kernel of the resolvent (Δ^{M} − s(1 − s))^{−1} to X_{1} × X_{1}. This is the interior part of the parametrix. For the exterior part consider the Laplacian on the cusp Z as linear operator in L^{2}(Z) with domain C_{0}^{∞}(Z). Impose Dirichlet boundary conditions at ∂Z and let Δ^{Z} be the corresponding self-adjoint extension. Put Q_{2}(s) = (Δ^{Z} − s(1 − s))^{−1}. This will be our parametrix at infinity.

Let ϕ_{2} ∈ C^{∞}(ℝ) be such that ϕ_{2}(u) = 0 for u ≤ 3/2 and ϕ_{2}(u) = 1 for u ≥ 2. We regard ϕ_{2} as function on [1, ∞) × S^{1} and extend it by zero to a smooth function on X. Let ϕ_{1} = 1 − ϕ_{2}. Pick ψ_{1} ∈ C_{0}^{∞}(X) such that ψ_{1} ≡ 1 on supp ϕ_{1} and let ψ_{2} ∈ C^{∞}(X) be such that ψ_{2} ≡ 1 on supp ϕ_{2} and ψ_{2} = 0 on X_{0}. Then we define the parametrix for (\overline{Δ} − s(1 − s))^{−1} in Re(s) > 1/2, s ∉ (1/2, 1] by

(3)  Q(s) = ϕ_{1}Q_{1}(s)ψ_{1} + ϕ_{2}Q_{2}(s)ψ_{2}.

It follows immediately from the definition that Q(s) extends to a bounded operator Q(s) : L^{2}(X) → H^{2}(X) where H^{2}(X) denotes the Sobolev space. To prove that Q(s) is indeed a parametrix for (\overline{Δ} − s(1 − s))^{−1} we have to investigate the resolvent of Δ^{Z}. First of all, it is easy to determine the spectral resolution of Δ^{Z}. Let L_{0}^{2}(Z) ⊂ L^{2}(Z) be the subspace of all cusp functions, that is, f ∈ L^{2}(Z) belongs to L_{0}^{2}(Z) if and only if

∫_{S^{1}} f(y, θ) dθ = 0 for almost all y ∈ [1, ∞).

The orthogonal complement of L_{0}^{2}(Z) in L^{2}(Z) is the space of functions on Z which are independent of θ. Thus L^{2}(Z) splits as

(4)  L^{2}(Z) = L_{0}^{2}(Z) ⊕ L^{2}([1, ∞); dy/y^{2}).

This decomposition is invariant under Δ^{Z}. Let Δ_{0}^{Z} be the restriction of Δ^{Z} to L_{0}^{2}(Z). Next consider −y^{2} d^{2}/dy^{2}, regarded as operator in L^{2}([1, ∞); dy/y^{2}) with domain C_{0}^{∞}((1, ∞)), and impose Dirichlet boundary conditions. Let A be the corresponding self-adjoint extension. Then with respect to the splitting (4), we have Δ^{Z} = Δ_{0}^{Z} ⊕ A. If we proceed as in the proof of Lemma 8.7 in [L-P], it follows that Δ_{0}^{Z} has pure point spectrum and, therefore, (Δ_{0}^{Z} − s(1 − s))^{−1} is a compact operator in L^{2}(Z). The spectrum of A is absolutely continuous and equals [1/4, ∞). Moreover, the kernel of (A − s(1 − s))^{−1} in the half–plane Re(s) > 1/2 can be explicitly determined. It is given by

(5)  q(y, w, s) = \frac{1}{2s − 1}(yw)^{1/2} \{ \begin{matrix}(y/w)^{1/2−s} − (yw)^{1/2−s}, & y > w; \\ (w/y)^{1/2−s} − (yw)^{1/2−s}, & w > y. \end{matrix}

Using the observation above together with (5), it is easy to see that

(6)  Q(s)(Δ − s(1 − s)) = Id + K(s), Re(s) > 1/2,

where K(s) is a compact operator in L^{2}(X). Hence, it is legitimate to call Q(s) a parametrix for (\overline{Δ} − s(1 − s))^{−1}.

Now we shall show that Q(s) admits a meromorphic continuation to ℂ. First, consider Q_{2}(s). To continue this operator analytically, we introduce weighted Sobolev spaces. Given δ ∈ ℝ, we define the L^{2} space with weight δ by

L_{δ}^{2}(Z) = {f : Z → ℂ | f measurable and ∫_{Z} |f(y, θ)|^{2}e^{2δy} dydθ < ∞}.

Let δ > 0. Then the following inclusions hold

(7)  L_{δ}^{2}(Z) ⊂ L^{2}(Z) ⊂ L_{−δ}^{2}(Z).

