Abstract and Applied Analysis

Volume 2013 (2013), Article ID 909782, 21 pages

http://dx.doi.org/10.1155/2013/909782

Research Article

On the Spectral Asymptotics of Operators on Manifolds with Ends

Sandro Coriasco and Lidia Maniccia

Dipartimento di Matematica, Università degli Studi di Torino, V. C. Alberto, n. 10, I-10123 Torino, Italy

Received 28 September 2012; Accepted 16 December 2012

Academic Editor: Changxing Miao

Copyright © 2013 Sandro Coriasco and Lidia Maniccia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We deal with the asymptotic behaviour, for , of the counting function of certain positive self-adjoint operators P with double order , > , defined on a manifold with ends M. The structure of this class of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined on . By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for and show how their behaviour depends on the ratio and the dimension of M.

1. Introduction

The aim of this paper is to study the asymptotic behaviour, for , of the counting function

where is the sequence of the eigenvalues, repeated according to their multiplicities, of a positive order, self-adjoint, classical, elliptic SG-pseudodifferential operator on a manifold with ends. Explicitly, SG-pseudodifferential operators on can be defined via the usual left-quantization

starting from symbols with the property that, for arbitrary multiindices , there exist constants such that the estimates

hold for fixed and all , where , . Symbols of this type belong to the class denoted by , and the corresponding operators constitute the class . In the sequel we will sometimes write and , respectively, fixing once and for all the dimension of the (noncompact) base manifold to .

These classes of operators, introduced on by Cordes [1] and Parenti [2], see also Melrose [3] and Shubin [4], form a graded algebra, that is, . The remainder elements are operators with symbols in ; that is, those having kernel in , continuously mapping to . An operator and its symbol are called SG-elliptic if there exists such that is invertible for and

In such case we will usually write . Operators in act continuously from to itself and extend as continuous operators from to itself and from to , where , , denotes the weighted Sobolev space

Continuous inclusions hold when and , compact when both inequalities are strict, and

An elliptic SG-operator admits a parametrix such that

for suitable , and it turns out to be a Fredholm operator. In 1987, Schrohe [5] introduced a class of noncompact manifolds, the so-called SG-manifolds, on which it is possible to transfer from the whole SG-calculus. In short, these are manifolds which admit a finite atlas whose changes of coordinates behave like symbols of order (see [5] for details and additional technical hypotheses). The manifolds with cylindrical ends are a special case of SG-manifolds, on which also the concept of SG-classical operator makes sense; moreover, the principal symbol of an SG-classical operator on a manifold with cylindrical ends , in this case a triple , has an invariant meaning on , see Egorov and Schulze [6], Maniccia and Panarese [7], Melrose [3], and Section 2. We indicate the subspaces of classical symbols and operators adding the subscript to the notation introduced above.

The literature concerning the study of the eigenvalue asymptotics of elliptic operators is vast and covers a number of different situations and operator classes, see, for example, the monograph by Ivrii [8]. Then, we only mention a few of the many existing papers and books on this deeply investigated subject, which are related to the case we consider here, either by the type of symbols and underlying spaces, or by the techniques which are used. We refer the reader to the corresponding reference lists for more complete informations. On compact manifolds, well-known results were proved by Hörmander [9] and Guillemin [10], see also the book by Kumano-go [11]. On the other hand, for operators globally defined on , see Boggiatto et al. [12], Helffer [13], Hörmander [14], Mohammed [15], Nicola [16], and Shubin [4]. Many other situations have been considered, see the cited book by Ivrii. On manifolds with ends, Christiansen and Zworski [17] studied the Laplace-Beltrami operator associated with a scattering metric, while Maniccia and Panarese [7] applied the heat kernel method to study operators similar to those considered here.

Here we deal with the case of manifolds with ends for , positive and self-adjoint, such that , , focusing on the (invariant) meaning of the constants appearing in the corresponding Weyl formulae and on achieving a better estimate of the remainder term. Note that the situation we consider here is different from that of the Laplace-Beltrami operator investigated in [17], where continuous spectrum is present as well. In fact, in view of Theorem 14, consists only of a sequence of real isolated eigenvalues with finite multiplicity.

