Dirichlet spectrum and Green function

In the first part of this article we obtain an identity relating the radial spectrum of rotationally invariant geodesic balls and an isoperimetric quotient $\sum 1/\lambda_{i}^{\rm rad}=\int V(s)/S(s)ds$. We also obtain upper and lower estimates for the series $\sum \lambda_{i}^{-2}(\Omega)$ where $\Omega$ is an extrinsic ball of a proper minimal surface of $\mathbb{R}^{3}$. In the second part we show that the first eigenvalue of bounded domains is given by iteration of the Green operator and taking the limit, $\lambda_{1}(\Omega)=\lim_{k\to \infty} \Vert G^k(f)\Vert_{2}/\Vert G^{k+1}(f)\Vert_{2}$ for any function $f>0$. In the third part we obtain explicitly the $L^{1}(\Omega, \mu)$-momentum spectrum of a bounded domain $\Omega$ in terms of its Green operator. In particular, we obtain the first eigenvalue of a weighted bounded domain in terms of the $L^{1}(\Omega, \mu)$-momentum spectrum, extending the work of Hurtado-Markvorsen-Palmer on the first eigenvalue of rotationally invariant balls.

A classical and important problem in Riemannian geometry is the study of the relations between the spectrum σ(Ω) and the geometry of Ω, see [7], [8]. In full generality, this is a difficult problem. A reasonable problem is, the study of the spectrum of rotationally invariant geodesic balls, i.e. balls with metrics invariant by rotations around the center. The spectrum σ(B(o, r)) of a rotationally invariant geodesic ball B(o, r) can be decomposed as a union of spectra σ l (B(o, r)) of a family of operators L l acting on smooth functions on [0, r] indexed by the eigenvalues of the sphere ν l = l(l + n − 2), l = 0, 1, . . ., this is, σ(B(o, r)) = ∪ ∞ l=0 σ l (B(o, r)), see [13, p. 41], [15,Chapters 7 & 8]. Our first results concern each spectrum σ l (B(o, r)) separately. More precisely, Theorem 2.5 & 2.7, states that λ −1 = c(l, n) · r 0 V (s)/S(s)ds, 0 < c(n, l) < 1, for l = 1, 2 . . . We also observe that a lemma due to Cheng-Li-Yau [14,Lemma 7] implies that if dim(V k ) ≥ 2 then the origin of B(o, r), belongs to the nodal set of every φ ∈ V k , see Theorem 2.2. To close this part of the article regarding the spectrum of rotationally invariant balls, i.e., Section (2), we construct examples of 4-dimensional non-rotationally invariant geodesic balls B(o, r) with the same spectrum σ 0 (B(o, r)) = σ 0 (B(o, r)) as the geodesic ballB(o, r) of the hyperbolic space H 4 (−1), see Example 2.3.
The momentum spectrum is intertwined with the spectrum of Ω. In the main result of this section, Theorem 5.2, we solve the hierarchy Dirichlet problem and explicitly give the momentum spectrum the in terms of the Green operator. In Section 6, the main result, see Theorem 6.1, gives necessary and sufficient conditions for the expansion of the Green function in L 2 -sense, is a orthonormal basis of L 2 (Ω, µ) formed by eigenfunctions. This result is the main tool to prove Theorems 2.5, 2.7, 3.1. In Section 7 we present the proofs of all results. Notation: in this article, the spectrum of Ω will be written either as a sequence of eigenvalues with repetition, according to their multiplicities, σ(Ω) = {0 ≤ λ 1 (Ω) < λ 2 (Ω) ≤ · · · } or as a sequence of eigenvalues without repetition σ(Ω) = {0 ≤ λ 1 (Ω) < λ 2 (Ω) < · · · }.