Recall that (Δ_{0}^{Z} −s(1−s))^{−1} : L^{2}(Z) → L^{2}(Z) is a meromorphic function of s ∈ ℂ. Hence, by composition with the inclusions (7), we get a meromorphic family of bounded linear operators

(Δ_{0}^{Z} − s(1 − s))^{−1} : L_{δ}^{2}(Z) → L_{−δ}^{2}(Z), s ∈ ℂ.

Next observe that the kernel of (A − s(1 − s))^{−1} which is given by (5), has an obvious analytic continuation. Let f ∈ L_{δ}^{2}(Z). Using (5), it follows that the integral

(8)  (q(s)f)(y) = ∫_{1}^{∞} q(y, w, s)f(w) \frac{dw}{w^{2}}

is absolutely convergent and q(s)f belongs to L_{−δ}^{2}(Z). In this way we get a meromorphic family, depending on s ∈ ℂ, of bounded linear operators

q(s) : L_{δ}^{2}(Z) → L_{−δ}^{2}(Z).

By (4), we obtain

Proposition 1. Let δ > 0. Then the resolvent (Δ^{Z} − s(1 − s))^{−1}, Re(s) > 1/2, extends to a meromorphic function s ∈ ℂ ↦ Q̃_{2}(s) of bounded linear operators from L_{δ}^{2}(Z) into L_{−δ}^{2}(Z).

To extend this result to Q(s), we introduce global weighted L^{2} spaces. Let ρ ∈ C^{∞}(X) be such that ρ(x) = 1 for x ∈ X_{0} and ρ(y, θ) = y for (y, θ) ∈ [2, ∞) × S^{1}. Given δ ∈ ℝ, let

L_{δ}^{2}(X) = {f : X → ℂ | f measurable and ∫_{X} |f(x)|^{2}e^{2δρ(x)} dx < ∞}.

For δ > 0, we have the inclusions

L_{δ}^{2}(X) ⊂ L^{2}(X) ⊂ L_{−δ}^{2}(X).

By construction, Q_{1}(s) has an obvious extension to a meromorphic function Q̃_{1}(s) on ℂ. Using Proposition 1, we get immediately

Proposition 2. Let δ > 0. Then Q(s) extends to a meromorphic function s ∈ ℂ ↦ Q̃(s) of bounded linear operators Q̃(s) : L_{δ}^{2}(X) → L_{−δ}^{2}(X).

2. Analytic continuation of the resolvent

Now consider the operator K(s) which is defined by the right hand side of (6). Let

Ω = {s ∈ ℂ | Re(s) > 1/2} − (1/2, 1]}.

Recall that K(s) : L^{2}(X) → L^{2}(X) is defined for all s ∈ Ω and s ∈ Ω ↦ K(s) is a holomorphic function of compact operators in L^{2}(X). Let L(L^{2}(X)) be the space of bounded linear operators in L^{2}(X).

Proposition 3. Let δ > 0. Then the holomorphic function s ∈ Ω ↦ K(s) ∈ L(L^{2}(X)) extends to a meromorphic function s ∈ ℂ ↦ K(s) of compact operators in L_{−δ}^{2}(X).

Proof. Let ψ′_{i} and ψ′′_{i} denote the first and second derivative, respectively, of ψ_{i}, i = 1, 2. By definition, we have supp ψ′_{i} ⊂ (1, 2) × S^{1}. Let D and V denote the operators in C^{∞}((1, 2) × S^{1}) which are defined by (Df)(v, θ) = v^{2}(∂f/∂v)(v, θ) and (V f)(v, θ) = v^{2}f(v, θ), respectively. Then, by definition of K(s), we have

(9)  K(s) = −ϕ_{1}Q_{1}(s)D ψ′_{1} − ϕ_{1}Q_{1}(s)V ψ′′_{1} − ϕ_{2}Q_{2}(s)D ψ′_{2} − ϕ_{2}Q_{2}(s)V ψ′′_{2}.

Now recall that Q_{1}(s) is the resolvent of the Laplacian Δ^{M} of the closed surface M. Therefore, by Rellich’s theorem, it follows that the first two terms on the right of (9) are meromorphic functions on ℂ with values in the compact operators on L^{2}(X). Furthermore, since ψ′_{1} and ψ′′_{1} have compact support, multiplication by ψ′_{1} and ψ′′_{1}, respectively, defines bounded linear operators L_{−δ}^{2}(X) → L^{2}(X). Using the inclusion L^{2}(X) ⊂ L_{−δ}^{2}(X), it follows that the first two terms on the right of (9) are meromorphic functions on ℂ with values in the compact operators in L_{−δ}^{2}(X).