As recalled above, a first result concerning the asymptotic behaviour of for operators including those considered in this paper was proved by Maniccia and Panarese in [7], giving, for ,

Note that the constants , , above depend only on the principal symbol of , which implies that they have an invariant meaning on the manifold , see Sections 2 and 3. On the other hand, in view of the technique used there, the remainder terms appeared in the form and for and , respectively. An improvement in this direction for operators on had been achieved by Nicola [16], who, in the case , proved that

while, for , showed that the remainder term has the form for a suitable . A further improvement of these results in the case has recently appeared in Battisti and Coriasco [18], where it has been shown that, for a suitable ,

Even the constant has an invariant meaning on , and both and are explicitly computed in terms of trace operators defined on .

In this paper the remainder estimates in the case are further improved. More precisely, we first consider the power of (see Maniccia et al. [19] for the properties of powers of SG-classical operators). Then, by studying the asymptotic behaviour in of the trace of the operator , , , defined via a Spectral Theorem and approximated in terms of Fourier Integral Operators, we prove the following.

Theorem 1. Let be a manifold with ends of dimension and let be a positive self-adjoint operator such that , , with domain . Then, the following Weyl formulae hold for :

where and .

The order of the remainder is then determined by the ratio of and and the dimension of , since

In particular, when , the remainder is always .

Examples include operators of Schrödinger type on , that is, , the Laplace-Beltrami operator in associated with a suitable metric , a smooth potential that, in the local coordinates on the cylindrical end growths as , with an appropriate related to . Such examples will be discussed in detail, together with the sharpness of the results in Theorem 1, in the forthcoming paper [20], see also [21].

The key point in the proof of Theorem 1 is the study of the asymptotic behaviour for of integrals of the form

where and satisfy certain growth conditions in and (see Section 3 for more details). The integrals represent in fact the local expressions of the trace of , obtained through the so-called “geometric optic method,” specialised to the SG situation, see, for example, Coriasco [22, 23], Coriasco and Rodino [24]. To treat the integrals we proceed similarly to Grigis and Sjöstrand [25], Helffer and Robert [26], see also Tamura [27].

The paper is organised as follows. Section 2 is devoted to recall the definition of SG-classical operators on a manifold with ends . In Section 3 we show that the asymptotic behaviour of , , for a positive self-adjoint operator , , is related to the asymptotic behaviour of oscillatory integrals of the form . In Section 4 we conclude the proof of Theorem 1, investigating the behaviour of for . Finally, some technical details are collected in the Appendix.

2. SG-Classical Operators on Manifolds with Ends

From now on, we will be concerned with the subclass of SG-operators given by those elements , , which are SG-classical, that is, with . We begin recalling the basic definitions and results (see, e.g., [6, 19] for additional details and proofs).

Definition 2. (i) A symbol belongs to the class if there exist , , positively homogeneous functions of order with respect to the variable , smooth with respect to the variable , such that, for a -excision function ,

(ii) A symbol belongs to the class if there exist , , positively homogeneous functions of order with respect to the variable , smooth with respect to the variable , such that, for a -excision function ,

Definition 3. A symbol is SG-classical, and we write , if

(i) there exist such that for a -excision function , and

(ii) there exist such that for a -excision function , and

We set .

Remark 4. The definition could be extended in a natural way from operators acting between scalars to operators acting between (distributional sections of) vector bundles. One should then use matrix-valued symbols whose entries satisfy the estimates (3).

Note that the definition of SG-classical symbol implies a condition of compatibility for the terms of the expansions with respect to and . In fact, defining and on and , respectively, as

It is possibile to prove that

Moreover, the composition of two SG-classical operators is still classical. For the triple is called the principal symbol of . The three components are also called the -, - and -principal symbol, respectively. This definition keeps the usual multiplicative behaviour, that is, for any , , , , with component-wise product in the right-hand side. We also set

for a fixed -excision function . Theorem 5 allows to express the ellipticity of SG-classical operators in terms of their principal symbol.

Theorem 5. An operator is elliptic if and only if each element of the triple is nonvanishing on its domain of definition.

