CUBO A Mathematical Journal
Vol.11, No¯ 05, (117–128). December 2009

Structure of Resolvents of Elliptic Cone

Differential Operators: A Brief Survey

Juan B. Gil

Penn State Altoona, 3000 Ivyside Park,

Altoona, PA 16601, USA.

email : jgil@psu.edu

ABSTRACT

The resolvent of an elliptic cone differential operator is surveyed under the aspect of its pseu-

dodifferential structure and its asymptotic behavior as the spectral parameter tends to infinity.

The exposition is descriptive and focuses on the case when the domain of the given operator

is stationary.

RESUMEN

Se examina la resolvente de un operador diferencial de tipo cónico, elíptico, bajo el aspecto de

su estructura pseudodiferencial y su comportamiento asintótico cuando el parámetro espectral

tiende a infinito. La exposición es descriptiva y se enfoca en el caso cuando el dominio del

operador dado es estacionario.

Key words and phrases: Resolvents, trace asymptotics, manifolds with conical singularities.

Math. Subj. Class.: 58J35, 35P05, 47A10.



118 Juan B. Gil CUBO
11, 5 (2009)

1 Introduction

The purpose of this paper is to give a brief descriptive account of joint work with Thomas Krainer

and Gerardo Mendoza on resolvents of general cone differential operators whose symbols satisfy

natural ellipticity conditions. Cone operators arise particularly in the study of differential equations

on a manifold with conical singularities – basic case of an incomplete Riemannian manifold.

The results presented here rely on the analytic and geometric approach developed in the series

of papers [3]–[7]. There the reader can find details and further information, including complete

proofs, examples, applications, as well as an extensive discussion on the existing literature in the

subject.

We start our survey by reviewing the necessary functional analytic framework.

Let H be a Hilbert space and let D0 ⊂ H be a dense subspace. Let A be a linear operator,

initially defined as an unbounded operator A : D0 ⊂ H → H.

We are interested in the closed extensions of A in H. In other words, we are looking for

domains D ⊂ H with D0 ⊂ D to which A can be extended as a closed operator. There are two

canonical such domains:

Dmin(A) = closure of D0 in H with respect to ‖ · ‖A,

Dmax(A) = {u ∈ H : Au ∈ H},

where ‖u‖A = ‖u‖H + ‖Au‖H. Both domains are dense in H and the extension

AD : D ⊂ H → H

is closed if and only if D is a closed subspace of Dmax(A) that contains Dmin(A). Thus, there

is a one-to-one correspondence between the closed extensions of A and the closed subspaces of

Dmax(A)/Dmin(A). If the operator A is fixed and there is no possible ambiguity, we will write

Dmin and Dmax instead of Dmin(A) and Dmax(A).

If AD is closed in H, so is AD − λ = AD − λI for every λ ∈ C. If AD − λ is invertible and

(AD − λ)
−1 is bounded in H, λ is said to be an element of res(AD), the resolvent set of AD. The

family (AD −λ)
−1 is called the resolvent of AD, and the set spec(AD) = C\ res(AD) is the spectrum

of AD.

A closed sector (or ray) Λ ⊂ C is called a sector (or ray) of minimal growth for AD : D → H

if there exists R > 0 such that AD − λ is invertible for every λ in

ΛR = {λ ∈ Λ : |λ| ≥ R},

and the resolvent satisfies either of the equivalent estimates

∥∥(AD − λ)−1
∥∥

L (H)
≤ C/|λ|,

∥∥(AD − λ)−1
∥∥

L (H,D)
≤ C,



CUBO
11, 5 (2009)

Structure of Resolvents of Elliptic Cone ... 119

for some C > 0 and all λ ∈ ΛR.

Our research on elliptic cone operators has been guided by two basic goals: One is to find

verifiable conditions on A and D for the resolvent of AD to exist and for a sector Λ ⊂ C to be a sector

of minimal growth for AD. This information is particularly relevant for nonselfadjoint operators.

Secondly, we are interested in describing the pseudodifferential structure and asymptotic properties

of the resolvent as the spectral parameter λ tends to infinity. In this paper, we will discuss our

progress and main difficulties around these goals.

We finish this introduction by mentioning that the asymptotic information obtained for the

resolvent can be directly applied, for instance, in the short-time asymptotic analysis of heat traces,

and in the study of the meromorphic structure of zeta functions. This follows from the standard

functional calculus, cf. [10], [15].