To deal with the remaining terms we observe that

(10)  Q_{2}(s) = (Δ_{0}^{Z} − s(1 − s))^{−1} ⊕ q(s)

where q(s) is the operator defined by (8). This gives a corresponding splitting of the third and fourth term on the right of (9). Repeating the argument used above, it follows that ϕ_{2}(Δ_{0}^{Z} −s(1−s))^{−1}D ψ′_{2} and ϕ_{2}(Δ_{0}^{Z} −s(1−s))^{−1}V ψ′′_{2} are meromorphic families of compact operators in L_{−δ}^{2}(X). Furthermore, by (4), we get

∫_{0}^{∞} ∫_{1}^{2} |q(u, v, s)|^{2} \frac{dv}{v^{2}} e^{−2δu} \frac{du}{u^{2}} < ∞.

This implies that ϕ_{2}q(s)V ψ′′_{2} and ϕ_{2}q(s)Dψ′_{2} define both Hilbert–Schmidt operators in L_{−δ}^{2}(X) and hence, compact operators in L_{−δ}^{2}(X). q.e.d.

Lemma 1. There exists C > 0 such that

∥ K(s) ∥≤ \frac{C}{\sqrt{Re(s)}}, Re(s) > 2.

Proof. Let Re(s) > 2. Then it follows that s(1 − s) ∈ ℂ − ℝ^{+} and the first two terms on the right hand side of (9) can be estimated by a constant times

∥ (Δ^{M} − s(1 − s))^{−1}(Δ^{M} + 1)^{1/2} ∥≤ \frac{1}{dist(s(1 − s), ℝ^{+})} ≤ \frac{1}{\sqrt{Re(s)}}.

To estimate the two remaining terms on the right of (9), we use (10) to split these terms correspondingly. The contribution of (Δ_{0}^{Z} −s(1−s))^{−1} can be estimated as above. Finally, by (5) it follows that

∫_{1}^{∞} ∫_{1}^{2} |q(u, v, s)|^{2} \frac{dv}{v^{2}} \frac{du}{u^{2}} ≤ \frac{C}{(2Re(s) − 1)^{2}} .

This implies ∥ ϕ_{2}q(s)V ψ′′_{2} ∥≤ C Re(s)^{−1}. Similarly, we get ∥ ϕ_{2}q(s)Dψ′′_{2} ∥≤ C Re(s)^{−1/2} and the claimed estimate follows. q.e.d.

Lemma 1 implies that for Re(s) ≫ 0, Id+K(s) is invertible as operator in L^{2}(X). Hence, by (6) we get

(11)  (\overline{Δ} − s(1 − s))^{−1} = (Id + K(s))^{−1} ∘ Q(s), Re(s) > C,

for some constant C > 0. Here all operators act in L^{2}(X). Now pick δ > 0. It follows from (9) and Lemma 1 that ∥ K(s) ∥_{−δ}≤ C_{1}Re(s)^{−1/2}, Re(s) > 2. Thus, for Re(s) ≫ 0, Id+K(s) is invertible as operator in L_{−δ}^{2}(X). Now, Theorem 1 of [St] together with Proposition 3 imply that s ∈ ℂ ↦ (Id + K(s))^{−1} is a meromorphic function of bounded operators in L_{−δ}^{2}(X). Combining this result with Proposition 2, it follows that the right hand side of (11), regarded as operator of L_{δ}^{2}(X) into L_{−δ}^{2}(X), has an analytic continuation to ℂ. Put R(s) = (Id + K(s))^{−1} ∘ Q̃(s) and let L(L_{δ}^{2}(X), L_{−δ}^{2}(X)) be the space of bounded linear operators of L_{δ}^{2}(X) into L_{−δ}^{2}(X). Summarizing, we have proved

Theorem 1. Let δ > 0. Then the resolvent of the Laplacian \overline{Δ} extends to a meromorphic function s ∈ ℂ ↦ R(s) ∈ L(L_{δ}^{2}(X), L_{−δ}^{2}(X)).

3. Generalized eigenfunctions

Let f ∈ C^{∞}(ℝ) and suppose that f(y) = 0 for y ≤ 1 and f(y) = 1 for y ≥ 2. Let s ∈ ℂ. We consider f(y)y^{s} as a function on [1, ∞) × S^{1} = Z in the obvious way and then extend it by zero to a C^{∞} function on X. Let

h(s) = (Δ − s(1 − s))(f(y)y^{s}).

Then we have h(s) ∈ C_{0}^{∞}(X). In particular, h(s) belongs to every weighted L^{2} space L_{δ}^{2}(X), δ ∈ ℝ. Let δ > 0 and put

(12)  E(z, s) = f(y)y^{s} − R(s)(h(s))(z), s ∈ ℂ.