Spectrum of rotationally invariant balls
A Riemannian m-manifold is a rotationally invariant m-manifold, also called model manifold, with radial sectional curvature −G(r) along the geodesics issuing from the origin, where G : R → R is a smooth even function, is defined as the quotient space and R h is the largest positive real number such that h| (0,R h ) > 0. If R h = ∞, the manifold M m h is geodesically complete. The geodesic ball B(o, r) centered at the origin o = {0} × S m−1 / ∼ with radius r < R h , i.e. the set [0, r) × S m−1 / ∼, is rotationally invariant. The volume V (r) of the ball B(o, r) and the volume S(r) of the boundary ∂B(o, r) are given by V (r) = ω m r 0 h m−1 (s)ds and S(r) = ω m h m−1 (r), respectively, where ω m = vol(S m−1 ). The Laplace operator on B(o, r), expressed in polar coordinates, is given by To search for the Dirichlet eigenvalues λ of B(o, r) it is enough to seek smooth functions of the form u(t, θ) = T (t)H(θ) satisfying △u + λu = 0 in B(o, r) and u|∂B(o, r) = 0, see [13, p. 42]. This is equivalent to the following eigenvalue problems For each value ν l = l(l + m − 2), the set of all λ such that the equation h 2 )T = 0 has a non-trivial solution satisfying the initial conditions consist of an increasing sequence of positive real numbers {λ l,j } ∞ j=1 , (2.4) 0 ≤ λ l,1 < λ l,2 < · · · ↑ +∞.
Moreover, each λ l,i determine a 1-dimensional space of solutions, say, generated by T l,i . The sequence (2.4) is called the ν l -spectrum, without repetitions, of B(o, r), denoted by σ l (B(o, r)).
It is well known that the set of eigenvalues of the sphere S m−1 are given by ν l = l(l + m − 2), l = 0, 1, 2, . . . and their multiplicity of each ν l is given by Thus, there exists an orthonormal basis formed by eigenfunctions H l,1 (θ), . . . , H l,δ (θ) of the vector space V ν l = {φ : △ θ φ+ ν l φ = 0}. This implies that the set of functions is an orthonormal basis of the vector space {ψ : △ψ + λ l,i ψ = 0}. Therefore, the multiplicity of each eigenvalue λ l,i of the sequence 2.4, in the spectrum σ(B(o, r) is δ(l, m). Since all of the eigenvalues of B(o, r) are obtained in this procedure above, the spectrum of B(o, r), without repetitions, is the union of the ν l -spectrum σ l (B(o, r)), l = 0, 1, . . .
The details of this discussion can be found in [13, pp. 40-42]. Observe that the eigenvalues associated to ν 0 = 0 are those whose eigenfunctions are radial functions u(t, θ) = c · T (t). We call them radial eigenvalues. More generally we have the following definition.
Definition 2.1. Let B(p, r) ⊂ M be a geodesic ball of radius r > 0 and center p, (not necessarily rotational invariant). The radial spectrum σ rad (B(p, r)) of B(p, r) is formed by those eigenvalues λ k of L = −△| W 2 0 (B(p,r)) whose associated eigenspace V k contains a radial eigenfunction.
The radial spectrum of a general geodesic ball may be empty, however, if B(p, r) is rotationally invariant, then its radial spectrum is σ rad (B(p, r)) = σ l=0 (B(p, r)). It should be remarked that there are non-rotationally invariant geodesic balls with non-empty radial spectrum, see Example 2.3. A criteria due to S.Y. Cheng, P. Li and S.T. Yau states that, for rotationally invariant geodesic balls, in each eigenspace V k = {φ : △φ + λ k φ = 0} either φ(p) = 0 for all φ ∈ V k or V k has a radial eigenfunction and λ k ∈ σ rad (Ω), see [14,Lem. 7]. As observed, the spectrum of rotationally invariant geodesic balls of model m-manifolds can be decomposed into the union of subsets called ν l -spectrum associated to eigenvalues ν l = l(l + m− 2) of the sphere S m−1 . Moreover, the multiplicity of each eigenvalue of ν l -spectrum is the multiplicity of the eigenvalue ν l of the sphere. In particular, the radial eigenvalues, associated to the ν 0 = 0, has multiplicity one. If we take in consideration the Cheng-Li-Yau criteria then we have the following theorem.