2 Cone Operators

Let M be a smooth compact n-dimensional manifold with boundary Y = ∂M . We fix a defining

function x for Y and choose a collar neighborhood [0, ε) × Y of the boundary of M . Let E be a

smooth vector bundle over M .

A cone differential operator of order m on sections of E is an element A = x−mP with P in

Diff
m
b (M ; E); the space of totally characteristic differential operators of order m, see [13]. Thus

A is a linear differential operator on C∞(
◦

M ; E) of order m, which near Y , in local coordinates

(x, y) ∈ (0, ε) × Y , takes the form

A = x−m
∑

k+|α|≤m

akα(x, y)(xDx)
kDαy (2.1)

with coefficients akα smooth up to x = 0.

These operators occur, for example, when introducing polar coordinates around a point or as

Laplace-Beltrami operators corresponding to cone metrics, cf. [1].

Every cone operator A ∈ x−m Diffmb (M ; E) has a principal c-symbol
cσσ(A) defined on the

c-cotangent cT ∗M of M . Over the interior of M , cσσ(A) is essentially the usual principal symbol

of A. Near the boundary Y , cσσ(A) is of the form
∑

k+|α|=m

akα(x, y)ξ
kηα,

see (2.1). The operator A is said to be c-elliptic if cσσ(A) is invertible on cT ∗M\0, and the family

A − λ is c-elliptic with parameter λ ∈ Λ ⊂ C if cσσ(A) − λ is invertible on ( cT ∗M × Λ)\0. With

A = x−mP one associates the (indicial) family

P̂ (σ) =
∑

k+|α|≤m

akα(0, y)σ
kDαy ,



120 Juan B. Gil CUBO
11, 5 (2009)

also called the conormal symbol of A. If A is c-elliptic, then P̂ (σ) is invertible for all σ ∈ C except

a discrete set, specb(A), called the boundary spectrum of A.

Fix a positive b-density m on M and let L2b (M ; E) denote the L
2 space with respect to a

Hermitian form on E and the density m. For s ∈ N let

Hsb (M ; E) = {u ∈ L
2
b (M ; E) : P u ∈ L

2
b (M ; E) ∀P ∈ Diff

s
b(M ; E)}.

Throughout this paper we will assume that A is a c-elliptic cone operator of order m > 0,

and as reference Hilbert space we choose, for instance, x−m/2L2b(M ; E). Consider A as a densely

defined unbounded operator

A : C∞c (
◦

M ; E) ⊂ x−m/2L2b (M ; E) → x
−m/2L2b (M ; E).

In [12] Lesch showed that, in the situation at hand, Dmax/Dmin is finite dimensional and every

closed extension of A,

AD : D ⊂ x
−m/2L2b (M ; E) → x

−m/2L2b(M ; E),

is Fredholm. Modulo Dmin the elements of Dmax are determined by their asymptotic behavior

near the boundary of M . The structure of these asymptotics depends on the elements σ in the

boundary spectrum of A with |ℑσ| < m/2.

In [9] it was shown that

Dmin = Dmax ∩
( ⋂

ε>0

xm/2−εHmb (M ; E)
)
,

and Dmin = x
m/2Hmb (M ; E) if and only if specb(A) ∩ {σ ∈ C : ℑσ = −

m
2
} = ∅. Moreover, there

exists ε > 0 such that

Dmax →֒ x
−m/2+εHmb (M ; E).

The embedding (Dmax, ‖·‖A) →֒ x
−m/2L2b (M ; E) is therefore compact. Thus, for every domain

D with Dmin ⊂ D ⊂ Dmax and all λ ∈ C,

AD − λ : D → x
−m/2L2b(M ; E)

is Fredholm with ind(AD − λ) = ind AD. Consequently,

spec(AD) 6= C ⇒ ind AD = 0.

Conversely, if ind AD = 0, then spec(AD) is either discrete or all of C.

Remark 2.2. The complexity of the spectrum of a cone operator can already be observed in the

simple case of the Laplacian on the interval [0, 1], see [6]. In that case, the following situations are

possible:



CUBO
11, 5 (2009)

Structure of Resolvents of Elliptic Cone ... 121

• Closed extensions with index zero whose spectrum is empty.

• Closed extensions with index zero whose spectrum is C.

• A family of domains Dβ with Dβ → D0 (in a suitable sense) such that spec(∆Dβ ) is discrete

and independent of β, but spec(∆D0 ) = C.