By Theorem 1, we have E(s) ∈ L_{−δ}^{2}(X), for all s ∈ ℂ which are not poles of R(s). In particular, E(., s) is locally integrable. Moreover, E(s) is a meromorphic function of s ∈ ℂ and for s ∈ Ω, E(z, s) is given by (2). Since E(z, s) is locally integrable, we can apply Δ in the distributional sense. From (2) follows that (Δ − s(1 − s))E(s) = 0 for s ∈ Ω. Since (Δ − s(1 − s))E(s) is a meromorphic function, it vanishes for all s ∈ ℂ. By elliptic regularity, it follows that E(s) ∈ C^{∞}(X) and we have proved

Theorem 2. The function E(z, s), defined by (12), has the following properties

1) E(z, s) is smooth in z ∈ X and meromorphic in s ∈ ℂ.

2) (Δ − s(1 − s))E(z, s) = 0, s ∈ ℂ.

3) For s ∈ Ω, f(y)y^{s} − E(z, s) is square integrable.

Since \overline{Δ} is self-adjoint, it follows that E(z, s) is uniquely determined by properties 1) - 3). Furthermore, 2) and 3) imply that on [1, ∞) × S^{1}, we have the following expansion

(13)  E((y, θ), s) = y^{s} + C(s)y^{1−s} + ∑_{n≠0} a_{n}(s)\sqrt{y} K_{s−1/2}(2π|y|y)e^{2πinθ}.

Here K_{w}(y) denotes the modified Bessel function which decays exponentially in y. Hence, if Re(s) < 1/2, then E(z, s) − C(s) f(y)y^{1−s} is square integrable. Put ω(s) = C(s)E(1 − s) − E(s) and let Re(s) > 1/2. Then we have ω(s) ∈ L^{2}(X) and (Δ − s(1 − s))ω(s) = 0. This implies ω(s) = 0, that is, C(s)E(1 − s) = E(s). Comparing the expansions (13) of the left and right hand side, we get

Theorem 3. For all s ∈ ℂ, the following functional equations hold

C(s)E(1 − s) = E(s) and C(s)C(1 − s) = 1.

By (2), we get \overline{E(z, s)} = E(z, \overline{s}), s ∈ ℂ, and therefore

(14)  \overline{C(s)} = C(\overline{s}).

Together with the functional equation, (14) implies that

|C(1/2 + iλ)|^{2} = C(1/2 + iλ)C(1/2 − iλ) = 1.

Hence, C(s) is regular on the line Re(s) = 1/2.

Remark. Our method has an obvious extension to the case of several cusps. A parametrix can be constructed as above by patching an interior parametrix to the exterior parametrix which is now given by a sum with contributions from each cusp. To obtain the analytic continuation of the resolvent, we may proceed in exactly the same way as in the case of a single cusp. If we carry out the construction of section 3 for each cusp individually, we obtain a set of generalized eigenfunctions attached to the various cusps.

4. Locally symmetric spaces of Q-rank one

In this section we shall describe how our method can be used to obtain the analytic continuation of Eisenstein series for all locally symmetric spaces of ℚ–rank one. Let G be a real reductive group of noncompact type, K a maximal compact subgroup of G, and Γ a rank one lattice in G such that the pair (G, Γ) satisfies the conditions of [OW, pp. 62-63]. Then X = Γ\G/K is a rank one locally symmetric space of finite volume. Examples are Hilbert modular varieties where G = SL(2, ℝ)^{m} and Γ is commensurable with Hilbert modular group SL(2, O_{F}) of some totally real number field F of degree m over ℚ [Hi]. As in the case of surfaces, we may work in the more general setting of manifolds X with cusps of rank one [Mü2]. Then X has a decomposition

X = X_{0} ∪_{i} W_{i}

where X_{0} is a compact manifold with boundary and W_{i}, i = 1, ..., m, are the ends of X. Moreover, each end W_{i} is isometric to a cusp in some locally symmetric space of rational rank one, which may depend on i. Thus, each cusp W_{i} is diffeomorphic to ℝ^{+} × Y_{i} where Y_{i}, i = 1, ..., k, are the components of ∂X_{0}. The metric on the cusp is described in [Bo, p.247]. We change coordinates by y = e^{r}. Then the structure is as follows: There is a fibre bundle

(15)  \begin{aligned}Γ_{i} ∩ N_{i}\N_{i} & → & Y_{i} \\ & & ↓ \\ & & Z_{i} \end{aligned}

where Γ_{i} ∩ N_{i}\N_{i} is a compact nilmanifold and Z_{i} is a compact locally symmetric space. Associated with this fibering is a local decomposition of the metric on ℝ^{+} × Y_{i}

(16)  ds^{2} = dr^{2} + dz_{i}^{2} + e^{−2b_{i}r}dn_{1}^{2}(z_{i}) + e^{−4b_{i}r}dn_{2}^{2}(z_{i})

where b_{i} > 0, dz_{i}^{2} is the metric on Z_{i} and dn_{1}^{2}(z_{i}), dn_{2}^{2}(z_{i}) are supported along the fibre Γ_{i} ∩ N_{i}\N_{i} over z_{i}. The volume element is given by

(17)  dvol = e^{−a_{i}r}dr dvol(z) dvol(n).