h be a geodesic ball of radius r > 0 and centred at the origin. Let V k = {φ : △φ + λ k φ = 0} be the eigenspace associated to the eigenvalue λ k . If dimV k ≥ 2 then zero belongs to the nodal set of each φ ∈ V k . This property allows to see the heat kernel as a probability distribution in the space of random paths on M . More precisely, for x ∈ M and U ⊂ M , open subset, U p t (x, y)dν(y) is the probability that a random path emanating from x lies in U at time t. Thus if we have strict inequality, M p t (x, y)dν(y) < 1, then there is a positive probability that a random path will reach infinity in finite time t. This motivates the following definition. for k = 1, 2, . . .. Moreover, (2.6) holds for any geodesically complete Riemannian with discrete spectrum and more general operators. In particular, the operator [12, eq. 2.85]. Since the radial spectrum  [19], [20], [26] If the model M m h is stochastically incomplete then When the model manifold M m h is the Euclidean space R m , we can show that the "harmonic series"of the eigenvalues, (without repetitions), {λ l,i } ∞ i=1 = σ l (B(o, r)), for l = 0, 1, . . ., also converges. Theorem 2.7. Let B(o, r) be the geodesic ball of R m with radius r centred at the origin o . Let σ l (B(o, r)) = {λ l,1 (B(o, r)) < λ l,2 (B(o, r)) < · · · } be the ν l -spectrum of B(o, r) without repetition, l = 0, 1, . . . . Then If we let σ(B(o, r)) = {0 < λ 1 < λ 2 ≤ · · · } be the spectrum of be B(o, r) ⊂ R m , repeating the eigenvalues according to their multiplicities, in this case, σ(B(o, r)) is a union of δ(l, m)-copies of σ l (B(o, r)) for l = 0, 1, . . ., then we can compute the whole sum In the particular case of dimension m = 2 or dimension m = 3, However, this series diverges for m ≥ 4 in agreement with Theorem 6.1 equation (6.4). The convergence of the series is related to the fact that the Green function belongs to L 2 only in the case of m = 1, 2, 3. We should remark that the divergence in higer dimension is because of the multiplicity of the eigenvalues. In fact, if we consider the spectrum σ(B(o, r)) = ∪ ∞ l=1 σ l (B(o, r)) = {0 < λ 1 < λ 2 < · · · } without repetition then

2.3.
Example. Let {∂/∂x, ∂/∂y, ∂/∂z} be a globally defined non-zero vector fields on S 3 satisfying these conditions and let dx, dy and dz be their dual 1-forms. Consider, on Ω = [0, r] × S 3 / ∼, where (t, θ) ∼ (s, β) ⇔ t = s = 0 or t = s and θ = β, the following metric The coefficients near t = 0 and every θ ∈ S 3 fixed are given by a(t, Let Ω = B(o, r) be the geodesic ball with center at the origin o = {0} × S 3 / ∼ and radius r with respect to the metric ds 2 . It is clear that Ω is not rotationally symmetric. The Laplace operator △ ds 2 is given by Let Ω = ([0, r)×S 3 / ∼, can H 4 ) be the geodesic ball of radius r centered at the origin in the Hyperbolic space H 4 (−1). The Laplace operator △ can H 4 on Ω is given by Observe that △ ds 2 = △ can H 4 on the set of the smooth radial functions u(t, θ) = u(t) with u ′ (0) = 0, in particular, Ω and Ω has the same radial eigenfunctions and radial eigenvalues.
Thus, (Ω, ds 2 ) is an example of a non-rotationally symmetric geodesic ball with the same radial spectrum of a rotationally symmetric ball (Ω, can H 4 ). We should notice that the volume functions t → V (t) and t → S(t) are the same for both metrics, that can be checked directly, coherently with the identity (2.7). This example is a variation of the example [9, Examp. 2] which, by its turn, was inspired by the example of G. Perelman in [33].

Spectrum of extrinsic balls of minimal submanifolds
Let ϕ : M → R n be a proper and minimal immersion of a complete m-dimensional Riemannian manifold into R n . Let Ω r = ϕ −1 (B(0, r)) be the extrinsic ball of radius r. It was proved in [6] and [14] that the first Dirichlet eigenvalue of Ω r is bounded below as where c m is a constant depending only on the dimension of M and λ 1 (B(0, r)) is the first Dirichlet eigenvalue of the ball of radius r in the Euclidean m-space R m . This inequality can be read as where C m is a constant depending only on the dimension. For a non-totally geodesic minimal submanifold we will give lower and upper bounds for this series in terms of the volume vol(Ω r ), of the radius r and the dimension m = 2, 3. We have the following result.