3 The Model Operator and Rays of Minimal Growth

By means of a Taylor expansion at x = 0, a cone operator A ∈ x−m Diff mb (M ; E) induces a

decomposition

xmA = P0 + xP̃1,

where P̃1 ∈ Diff
m
b (M ; E) and P0 is an operator with coefficients independent of x near Y . We let

Y ∧ = [0, ∞) × Y and consider P0 as an element of Diff
m
b (Y

∧
; E).

We call the operator x−mP0 ∈ x
−m

Diff
m
b (Y

∧
; E) the model operator of A and denote it by

A∧. If A is written as in (2.1) near the boundary, then

A∧ = x
−m

∑

k+|α|≤m

akα(0, y)(xDx)
kDαy .

This operator acts on C∞c (
◦

Y ∧; E) and can be extended as a densely defined closed operator in

x−m/2L2b (Y
∧
; E). The domains of the minimal and maximal closed extensions of A∧ are denoted

by D∧,min and D∧,max, and like for A, the space D∧,max/D∧,min is finite dimensional. In fact, there

is a natural isomorphism

θ : Dmax/Dmin → D∧,max/D∧,min

that allows passage from domains over M to domains over Y ∧. With a domain D for A we associate

a domain D∧ for A∧ defined via

D∧/D∧,min = θ(D/Dmin). (3.1)

The model operator and its canonical domains D∧,min and D∧,max exhibit an important in-

variance property with respect to the natural R+-action on Y
∧. This property is crucial for the

characterization of domains and in the geometric study of resolvents of elliptic cone operators. For

this reason, it has been incorporated in our systematic approach and is worth reviewing: Let

R+ ∋ ̺ 7→ κ̺ : x
−m/2L2b (Y

∧
; E) → x−m/2L2b (Y

∧
; E)

be the one-parameter group of isometries which on functions is defined by

(κ̺f )(x, y) = ̺
m/2f (̺x, y).



122 Juan B. Gil CUBO
11, 5 (2009)

It is easily verified that A∧ satisfies

κ̺A∧ = ̺
−mA∧κ̺, (3.2)

thus the domains D∧,min and D∧,max are both κ-invariant. In particular, κ induces an action on

D∧,max/D∧,min. A domain D for a cone operator A is said to be stationary if its associated domain

D∧, see (3.1), is κ-invariant.

The relation (3.2) implies

A∧ − ̺
mλ = ̺mκ̺(A∧ − λ)κ

−1
̺ (3.3)

for every ̺ > 0 and λ ∈ C. This property is called κ-homogeneity, see e.g. [14].

Any intermediate space D∧ with D∧,min ⊂ D∧ ⊂ D∧,max gives rise to a closed extension

A∧,D∧ : D∧ ⊂ x
−m/2L2b (Y

∧
; E) → x−m/2L2b (Y

∧
; E).

As opposed to A, even if the c-symbol of A∧ is invertible, not every such extension is Fredholm.

However, for certain values of λ ∈ C, A∧ − λ is better behaved: We define the background resolvent

set of A∧ as

bg-res(A∧) = {λ ∈ C : A∧,min − λ injective and A∧,max − λ surjective}.

Using the κ-homogeneity (3.3) one can prove that this set is a union of open sectors. Moreover,

if λ ∈ bg-res(A∧), then

A∧,D∧ − λ : D∧ ⊂ x
−m/2L2b (Y

∧
; E) → x−m/2L2b (Y

∧
; E)

is Fredholm with ind(A∧,D∧ − λ) = ind(A∧,min − λ) + dim D∧/D∧,min. The index is constant on

connected components of bg-res(A∧).

Let Λ be a sector in bg-res(A∧) and consider the Grassmannian

G = {D∧/D∧,min : ind(A∧,D∧ − λ) = 0 for λ ∈ Λ} (3.4)

of d-dimensional subspaces of D∧,max/D∧,min, where d = − ind(A∧,min − λ).

One of the main reasons for considering the model operator in the context of spectral theory

for cone operators is the following result:

Theorem 3.5. Let A ∈ x−m Diff mb (M ; E) be c-elliptic with parameter in Λ. Let D be a domain

for A and let D∧ be its associated domain. If Λ is a sector of minimal growth for A∧,D∧ , then it

is a sector of minimal growth for AD.