Here dvol(z) and dvol(n) are r–independent volume elements along the base and fibre, and a_{i} > 0.

Let Δ be the Laplacian of X. We may now proceed exactly as in §1 and construct a paramatrix Q(λ) for (\overline{Δ} − λ)^{−1}. For simplicity, we again assume that X has a single cusp W = ℝ^{+} × Y with associated fibration π : Y → Z. The metric on W determines via (16) a number b > 0 and the volume element (17) involves a > 0. Let Δ_{W} be the Dirichlet Laplacian of the cusp W and set Q_{2}(λ) = (Δ_{W} − λ)^{−1}, λ ∈ ℂ − ℝ^{+}, i = 1, ..., m. Furthermore, let

X_{1} = X_{0} ∪ ([1, ∞] × Y)

and let Q_{1}(λ) be the restriction to X_{1} of the resolvent of the Laplacian of a closed Riemannian manifold containing X_{1} as isometric submanifold. Picking appropriate patching functions φ_{i}, ψ_{i}, i = 1, 2, we put

Q(λ) = ∑_{i=1}^{2} φ_{i}Q_{i}(λ)ψ_{i}.

To study Q(λ), we have to analyze the Laplacian on the cusp W. Let φ be a function on ℝ^{+} × Y which is constant along the fibres of (15). We may identify φ with a function on ℝ^{+} × Z. Set dρ = e^{−ar}dr dvol(z) and let L^{2}(ℝ^{+} × Z; dρ) be the space of measurable functions on ℝ^{+} × Y which are square integrable with respect to the measure dρ. It follows from (17) that we may identify L^{2}(ℝ^{+} × Z; dρ) with a subspace of L^{2}(ℝ^{+} × Y). The orthogonal complement of this subspace consists of the space of cuspidal functions L_{0}^{2}(ℝ^{+} × Y). Given φ ∈ L^{2}(ℝ^{+} × Y), set

(18)  φ^{0}(x) = ∫_{Γ∩N\N} φ(nx) dn.

Then φ belongs to L_{0}^{2}(ℝ^{+} × Y) if and only if φ^{0}(x) = 0 for almost all x ∈ ℝ^{+} × Y. From this description we immediately obtain the orthogonal splitting

(19)  L^{2}(ℝ^{+} × Y) = L^{2}(ℝ^{+} × Z; dρ) ⊕ L_{0}^{2}(ℝ^{+} × Y).

Let Δ_{Z} denote the Laplacian of Z. If φ ∈ C_{0}^{∞}(ℝ^{+} × Z), then the structure of the metric (16) implies that

(20)  Δ_{W} (φ) = (−\frac{∂^{2}}{∂r^{2}} + a\frac{∂}{∂r} + Δ_{Z})φ

where a > 0 is determined by (17). Now one can easily see that (19) is an orthogonal splitting into invariant subspaces of Δ_{W}, that is,

Δ_{W} = Δ_{W,0} ⊕ Δ_{W,1}

where Δ_{W,0} is Friedrichs’ extension of the positive symmetric operator

−e^{ar} \frac{∂}{∂r} (e^{−ar} \frac{∂}{∂r} ) + Δ_{Z} : C_{0}^{∞}(ℝ^{+} × Z) → L^{2}(ℝ^{+} × Z; dρ)

and Δ_{W,1} is the restriction of Δ_{W} to the subspace L_{0}^{2}(ℝ^{+} × Y).

Let E_{W,1}(x, y, t) be the kernel of the heat operator exp(−tΔ_{W,1}). This kernel can be studied in the same way as in [Do]. The only difference is that we work with Dirichlet boundary conditions instead of Neumann boundary conditions. The estimates, however, remain the same. In particular, Theorem 7.2 of [Do] holds for E_{W,1}(x, y, t) as well. This implies that the heat operator exp(−tΔ_{W,1}) is trace class for t > 0. Thus, we proved

Lemma 2. The operator Δ_{W,1} has pure point spectrum.

By Lemma 2, (Δ_{W,1} − λ)^{−1} is a meromorphic function of λ ∈ ℂ. The nature of the spectrum of Δ_{W,0} is described by

Lemma 3. The operator Δ_{W,0} has pure absolutely continuous spectrum.

Proof. We define an isometry B : L^{2}(ℝ^{+} × Z; dρ) → L^{2}(ℝ^{+} × Z) by (Bφ)(r, z) = e^{−ar/2}φ(r, z). Then we have

(21)  Δ_{W,0} = B^{−1} ∘ (−\frac{∂^{2}}{∂r^{2}} + Δ_{Z}) ∘ B + \frac{a^{2}}{4} Id,

that is, B intertwines Δ_{W,0} and the Dirichlet Laplacian of ℝ^{+} × Z, shifted by a^{2}/4. The latter has pure absolutely continuous spectrum. q.e.d.