As a consequence, denoting by the sequence of eigenvalues of , ordered such that , with each eigenvalue repeated accordingly to its multiplicity, the counting function is well defined for a SG-classical elliptic self-adjoint operator see, for example, [16, 18, 20, 21]. We now introduce the class of noncompact manifolds with which we will deal.

Definition 6. A manifold with a cylindrical end is a triple , where is a -dimensional smooth manifold and

(i) is a smooth manifold, given by with a -dimensional smooth compact manifold without boundary , a closed disc of , and a collar neighbourhood of in ;(ii) is a smooth manifold with boundary , with diffeomorphic to ;(iii), , is a diffeomorphism, and , , is diffeomorphic to ; (iv)the symbol means that we are gluing and , through the identification of and ; (v)the symbol represents an equivalence class in the set of functions

where if and only if there exists a diffeomorphism such that

for all and .

We use the following notation:

(i);(ii), where . The equivalence condition (22) implies that is well defined;(iii);(iv) is a parametrisation of the end. Let us notice that, setting , the equivalence condition (22) implies

We also denote the restriction of mapping onto by .

The couple is called the exit chart. If is such that the subset is a finite atlas for and , then , with the atlas , is a SG-manifold (see [4]). An atlas of such kind is called admissible. From now on, we restrict the choice of atlases on to the class of admissible ones. We introduce the following spaces, endowed with their natural topologies,

Definition 7. The set consists of all the symbols which fulfill (3) for only. Moreover, the symbol belongs to the subset if it admits expansions in asymptotic sums of homogeneous symbols with respect to and as in Definitions 2 and 3, where the remainders are now given by SG-symbols of the required order on .

Note that, since is conical, the definition of homogeneous and classical symbol on makes sense. Moreover, the elements of the asymptotic expansions of the classical symbols can be extended by homogeneity to smooth functions on , which will be denoted by the same symbols. It is a fact that, given an admissible atlas on , there exists a partition of unity and a set of smooth functions which are compatible with the SG-structure of , that is,

(i), , , ; (ii) and for all .

Moreover, and can be chosen so that and are homogeneous of degree on . We denote by the composition of with the coordinate patches , and by the composition of with , . It is now possible to give the definition of SG-pseudodifferential operator on .

Definition 8. Let be a manifold with a cylindrical end. A linear operator is an SG-pseudodifferential operator of order on , and we write , if, for any admissible atlas on with exit chart :

(1)for all and any , there exist symbols such that

(2)for any of the type described above, there exists a symbol such that

(3), the Schwartz kernel of , is such that

where is the diagonal of and with any conical neighbourhood of the diagonal of .

The most important local symbol of is . Our definition of -classical operator on differs slightly from the one in [7].

Definition 9. Let . is an -classical operator on , and we write , if and the operator , restricted to the manifold , is classical in the usual sense.

The usual homogeneous principal symbol of an -classical operator is of course well defined as a smooth function on . In order to give an invariant definition of the principal symbols homogeneous in of an operator , the subbundle was introduced. The notions of ellipticity can be extended to operators on as well.

Definition 10. Let and let us fix an exit map . We can define local objects as

Definition 11. An operator is elliptic, and we write , if the principal part of satisfies the -ellipticity conditions on and the operator , restricted to the manifold , is elliptic in the usual sense.

Proposition 12. The properties and , as well as the notion of SG-ellipticity, do not depend on the (admissible) atlas on . Moreover, the local functions and give rise to invariantly defined elements of and , respectively.

Then, with any , it is associated an invariantly defined principal symbol in three components . Finally, through local symbols given by , , and , , we get a SG-elliptic operator and introduce the (invariantly defined) weighted Sobolev spaces as

The properties of the spaces extend to without any change, as well as the continuity of the linear mappings induced by , mentioned in Section 1.

3. Spectral Asymptotics for SG-Classical Elliptic Self-Adjoint Operators on Manifolds with Ends

In this section we illustrate the procedure to prove Theorem 1, similar to [13, 25, 27]. The result will follow from the Trace formula (39), (41), the asymptotic behaviour (42), and the Tauberian Theorem 19. The remaining technical points, in particular the proof of the asymptotic behaviour of the integrals appearing in (41), are described in Section 4 and in the Appendix.