So, the question on the existence of rays of minimal growth for a cone operator AD is reduced

to studying rays of minimal growth for the corresponding A∧,D∧ . The simplest case to study is

when the domain D∧ is κ-invariant.



CUBO
11, 5 (2009)

Structure of Resolvents of Elliptic Cone ... 123

Proposition 3.6. Suppose D∧ is κ-invariant. A sector Λ is a sector of minimal growth for A∧,D∧
if and only if

Λ\{0} ⊂ bg-res(A∧) and A∧,D∧ − λ0 is invertible for some λ0 ∈ Λ\{0}.

If D∧ is not κ-invariant, it generates an orbit on the Grassmannian G, see (3.4). In this case,

we consider the attracting set of its κ-orbit as ̺ → 0:

Ω
−

(D∧) = {D ∈ G : ∃ ̺k → 0 such that D = lim
k→∞

κ̺k (D∧/D∧,min)}.

Theorem 3.7. Let λ0 ∈ bg-res(A∧). The ray Γ through λ0 is a ray of minimal growth for A∧,D∧
iff A∧,D − λ0 is invertible for all D such that D/D∧,min ∈ Ω

−
(D∧).

Remark 3.8. The above invertibility condition can be expressed in terms of the nonvanishing of

a suitable finite determinant. The limiting set Ω−(D∧) can be interpreted as the “principal object”

associated with the domain of A.

A nice and explicit application of the previous theorem to second order regular singular oper-

ators on a metric graph can be found in [6].

4 Structure of Resolvents

Let Λ be a closed sector in C and assume that A ∈ x−m Diffmb (M ; E) is c-elliptic with parameter

in Λ. Let AD be a closed extension of A in x
−m/2L2b(M ; E) and let D∧ be the associated domain

of D. By Theorem 3.5 we know that if Λ is a sector of minimal growth for A∧,D∧ , then it is a

sector of minimal growth for AD. In particular, in such a sector the resolvent (AD − λ)
−1 exists,

thus

ADmin − λ : Dmin → x
−m/2L2b is injective and

ADmax − λ : Dmax → x
−m/2L2b is surjective.

Let

Kλ = ker(ADmax − λ) and Rλ = rg(ADmin − λ).

If λ ∈ res(AD), then

Dmax = Kλ ⊕ D. (4.1)

Let Bmin(λ) be the left-inverse of ADmin −λ with kernel R
⊥
λ and let Bmax(λ) be the right-inverse

of ADmax − λ with range K
⊥
λ . We then have (see [3, Section 5])

(AD − λ)
−1

= Bmax(λ) +
[
1 − Bmin(λ)(A − λ)

]
πKλ,D Bmax(λ), (4.2)

where πKλ,D is the projection on Kλ with kernel D according to (4.1). In fact, the projection can

be replaced by πmaxπKλ,Dπmax, where πmax is the projection onto the orthogonal complement of



124 Juan B. Gil CUBO
11, 5 (2009)

Dmin in Dmax. With similar computations one can also analyze the resolvent of the model operator

A∧ on D∧, see [3, Section 8].

If we are interested in the asymptotic properties of the resolvent, it is accustomed to use a

suitable parameter-dependent pseudodifferential calculus to approximate the resolvent by means

of a “good” parametrix. In [4] we showed:

Theorem 4.3. If Λ is a sector of minimal growth for A∧,D∧ , then

(AD − λ)
−1

= B(λ) + GD(λ) for λ ∈ Λ,

where B(λ) is a parametrix of ADmin − λ with B(λ)(ADmin − λ) = 1 for λ sufficiently large, and

GD(λ) is a smoothing operator of finite rank.

In the proof of this theorem the first major step is the construction of the parametrix B(λ).

An important aspect of our parametrix is that it is an actual left-inverse for λ sufficiently large.

The family GD(λ) is then constructed as follows. Under the given assumptions, there is an operator

family K(λ) : Cd → x−m/2L2b , with d = − ind ADmin , such that

(
(A − λ) K(λ)

)
:

Dmin

⊕

C
d

→ x−m/2L2b

is invertible for λ ∈ ΛR for some R > 0. Its inverse can be written as

(
(ADmin − λ) K(λ)

)−1
=

(
B(λ)

T (λ)

)
,

where B(λ) is the parametrix of ADmin − λ, and T (λ) : x
−m/2L2b → C

d is a smooth family of

operators with “nice” asymptotic properties.