By (21), it is easy to determine the kernel of the resolvent of Δ_{W,0}. Let 0 = μ_{1} ≤ μ_{2} ≤ ⋯ be the sequence of eigenvalues, counted with multiplicity, of Δ_{Z}. Let {ϕ_{j}}_{j∈ℕ} be a corresponding orthonormalized sequence of eigenfunctions. For each j ∈ ℕ, let λ ↦ \sqrt{λ − μ_{j} − a^{2}/4} be the branch of the square root whose imaginary part is negative on ℂ − ℝ^{+}. Then for r ≠ r′, the kernel of (Δ_{W,0} − λ)^{−1} is given by

(22) Q_{W,0}((r, z), (r′, z′), λ) = e^{a(r+r′)} ∑_{j∈ℕ} \frac{e^{−i\sqrt{λ−μ_{j}−a^{2}/4} |r−r′|}}{\sqrt{λ − μ_{j} − a^{2}/4}} ϕ_{j}(z) ⊗ ϕ_{j}(z′).

The series is absolutly convergent for r ≠ r′. As in §1, this explicit description of the resolvent kernel can be used to show that Q(λ)(Δ − λ) = Id + K(λ), λ ∈ ℂ − ℝ^{+} where K(λ) is a compact operator in L^{2}(X).

To obtain an analytic continuation of Q(λ), we introduce again weighted L^{2} spaces. Since we use different coordinates at the cusps, we have to modify the definition of the weighted L^{2} spaces. Given δ ∈ ℝ, we set now

(23)  L_{δ}^{2}(W) = {f : W → ℂ | f measurable and ∫_{W} |f(r, y)|^{2} e^{2δr^{2}} drdy < ∞}.

Furthermore, we have to introduce the Riemann surface Σ to which the square roots λ ↦ \sqrt{λ − μ_{j} − a^{2}/4}, j ∈ ℕ, extend holomorphically. Let 0 = μ̃_{0} < μ̃_{1} < ⋯ be the strictly increasing sequence of eigenvalues of Δ_{Z}. To construct Σ we start with the Riemann surface Σ_{0} which corresponds to the square root \sqrt{λ − a^{2}/4}. This is a double covering π_{0} : Σ_{0} → ℂ with ramification at a^{2}/4. Next observe that ±\sqrt{λ − μ̃_{1} − a^{2}/4} lifts from ℂ − [μ̃_{1} + a^{2}/4, ∞) to Σ_{0} − π_{0}^{−1}([μ̃_{1} + a^{2}/4, ∞)). Taking two copies of Σ_{0}, cut along π_{0}^{−1}([μ̃_{1} + a^{2}/4, ∞)), and identifying pairs of cuts appropriately gives a four fold covering Σ_{1} of ℂ to which ±\sqrt{λ − a^{2}/4} and ±\sqrt{λ − μ̃_{1} − a^{2}/4} extend holomorphically. In this way we construct successively Riemann surfaces Σ_{j}. This construction leaves an increasingly large compact subset of the corresponding Riemann surface unchanged and, therefore, in the limit we get the desired Riemann surface Σ. Using (22) and (23), we obtain

Proposition 4. Let δ > 0 be given. Then the resolvent (Δ_{W} −λ)^{−1} extends from ℂ−ℝ^{+} to a meromorphic function λ ∈ Σ ↦ Q̃_{2}(λ) of bounded linear operators from L_{δ}^{2}(W) into L_{−δ}^{2}(W).

The rest is now completely analogous to §1. Let L_{δ}^{2}(X) be the global weighted L^{2} space defined in the obvious way. Then we obtain

Theorem 5. Let δ > 0. Then the resolvent (\overline{Δ} − λ)^{−1} extends from ℂ − ℝ^{+} to a meromorphic function λ ∈ Σ → R(λ) of bounded linear operators from L_{δ}^{2}(X) into L_{−δ}^{2}(X).

We can now use Theorem 5 to construct generalized eigenfunctions for Δ. Let ϕ ∈ ℂ^{∞}(Z) be an eigenfunction of Δ_{Z} with eigenvalue μ. Let \sqrt{λ − μ − a^{2}/4} be the branch of the square root whose imaginary part is positive on ℂ − ℝ^{+}. Furthermore, let f ∈ ℂ^{∞}(ℝ) such that f(r) = 0 for r ≤ 1 and f(r) = 1 for r ≥ 2. Set

h(ϕ, λ, x) = \{ \begin{matrix}f(r)e^{(a/2−i\sqrt{λ−μ−a^{2}/4})r} ϕ(π(y)), & if x = (r, y) ∈ ℝ^{+} × Y; \\ 0, & otherwise. \end{matrix}

It follows from the definition that (Δ−λ)h(ϕ, λ) ∈ C_{0}^{∞}(X) and, therefore, belongs to every weighted L^{2} space L_{δ}^{2}(X), δ ∈ ℝ. Let δ > 0 and put

E(ϕ, λ) = h(ϕ, λ) − R(λ)((Δ − λ)(h(ϕ, λ)).