If m = 2, 3, then and B m = e 4/m 16π 2 are constants, ω m is the volume of the geodesic ball of radius 1 in R m and ζ(4/m) = ∞ k=1 1 then M has finite number of ends {End 1 , · · · , End k }, see [2,27]. Moreover, the function r → vol(Ω r ) ω m r m is increasing, [31,32] and where E is related with the finite number of ends of M in the following way: where I i is the geometric index of the end End i , see [2,27,34]. If m = 2, 3 then

Weighted Green operator and the Dirichlet spectrum
A weighted manifold is a triple (M, ds 2 , µ) consisting of a Riemannian manifold (M, ds 2 ) and a measure µ with positive density function ψ ∈ C ∞ (M ), i.e., dµ = ψdν, where dν is the Riemannian density of (M, ds 2 ). Let Ω ⊂ M be a relatively compact open subset with smooth boundary ∂Ω = ∅ and consider the weighted Laplace operator △ µ : , the space of smooth functions with compact support on Ω, defined by The weighted Laplace operator is densely defined and symmetric with respect to L 2 (Ω, µ)-inner product, but it is not self-adjoint. As in the classic case, we may consider the Sobolev spaces W 1 0 (Ω, µ) as the closure of C ∞ 0 (Ω) with respect to the norm and W 2 0 (Ω, µ), formed by those functions u ∈ W 1 0 (Ω, µ) whose weak Laplacian △ µ u exists and belongs to L 2 (Ω, µ), i.e.
is a self-adjoint, non-negative definite elliptic operator and its spectrum, denoted by σ(Ω, µ), is a discrete increasing sequence of non-negative real numbers {λ k } ∞ k=1 ⊂ [0, ∞), (counted according to multiplicity), with lim k→∞ λ k = ∞, see [22,Thm.10.3]. The weighted Laplace operator △ µ extends the classical Laplace operator △, in the sense that if the density function ψ ≡ 1 then △ µ = △. Let (Ω, µ) be a weighted bounded open subset with smooth boundary ∂Ω = ∅ of a Riemannian weighted manifold (M, ds 2 , µ). The (weighted) Green operator G Ω : L 2 (Ω, µ) → L 2 (Ω, µ) is given by where p Ω t (x, y) is the heat kernel of the operator L = −△ µ | W 2 0 (Ω,µ) and is the Green function of Ω. The Green operator is a bounded self-adjoint operator in L 2 (Ω, µ) and it is the inverse of L, i.e., G Ω = L −1 . Thus for any [22,Thm. 13.4]. Applying G Ω to the equation we obtain that the i-th eigenvalue λ i (Ω) of Ω is given by λ i (Ω) = u/G Ω (u)· The difficulty to know precisely the i th -eigenvalue applying the Green operator is that one needs to know an i th eigenfunction to start. For simplicity of notation, if no ambiguity arises, we will suppress the reference to Ω in the gadgets associated to Ω. Thus, g and G will be, respectively, the Green function and the Green operator of the operator L = −△ µ | W 2 0 (Ω,µ) . Our main result in this section is that in order to obtain the first Dirichlet eigenvalue, assuming λ 1 (Ω) > 0, one picks a positive function f ∈ L 2 (Ω, µ) and compute the limit where u 2 = Ω |u| 2 dµ is the L 2 (Ω, µ)-norm and G k = k−times G • · · · • G. More generally, let f ∈ L 2 (Ω, µ) and let ℓ be the smallest positive integer such that Ω f (x)u i (x) dµ(x) = 0 for i = 1, 2, . . . , ℓ − 1 and Ω f (x)u ℓ (x)dµ(x) = 0, where {u ℓ } is an orthonormal basis of L 2 (Ω, µ) formed by eigenfunctions u ℓ with eigenvalues λ ℓ . Then the ℓ theigenvalue is given by where ℓ is the first positive integer such that, Moreover, In particular, for any positive f ∈ L 2 (Ω, µ), In addition, if f ∈ L 2 (Ω, µ) satisfying (4.4) and denoting by f 1 the function defined by then, And   , r)) and as k → ∞. With φ l ∈ ker(△ + λ l ), λ l = λ l (B(o, r)) is l th radial eigenvalue, where l is the first integer such that  (B(o, r)) we have as k → ∞.