Let E be any d-dimensional complement of Dmin in D. If we split D = Dmin ⊕ E and write

AD − λ =
(
(ADmin − λ) (A − λ)|E

)
,

then (
B(λ)

T (λ)

)(
(ADmin − λ) (A − λ)|E

)
=

(
1 B(λ)(A − λ)|E

0 T (λ)(A − λ)|E

)
,

so AD − λ is invertible if and only if T (λ)(A − λ) : E → C
d is invertible. Now, since T (λ)(A − λ)

vanishes on Dmin, it induces an operator on the quotient:

F (λ) = [T (λ)(A − λ)] : Dmax/Dmin → C
d,

and AD − λ is invertible if and only if FD(λ) = F (λ)|D/Dmin is invertible.

On the other hand, 1 − B(λ)(A − λ) also vanishes on Dmin, so it induces a map

[
1 − B(λ)(A − λ)

]
: Dmax/Dmin → x

−m/2L2b ,



CUBO
11, 5 (2009)

Structure of Resolvents of Elliptic Cone ... 125

and we end up with the decomposition

(AD − λ)
−1

= B(λ) +
[
1 − B(λ)(A − λ)

]
FD(λ)

−1T (λ). (4.4)

This decomposition and the asymptotic properties of its components are crucial for the results

presented in the next section.

Observe that both representations of the resolvent, (4.2) and (4.4), give a more refine picture

of how the domain D affects it. In each case, the domain-dependent contribution is reduced to a

family of linear operators acting on finite dimensional spaces. From these representations one can

derive explicit Krein-like formulas.

5 Trace Expansions

Under the assumptions of the previous section, if Λ is a sector of minimal growth for AD, then for

ℓ ∈ N sufficiently large, (AD −λ)
−ℓ is an analytic family of trace class operators in x−m/2L2b (M ; E).

In this section we give a complete asymptotic expansion of Tr(AD − λ)
−ℓ, as |λ| → ∞, in the case

when the domain is stationary.

Theorem 5.1. Suppose D is stationary. Then, for any ϕ ∈ C∞(M ; End(E)) and ℓ ∈ N with

mℓ > n,

Tr
(
ϕ(AD − λ)

−ℓ
)
∼

∞∑

j=0

mj∑

k=0

αjkλ
n−j

m
−ℓ

log
k λ as |λ| → ∞,

with a suitable branch of the logarithm, with constants αjk ∈ C. The numbers mj vanish for j < n,

and mn ≤ 1. In general, the αjk depend on ϕ, A, D, and ℓ, but the coefficients αjk for j < n

and αn,1 do not depend on D. If both A and ϕ have coefficients independent of x near ∂M , then

mj = 0 for all j > n.

As mentioned in the introduction, this result has direct consequences in the asymptotic analysis

of spectral functions defined by means of the resolvent.

For some 0 < ε0 < π/2, let

Λ = {λ ∈ C : | arg λ| ≥ π
2
− ε0}

be a sector of minimal growth for AD. Then it is known that −AD generates an analytic semigroup

in H given by

e−tAD =
i

2π

∫

Γ

e−tλ(AD − λ)
−1 dλ for t > 0, (5.2)

where Γ is a contour in Λ such that for λ large, | arg λ| = π
2
− δ for some 0 < δ < ε0.

If, in addition, the resolvent set of AD contains an open neighborhood V of the origin, then

for z ∈ C with ℜz < 0, we define

AzD =
i

2π

∫

Γ

λz (AD − λ)
−1 dλ, (5.3)



126 Juan B. Gil CUBO
11, 5 (2009)

where Γ is an infinite path in Λ ∪ V that runs along a ray of minimal growth to a small circle

centered at the origin and contained in V , then clockwise about the origin avoiding the negative

real axis, and out of V along a ray of minimal growth.

In both equations (5.2) and (5.3), the path Γ is chosen to be positively oriented with respect

to the spectrum of AD.

Now, Theorem 5.1 together with (5.2) give the asymptotic expansion

Tr(ϕe−tAD ) ∼

∞∑

j=0

aj t
j−n

m +

∞∑

j=0

mj∑

k=0

ajkt
j
m log

k t as t → 0+.