By Theorem 5, E(ϕ, λ) is a meromorphic function of λ ∈ Σ with values in L_{−δ}^{2}(X). Furthermore, for λ ∈ ℂ − ℝ^{+}, which is identified with one sheet of the Riemann surface Σ, R(λ) = (\overline{Δ} − λ)^{−1}. Hence, it follows that

(24)  ΔE(ϕ, λ) = λE(ϕ, λ).

Since both sides are meromorphic functions, this equality holds for all λ ∈ Σ which are not poles. Moreover, for λ ∈ ℂ − ℝ^{+}, it follows that h(ϕ, λ) − E(ϕ, λ) is square integrable. Let E^{0}(ϕ, λ) ∈ ℂ^{∞}(ℝ^{+} × Y) be the constant term of E(ϕ, λ) defined by (18). Moreover, let E(μ′) be the eigenspace of Δ_{Z} corresponding to the eigenvalue μ′ ∈ Spec(Δ_{Z}). Then for λ ∈ ℂ − ℝ^{+}, we get the following expansion

(25)  \begin{matrix}E^{0}(ϕ, λ, (r, y)) = & e^{(a/2−i\sqrt{λ−μ−a^{2}/4})r}ϕ(π(y)) \\ + ∑_{μ′∈Spec(Δ_{Z})} e^{(a/2+i\sqrt{λ−μ′−a^{2}/4})r}S_{μμ′}(λ)(ϕ)(π(y)) \end{matrix}

where S_{μμ′}(λ) : E(μ) → E(μ′) is a linear operator which depends holomorphically on λ ∈ ℂ − ℝ^{+}. Obviously, each S_{μμ′} extends to a meromorphic function on Σ with values in Hom(E(μ), E(μ′)). Let λ ∈ ℝ^{+}. Let χ_{W} be the characteristic function of W. Since E(ϕ, λ) − χ_{W} E^{0}(ϕ, λ) is square integrable, it follows that

(26)  \begin{matrix}E(ϕ, λ, (r, y)) = & e^{(a/2−i\sqrt{λ−μ−a^{2}/4})r}ϕ(π(y)) \\ + ∑_{μ′≤λ−a^{2}/4} e^{(a/2+i\sqrt{λ−μ′−a^{2}/4})r}S_{μμ′}(λ)(ϕ)(π(y)) \\ + F (ψ, λ, (r, y)) \end{matrix}

where F (ϕ, λ) is square integrable on ℝ^{+} × Y. Lemma 2 implies that the wave operators W_{±}(\overline{Δ}, Δ_{W,0}) exist and are complete. Moreover, they can be expressed in terms of the generalized eigenfunctions (cf.Ch.8 of [Mü2]). Put

Ψ_{μμ′}(λ) = (\frac{λ − μ − a^{2}/4}{λ − μ′ − a^{2}/4})^{1/2}S_{μμ′}(λ), μ, μ′ ∈ Spec(Δ_{Z}).

Then it follows from the expression of the wave operators in terms of the generalized eigenfunctions that the scattering matrix is given by

Ψ(λ) = (Ψ_{μμ′}(λ)), μ, μ′ ≤ λ − a^{2}/4.

Now consider the case where X = Γ \ G/K is a finite volume locally symmetric space of rational rank one (or, more generally, we may assume that Γ is a rank one lattice in G). We shall briefly discuss Eisenstein series. For details see [HCh].

For simplicity, we assume again that X has a single cusp. Then there is a unique Γ–conjugacy class of rational rank one parabolic subgroups of G. Let P be a representative of this unique conjugacy class. Then P admits a Langlands decomposition

P = N · A · M

where N is nilpotent, M is reductive and A ≅ ℝ^{+} (cf. [Bo]). Moreover, P acts transitively on the symmetric space X̃ = G/K so that G = P ·K. Therefore, each g ∈ G can be written as

(27)  g = namk, n ∈ N, a ∈ A, m ∈ M, k ∈ K.

The discrete group Γ ∩ P is contained in NM and, under the canonical map NM ⟶ M, Γ ∩ P projects to a co–compact discrete subgroup Γ_{M} of M. Furthermore, we have P ∩ K = M ∩ K which we denote by K_{M}. Then K_{M} is a maximal compact subgroup of M and

Y = Γ ∩ P \ NM/K_{M} ⟶ Γ_{M} \ M/K_{M} = Z

is the fibration (15).

Let 𝔤 and 𝔪 be the Lie algebras of G and M, respectively, and let ℨ(𝔤) and ℨ(𝔪) be the centers of the universal enveloping algebras of 𝔤 ⊗ ℂ and 𝔪 ⊗ ℂ, respectively.