To show how efficient Theorem 4.2 is, we let B(o, r) be the geodesic ball centered at the origin o and radius r in rotationally symmetric manifold M n h . Consider the following two maps: and the following family of functions arising from the φ 0 = 1 (4.14) Applying Theorem 4.2 we can obtain a subset of the radial spectrum, namely, , where j 0,k is the k th -zero of the Bessel function J 0 . The following table shows the T j (φ i ), j = 1, 2, 3, 9 and i = 0, 1, 2. Observe that T 9 (φ 1 ) agrees with j 2 0,1 to the 5 th -decimal place. 3. This bootstrapping argument using the Green operator in Theorem 4.1 was discovered, and applied to a particular kind of functions, by S. Sato [35] to obtain explicit estimates for the first eigenvalue of spherical caps of S 2 (1). It was applied by Barroso-Bessa [4] to obtain estimates for the first eigenvalue of balls in rotationally symmetric manifolds.
The solution φ 1 (x) is the mean time for the first exit of Ω of a Brownian motion t → X t in Ω, with X 0 = x, see [20].
The number A 1 (Ω) is called the torsion rigidity of Ω because, when Ω ⊂ R 2 , A 1 (Ω) is the torque required for a unit angle of twist per unit length when twisting an elastic beam of uniform cross section Ω, see [25], [29] and [30].
Observe that if Ω = B(o, r) is a rotationally invariant geodesic ball then the solutions φ k (x) = φ k (|x|) of (5.1) are radial functions since φ 1 is radial. In [25], A. Hurtado, S. Markvorsen and V. Palmer, showed that the first eigenvalue of a rotationally invariant ball can be given in terms of the momentum spectrum. They proved that and φ ∞ (r) = lim k→∞ φ k (r)/φ k (0) is a radial first eigenfunction. In our next result we solve explicitly the problem ( Let φ k be the sequence of functions given by the boundary problem (5.1) and G the △ µ -Green operator of Ω. Then Hence the exit moment spectrum is related to the Dirichlet spectrum as follows where a i = Ω u i (x)dµ(x) and {u i } is an orthonormal basis of L 2 (Ω, µ) formed by eigenfunctions of L = −△ µ | W 2 0 (Ω,µ) . Corollary 5.3. Let (Ω, ds 2 , µ) be weighted bounded open subset of a Riemannian weighted Riemannian manifold manifold (M, ds 2 , µ) with smooth boundary ∂Ω = ∅. Then, . repeated accordingly to their multiplicity. Let {u k } ∞ k=1 be an orthonormal basis in L 2 (Ω, µ) such that each function u k is an eigenfunction of L with eigenvalue λ k . In this basis, the heat kernel p Ω t (x, y) of the operator L admits an expansion This series converges absolutely and uniformly in the domain t ≥ ǫ, x, y ∈ Ω for any ǫ > 0 as well as in the topology of C ∞ (R + × Ω × Ω), see the details in [22,Thm. 10.13]. If we could integrate the heat kernel expansion (6.1) in the variable t we would obtain that It is known that identity (6.2) holds in the sense of distributions, see [22, exercise 13.14] and we may ask whether this identity holds in stronger topology, for instance in L 1 (Ω × Ω, µ) or L 2 (Ω × Ω, µ)? It is shown in [22,Exercise 13.7] that g Ω (x, y) ∈ L 1 (Ω × Ω, µ), however, g Ω (x, ·) does not need to belong to L 2 (Ω, µ). Indeed, let Ω = B R 4 (1) ⊂ R 4 to be the geodesic ball of the 4-dimensional Euclidean space R 4 , with its canonical metric, of radius r = 1 and center at the origin 0. The Green function of Ω is given by which obviously does not belong to L 2 (B R 4 (1)). Here ω 4 is the volume of the unit 3-sphere in R 4 . On the other hand, if Ω = B R (1) is the geodesic ball of radius r = 1 in the real line R then the Green function is g Ω (x, y) = (|x − y| − x · y + 1)/2 which clear is in L 2 (Ω × Ω). We start showing that a necessary and sufficient condition to g Ω ∈ L 2 (Ω × Ω, µ), for any measure µ, is that dim(M ) = 1, 2, 3 and in these dimensions identity (6.2) holds in L 2 (Ω × Ω, µ). Precisely, we have the following result.