Moreover, if A is bounded from below on the minimal domain, then the ζ-function

ζAD (s) = Tr(A
−s
D

)

of any selfadjoint extension with stationary domain (e.g. the Friedrichs extension) is holomorphic

for ℜs > n/m and has a meromorphic extension to all of C with poles contained in the set

{ n−j
m

: j ∈ N0}. (5.4)

This follows from Theorem 5.1 together with (5.3), or via the formula

A−s
D

=
1

Γ(s)

∫ ∞

0

ts−1e−tAD dt, ℜs > 0,

which implies

ζAD (s) =
1

Γ(s)
M(hD)(s),

where M(hD) denotes the Mellin transform of the function hD(t) = Tr(e
−tAD ).

If D is nonstationary, the analysis for the asymptotics passed the n-th term is considerably

more involved. For instance, at the level of resolvents, these asymptotics may include rational

functions in log λ and complex powers of λ. This case is discussed in [8]. With the results from

[7], one gets the partial expansion

Tr
(
ϕ(AD − λ)

−ℓ
)
∼

n−1∑

j=0

αj,0λ
n−j

m
−ℓ

+ αn,1λ
−ℓ

log λ + O(|λ|−ℓ) as |λ| → ∞,

which implies

Tr(ϕe−tAD ) ∼
n−1∑

j=0

aj t
j−n

m + an,1 log t + O(1) as t → 0
+.

Consequently, we get that ζAD (s) extends meromorphically to ℜs > 0, but we do not know in

general how this function behaves in all of C.

The complexity of the nonstationary case has already been observed in simple situations.

There are examples on the half-line (see [2]) where the ζ-function extends meromorphically with



CUBO
11, 5 (2009)

Structure of Resolvents of Elliptic Cone ... 127

additional poles not contained in the set (5.4). Moreover, for partial differential operators of

Laplace type (with coefficients independent of the radial variable x), the ζ-function may not admit

a meromorphic extension to all of C due to the presence of logarithmic singularities, see e.g. [11].

Acknowledgment

The present exposition is based on a talk given at the “Second Symposium on Scattering and

Spectral Theory” in Serrambi, Pernambuco, Brazil, August 2008. Their financial support is greatly

appreciated.

Received: February, 2009. Revised: May, 2009.

References

[1] Cheeger, J., On the spectral geometry of spaces with cone-like singularities, Proc. Nat.

Acad. Sci., 76 (1979), 2103–2106.

[2] Falomir, H., Pisani, P.A.G. and Wipf, A., Pole structure of the Hamiltonian ζ-function

for a singular potential, J. Phys. A, 35 (2002), 5427–5444.

[3] Gil, J., Krainer, T. and Mendoza, G., Geometry and spectra of closed extensions of

elliptic cone operators, Canad. J. Math., 59 (2007), no. 4, 742–794.

[4] , Resolvents of elliptic cone operators, J. Funct. Anal., 241 (2006), no. 1, 1–55.

[5] , On rays of minimal growth for elliptic cone operators, Oper. Theory Adv. Appl., 172

(2007), 33–50.

[6] , A conic manifold perspective of elliptic operators on graphs, J. Math. Anal. Appl.,

340 (2008), 1296–1311.

[7] , Trace expansions for elliptic cone operators with stationary domains, preprint, 2008.

[8] , Dynamics on Grassmannians and resolvents of cone operators, preprint, 2009.

[9] Gil, J. and Mendoza, G., Adjoints of elliptic cone operators, Amer. J. Math., 125 (2003),

no. 2, 357–408.

[10] Gilkey, P., Invariance theory, the heat equation, and the Atiyah-Singer index theorem, CRC

Press, Boca Raton, Ann Arbor, 1996, second edition.

[11] Kirsten, K., Loya, P. and Park, J., The very unusual properties of the resolvent, heat

kernel, and zeta function for the operator −d2/dr2 − 1/(4r2), J. Math. Phys., 47 (2006), no.

4, 043506, 27 pp.



128 Juan B. Gil CUBO
11, 5 (2009)

[12] Lesch, M., Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-

Texte zur Math. vol 136, B.G. Teubner, Stuttgart, Leipzig, 1997.

[13] Melrose, R., The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics,

A K Peters, Ltd., Wellesley, MA, 1993.

[14] Schulze, B.–W., Pseudo-differential operators on manifolds with singularities, Studies in

Mathematics and its Applications, 24. North-Holland Publishing Co., Amsterdam, 1991.

[15] Seeley, R., Complex powers of an elliptic operator, Singular Integrals, AMS Proc. Symp.

Pure Math. X, 1966, Amer. Math. Soc., Providence, 1967, pp. 288–307.


	B8-CuboProc