Recall that each D ∈ ℨ(𝔪) defines an invariant differential operator on M/K_{M} which we also denote by D. Similarly, each element of ℨ(𝔤) defines an invariant differential operator on G/K. For a given character χ: ℨ(𝔪) ⟶ ℂ, set

A(χ) = {ϕ ∈ C^{∞}(Z)|Dϕ = χ(D)ϕ for all D ∈ ℨ(𝔪)}.

This is a space of Γ_{M}–automorphic forms for M. Let ϕ ∈ A(χ). We identify ϕ with a C^{∞}–function on M which satisfies ϕ(γmk) = ϕ(m) for γ ∈ Γ_{M}, m ∈ M, and k ∈ K_{M}. Let s ∈ ℂ. Using (27), we define a function

φ_{s} ∈ C^{∞}(Γ ∩ P \ G/K)

by

φ_{s}(na_{r}mk) = e^{(s+a/2)r}ϕ(m),

where a is determined by (17). Then φ_{s} is well–defined and satisfies

φ_{s}(γgk) = φ_{s}(g) for γ ∈ Γ ∩ P, g ∈ G, k ∈ K.

Hence, φ_{s} can be identified with some function in C^{∞}(Γ ∩ P \ G/K). Put

(28)  E(ϕ, s, x) = ∑_{γ∈Γ∩P \Γ} φ_{s}(γx).

The series is absolutely convergent for Re(s) > a/2 and it is a common eigenfunction of ℨ(𝔤). In particular, E(ϕ, s) is an eigenfunction of Δ_{X}. Suppose that Δ_{Z}ϕ = μϕ. Then we have

Δ_{X}E(ϕ, s) = (−s^{2} + a^{2}/4 + μ)E(ϕ, s)

Furthermore, the constant term E^{∘}(ϕ, s) of E(ϕ, s) along the cusp takes the form

(29)  E^{∘}(ϕ, s(r, m)) = e^{(s+a/2)r}ϕ(m) + e^{(−s+a/2)r}(C_{χ}(s)ϕ)(m).

where C_{χ}(s): A(χ) ⟶ A(χ) is the intertwining operator. Moreover, E(ϕ, s) − χ_{W} E^{∘}(ϕ, s) is square integrable. Put

u(ϕ, s) = E(ϕ, −s^{2} + a^{2}/4 + μ) − E(ϕ, s), Re(s) > a/2.

Then it follows from (26) and (29) that u(ϕ, s) is square integrable. Since Δu(ϕ, s) = (−s^{2} + a^{2}/4 + μ)u(ϕ, s), we obtain

(30)  E(ϕ, s) = E(ϕ, −s^{2} + a^{2}/4 + μ), Re(s) > a/2.

Let π_{1}: ∑_{1} ⟶ ℂ be the Riemann surface with ramification at μ′+a^{2}/4, μ′ ∈ Spec (Δ_{Z})− {μ}, which is constructed as above. Note that ∑ covers ∑_{1} . Then the right hand side of (30) provides the analytic continuation of the Eisenstein series E(ϕ, s) to a meromorphic function E(ϕ, Λ) on ∑_{1} . Comparing (25) and (29), it follows that

S_{μ,μ′} = 0 for μ′ ≠ μ, and S_{μμ} = C_{χ} on A(χ).

For μ′ ∈ Spec(Δ_{Z}), let γ_{μ′} be the automorphism of order 2 of the covering π_{1}: ∑_{1} ⟶ ℂ, generated by a simple closed curve surrounding μ′ + a^{2}/4. Suppose that Re(s) > a/2. We indentify s wiht a point in the physical sheet of ∑_{1} ⟶ ℂ. If μ′ ≠ μ, then the constant term of E^{∘}(ϕ, γ_{μ′}(s)) is given by

E^{∘}(ϕ, γ_{μ′}(s), (r, m)) = e^{(s+a/2)r}ϕ(m) + e^{(−s+a/2)r}(C_{χ}(γ_{μ′}(s))ϕ)(m).

Arguing as above, it follows that E(ϕ, γ_{μ′}(s)) = E(ϕ, s) for all μ′ ∈ Spec(Δ_{Z}) − {μ}. Hence, E(ϕ, s) is a meromorhic function on ℂ. Now observe that γ_{μ}s = −s. If we proceed as above, we get the functional equations satisfied by the Eisenstein series. Summarizing, we have proved Theorem 6. Let ϕ ∈ C^{∞}(Z) be an eigenfunction of ℨ(𝔪) with character χ. Then the Eisenstein series E(ϕ, s), defined by (28), has a meromorphic continuation to ℂ. Moreover, the following functional equations hold

\begin{aligned}C_{χ}(s)C_{χ}(−s) & = Id \\ E(C_{χ}(s)ϕ, −s) & = E(ϕ, s). \end{aligned}

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