where the series converges in L 2 (Ω × Ω, µ), and {u i } ∞ k=1 is an orthonormal basis formed by eigenfunctions associated to the eigenvalues {λ k (Ω)} ∞ k=1 of Ω. Moreover If n = 1 then It should be remarked that this result above is a direct consequence of Weyl's assymptotic formula for the eigenvalues λ i (Ω). The first consequence of Theorem 6.1 regards a result due to Grüter and Widman, see [24,Thm. 1.1]. They proved that for any y ∈ Ω ⊂ R n , n ≥ 3, the function It is easy to see that L p (Ω) ⊂ L * p (Ω), thus coupling Theorem 6.1 with Grüter and Widman's result we obtain a more precise statement.
Let Ω ⊂ R n , n ≥ 3 be a bounded open subset with smooth boundary. Let g Ω its Green function (associated to the Laplacian △ ). •

Proofs of the results
The structure of this section is the following. First, we will prove Theorem 6.1 and then Theorem 3.1. Then we will prove Theorems 2.5 and 2.7. be set of eigenvalues of L = −△ µ | W 2 0 (Ω,µ) , repeated according to multiplicity. One has that the Weyl's asymptotic formula holds for the eigenvalues λ k , where c m > 0 is the same constant as in R m , depending only on the dimension m = dim(M ), see [21, p.7]. Thus Let {g k : Ω × Ω → R} ⊂ L 2 (Ω × Ω, µ) be a sequence of functions defined by Let k 2 > k 1 and compute .
On the other hand, it is known that the Green function g Ω satisfies the functional identity in the sense of distributions, see [22, p. 348]. Therefore, g Ω = g ∞ ∈ L 2 (Ω × Ω, µ) if and only if m = 1, 2, 3 and  (B N (p, r)) be an extrinsic ball that contains q, where B N (p, r) is the geodesic ball of N with center at p and radius r < min{inj(p), π/ √ k}, k = sup K N and where π/ √ k = ∞ if k ≤ 0. Let E r (x) be the mean time of the first exit from Ω r for a Brownian motion particle starting at x ∈ Ω r . A fundamental observation of Dynkin [18,vol.II,p.51] states that the function E r satisfies the Poisson equation If g Ωr (x, y) = g(x, y) is the Green function of Ω r , with Dirichlet boundary data, then E r (x) = Ωr g(x, y)dν(y).
Applying Cauchy-Schwarz and assuming that m = 2, 3 we obtain Assume that N = R n , p = o ∈ R n . Let E r : B m (0, r) → R be the mean exit time of the first exit from the geodesic ball B m (o, r) ⊂ R m . It is known that E r is radial, i.e. E r (y) = E r (t(y)), t(y) = |y − o|, y ∈ R m . Denote by E r the transplant of E r to B n (o, r), i.e., the function E r : B n (o, r) ⊂ R n → R defined by E r (z) = E r (t(z)). Consider the restriction of E r (z) to the immersion ϕ(M ), i.e. x ∈ Ω r → E r (ϕ(x)). In [28], Steen Markvorsen proved that if the immersion ϕ : M → R m+1 is a minimal hypersurface then E r (x) = E r (ϕ(x)) = E r (t(ϕ(x))). Solving problem (7.2), we have therefore, by inequality (7.3), we have where we identified t(ϕ(x)) = t(x). Applying co-area formula for the extrinsic distance function t : M → R + we obtain On the other hand, we have that vol(∂Ω s ) ≥ mω m s m−1 , see [32]. Then In order to simplify the notation let us denote by A m := 1 + m 4+m − 2m 2+m . Hence, That proves the lower bound in (3.3). To prove the upper bound recall that Cheng, Li and Yau proved in [14] that Observe that ζ(2) = π 2 /6. Putting together inequalities (7.6) and (7.8) we obtain In order to obtain inequality (3.4) we have by the monotonicity formula, see [31,32], that the function r → vol(Ω r ) ω m r m is an increasing function. Moreover by the classical results of Jorge-Meeks in [27], see also [2,34], lim r→∞ vol(Ω r ) ω m r m = E. Therefore vol(Ω r ) ≤ ω m Er m and the theorem follows taking in consideration that where I i is the geometric index of the end E i , see details in [27].