Let the operator be considered as an unbounded operator . The following proposition can be proved by reducing to the local situation and using continuity and ellipticity of , its parametrix, and the density of in the spaces.

Proposition 13. Every , considered as an unbounded operator , admits a unique closed extension, still denoted by , whose domain is .

From now on, when we write we always mean its unique closed extension, defined in Proposition 13. As standard, we denote by the resolvent set of , that is, the set of all such that maps bijectively onto . The spectrum of is then . The next theorem was proved in [7].

Theorem 14 (Spectral theorem). Let be regarded as a closed unbounded operator on with dense domain . Assume also that and . Then

(i) is a compact operator on for every . More precisely, is an extension by continuity from or a restriction from of an operator in .(ii) consists of a sequence of real isolated eigenvalues with finite multiplicity, clustering at infinity; the orthonormal system of eigenfunctions is complete in . Moreover, for all .

Given a positive self-adjoint operator , ,  , we can assume, without loss of generality (considering, if necessary, in place of , with a suitably large constant), . Define the counting function , , as

Clearly, is nondecreasing, continuous from the right and supported in . If we set , (see [19] for the definition of the powers of ), turns out to be a SG-classical elliptic self-adjoint operator with . We denote by the sequence of eigenvalues of , which satisfy . We can then, as above, consider . It is a fact that , see [7].

From now on we focus on the case . The case can be treated in a completely similar way, exchanging the role of and . So we can start from a closed positive self-adjoint operator with domain , . For , , we set

and the series converges in the norm (cf., e.g., [25]). Clearly, for all , is a unitary operator such that

Moreover, if for some , and, for , we have , , which implies that is a solution of the Cauchy problem

Let us fix . We can then define the operator either by using the formula

or by means of the vector-valued integral , . Indeed, there exists such that , so the definition makes sense and gives an operator in with norm bounded by . The following lemma, whose proof can be found in the Appendix, is an analog on of Proposition  1.10.11 in [13].

Lemma 15. is an operator with kernel .

Clearly, we then have

By the analysis in [22–24, 28] (see also [29]), the above Cauchy problem (33) can solve modulo by means of a smooth family of operators , defined for , suitably small, in the sense that is a family of smoothing operators and is the identity on . More explicitly, the following theorem holds (see the Appendix for some details concerning the extension to the manifold of the results on proved in [22–24, 28]).

Theorem 16. Define , where and are as in Definition 8, with , , while the are SG FIOs which, in the local coordinate open set and with , are given by

Each solves a local Cauchy problem , , with and local (complete) symbol of associated with , , with phase and amplitude functions such that

Then, satisfies

and .

Remark 17. Trivially, for , and can be considered -classical, since, in those cases, they actually have order with respect to , by the fact that vanishes for outside a compact set.

Remark 18. Notation like , , and similar, in Theorem 16 and in the sequel, also mean that the seminorms of the involved elements in the corresponding spaces (induced, in the mentioned cases, by (3)), are uniformly bounded with respect to .

If we write in place of , for a chosen , the trace formula (35) becomes

Let us denote the kernel of by . Then, the distribution kernel of is

where the local coordinates in the right-hand side depend on and, to simplify the notation, we have omitted the corresponding coordinate maps. By the choices of , and we obtain

Let , , be such that and , (e.g., set with a suitable ). By the analysis of the asymptotic behaviour of the integrals appearing in (41), described in Section 4, we finally obtain

with . The following Tauberian theorem is a slight modification of Theorem  4.2.5 of [13] (see the Appendix).

Theorem 19. Assume that

(i) is an even function satisfying , , ;(ii) is a nondecreasing function, supported in , continuous from the right, with polynomial growth at infinity and isolated discontinuity points of first kind , , such that ;(iii)there exists such that

with , .

Then

Remark 20. The previous statement can be modified as follows: with , , and as in Theorem 19, when

with , then , for .

4. Proof of Theorem 1

In view of Theorem 19 and Remark 20, to complete the proof of Theorem 1 we need to show that (42) holds. To this aim, as explained previously, this section will be devoted to studying the asymptotic behaviour for of

where , , ,  , , and

such that

(i), ;(ii), for a suitable constant ;(iii), , SG-elliptic.