7.0.3. Proof of Theorem 2.5. The metric on the geodesic ball B(o, r) ⊂ M n h is expressed, in polar coordinates, as ds 2 = dt 2 + h 2 (t)dθ 2 . The Laplacian △ of this metric is given by and △ θ is the Laplacian on S n−1 . Observe that the radial eigenvalues of B(o, r) are the eigenvalues of the operator L 0 in the following eigenvalue problem.
In order to study this eigenvalue problem, define the following space of functions   B(o, r)). Define the bilinear form E bf acting on Lipschitz functions and let F be the closure of Λ in L 2 ([0, r], µ) with respect to the norm The bilinear form E bf acting on F , in the distributional sense, is a Dirichlet form, i.e. it has the following properties.
(2) Closedness: the space F is a Hilbert space with respect to the following product f, g = (f, g) + E bf (f, g). Proof: For any f, g ∈ Λ The generator L determines the heat semigroup P t = e −Lt , t ≥ 0 which posses a heat kernel p(t, x, y) and Green function g(x, y) = ∞ 0 p(t, x, y)dt. Observe that L| D is a self-adjoint extension of L 0 | Λ . Thus, the solution of eigenvalue problem (7.9) is an infinite sequence of eigenvalues 0 < λ rad 1 < λ rad 2 < · · · , (the radial spectrum of B(o, r)). Moreover, L 0 = △ µ , then we have by Theorem 6.1 that We need to determine the Green function g(x, y) for the operator L.
Proposition 7.2. The Green function g(x, y) for the operator L is given by Moreover, Here T is the operator defined in (4.10).
Proof: We need to prove that g(r, y) = 0, lim x→0 + ∂ ∂x g(x, y) = 0, L x (x, y) = 0 for x = y and Lg(x, ·) = −δ x . The two first properties are straight forward from equation (7.16). When x = y we have Then, The last requirement Lg(x, ·) = −δ x is proven now. We have to show that if f ∈ F then On the other hand, Hence we have to prove that L • T = −id. Since T : Λ → Λ, for any u ∈ Λ However, Then L • T (u)(x) = −u(x) and the proposition follows. To prove Theorem 2.5 we have from 7.15 that This proves identity (2.7). If M n h is stochastically incomplete, then its spectrum is discrete, say σ(M n h ) = {λ 1 (M n h ) < λ 2 (M n h ) ≤ · · · }. Taking the limits in (7.24) we obtain To prove identity (2.8) we recall that λ rad i (M m h ) = lim r→∞ λ rad i (B(o, r)). This proves that with initial conditions T (t) ∼ c · t l as t → 0 when l = 1, 2 . . . and T (r) = 0. The procedure to show that the operator L l has a self-adjoint extension L l and a Green function g l is similar to the procedure in the proof of Theorem 2.5. By Theorem 6.1 we have that We need to find the Green function g l . is given by Proof. Observe first of all that On the other hand For any function f : [0, r] → R, we have then where we have applied l + α − β − 1 = 0. Therefore we conclude that, and g l is a Green function for our problem. Now This proves (2.9). To prove (2.10) we proceed as follows.
Proof. We only have to apply the above proposition taking into account that by equality (7.28) Observe that since u 1 does not change its sign in B M h (r), f (x)u 1 (x)dµ(x) = 0 positive or negative) function f . Hence, using Proposition 7.5, we have Then, for any positive (or negative) f ∈ L 2 (Ω, µ), Theorem 5.2 states that this problem admits a unique family of solutions {φ k } ∞ k=1 , given by (7.48) φ k (x) = k! G k (1)(x).
But that is straightforward because the Green operator is the inverse of −△. To prove (7.48) let us use the induction method. Observe that φ 1 = G(1)(x). Suppose that equation (7.48) is true and let us compute φ k+1 (x).
This proves Theorem 5.2. The proof of Corollary 5.3 is straightforward.