Since for , it is not restrictive to assume that this estimate holds on the whole phase space, so that, for a certain constant ,

Remark 21. The assumption on above amounts, at most, to modifying by adding and subtracting a compactly supported symbol, that is, an element of . The corresponding solutions and of the eikonal and transport equations, respectively, would then change, at most, by an element of , see [23, 24, 28]. It is immediate, by integration by parts with respect to t, that an integral as (46) is for . Then, the modified obviously keeps the same sign everywhere.

For two functions , defined on a common subset of and depending on parameters , we will write or to mean that there exists a suitable constant such that for all . The notation or means that both and hold.

Remark 22. The ellipticity of yields, for ,

which, by integration by parts, implies when .

From now on any asymptotic estimate is to be meant for .

We will make use of a partition of unity on the phase space. The supports of its elements will depend on suitably large positive constants . We also assume, as it is possible, , again with an appropriate . As we will see below, the values of , , and depend only on and its associated seminorms.

Proposition 23. Let be any function in such that , and on , where is a suitably chosen, large positive constant. Then

Proof. Write

and observe that, by , , we find

Thus, if we have on the support of , and the assertion follows integrating by parts with respect to in the first integral of (51).

Remark 24. We actually choose , since this will be needed in the proof of Proposition 28; see also Section C in the Appendix.

Let us now pick such that , for and for , where is a constant which we will choose big enough (see below). We can then write

In what follows, we will systematically use the notation , , , to generally denote functions depending smoothly on and and satisfying SG-type estimates of order in . In a similar fashion, will stand for some function of the same kind which, additionally, depends smoothly on , and, for all , satisfies SG-type estimates of order in , uniformly with respect to .

To estimate , we will apply the stationary phase theorem. We begin by rewriting the integral , using the fact that is solution of the eikonal equation associated with and that is a classical SG-symbol. Note that then , since

In view of the Taylor expansion of at , recalling the property , a fixed -excision function, we have, for some ,

where the subscript denotes the -homogeneous (exit) principal parts of the involved symbols, which are all SG-classical and real-valued, see [28].

Observe that on the support of the integrand in , so that we can, in fact, assume there. Indeed, recalling that, by definition, , for , for , with a fixed constant , it is enough to observe that

which of course implies , provided that is large enough. Moreover, by the ellipticity of , writing , , with the constant of (48),

taking the limit for . Then, setting , , , , in , by homogeneity and the previous remarks, we can write

and find, in view of the compactness of the support of the integrand (see the proof of Proposition 25 below) and the hypotheses

with , . We can now prove the following.

Proposition 25. Choosing the constants large enough and suitably small, one has, for any and for a certain sequence , ,

that is, , with

Proof. It is easy to see that, on the support of , the phase function admits a unique, nondegenerate, stationary point , that is, for all such that , provided that is chosen suitably small (see, e.g., [25, page 136]), and the Hessian equals . Moreover, the amplitude function

is compactly supported with respect to the variables and and satisfies, for all ,

for all , , . In fact,

(1), , , and

where ;(2)all the factors appearing in the expression of are uniformly bounded, together with all their -derivatives, for , , and .

Of course, (2) trivially holds for the cutoff functions and , and for the factor . Since , on we have, for all and ,

Moreover, since is actually in on , the same holds for , by an application of the Faà di Bruno formula for the derivatives of compositions of functions, so also this factor fulfills the desired estimates. Finally, another straightforward computation shows that, for all and ,

on .  The proposition is then a consequence of the stationary phase theorem (see [30, Proposition  1.2.4], [31, Theorem  7.7.6]), applied to the integral with respect to . In particular, the leading term is given by times the integral with respect to of , that is

recalling that , for all .

Indeed, having chosen , , (58) implies

uniformly for , , . This concludes the proof.

Let us now consider . We follow a procedure close to that used in the proof of    of [31]. However, since here we lack the compactness of the support of the amplitude with respect to , we need explicit estimates to show that all the involved integrals are convergent, so we give the argument in full detail in what follows.

We initially proceed as in the analysis of mentioned previously. In view of the presence of the factor in the integrand, we can now assume , the radius of the smallest ball in including , so that . Then, with some ,

Setting , , , , we can rewrite as

, , where we have set

On the support of , we have

so that

For any fixed , we then have belonging to a compact set, uniformly with respect to , say , for a suitable .

Remark 26. Incidentally, we observe that a rough estimate of is

An even less precise result would be the bound , using the convergence of the integral with respect to in the whole , given by .

The next lemma is immediate, and we omit the proof.

Lemma 27. for any , , , , , and, for all ,

The main result of this section is as follows.

Proposition 28. If are chosen large enough, one has

Explicitly,

We will prove Proposition 28 through various intermediate steps. First of all, arguing as in the proof of (58), exchanging the role of and , we note that, for all , ,

. We now study

, , , where we have used Lemma 27. By the symbolic calculus, remembering that on , we can rewrite the expressions mentioned previously as

It is clear that implies and , so that we finally have

We now prove that, modulo an term, we can consider an amplitude such that, on its support, the ration is very close to . To this aim, take such that , for and for , with an arbitrarily fixed, small enough , and set

Proposition 29. With the choices of , for any , one can find large enough such that .

Proof. Since , in view of (3), (74), and (79), we can choose so large that, for an arbitrarily fixed , for any , satisfying ,

uniformly with respect to . Then, is nonstationary on , since there we have , while

which implies . Observing that, on , , as well as , the assertion follows by repeated integrations by parts with respect to , using the operator

and recalling Remark 26.

Proposition 30. With the choices of , , one can assume, modulo an term, that the integral with respect to in is extended to the set , with

Proof. Indeed if , we can split into the sum

since the inequality is true when is sufficiently large.

Observing that, on supp,

switching back to the original variables, the first integral in (88) can be treated as , and gives, in view of Proposition 25, an term, as stated.

Now we can show that admits a unique, nondegenerate stationary point belonging to for . Under the same hypotheses, lies in a circular neighbourhood of of arbitrarily small radius.

Proposition 31. With , , fixed previously, vanishes on only for , that is, for all such that . Moreover,

holds on .

Proof. We have to solve

. By (79) and (84a) and (84b), with the choices of , , the coefficient of in the second equation does not vanish at any point of . Then , and must satisfy

Since, by the choice of , , uniformly with respect to , , has a unique fixed point , smoothly depending on the parameters; see the Appendix for more details. Since

we can assume that and the choices of the other parameters imply, on ,

So we have proved that, on ,

By (3), (92), and , , we also find

uniformly with respect to . The proof is complete.

Remark 32. The choice of depends only on the properties of and on the values of and ; that is, we first fix and , then small enough as explained at the beginning of the proof of Proposition 25, then as explained in the proofs of Propositions 29 and 31, then, finally, .

The next lemma says that the presence in the amplitude of factors which vanish at implies the gain of negative powers of .

Lemma 33. Assume , ,

is smooth, , , and has a -behaviour as the factors appearing in the expression of . Then

where has the same -behaviour, support and -order of , including the powers of .

Proof. By arguments similar to those used in the proof of Proposition 29, on

Assume that the first condition in (97) holds. Under the hypotheses, if , we can first insert in the left-hand side of (98), where , and integrate by parts times. Similarly, if , we subsequently use , , and integrate by parts times. The assertion then follows, remembering that -derivatives of produce either an additional factor or a lowering of the exponent of , and that on . The proof in the case that the second condition in (97) holds is the same, using first and then .

Proof of Proposition 28. Define,

and, for ,

Remembering that , , is the Taylor polynomial of degree two of at , so that vanishes of order at . Obviously, and . Write

, and consider the Taylor expansion of of order , , so that

Since

Remark 26 and Lemma 33 imply that , , . Indeed, it is easy to see, by direct computation, that can be bounded by linear combinations of expressions of the form

with , , having the required properties. Then, the bound of will always contain a term of the type , which corresponds to the (minimum) value in (97).

Each term , , has the quadratic phase function , which of course also satisfies

Then, denoting by the Taylor expansion of at of order , we observe that can be bounded by polynomial expressions in of the kind appearing in the right-hand side of (97), with (cf. the proof of Theorem in [31]). Setting

Lemma 33 implies

We now apply the stationary phase method to and prove that

which is a consequence of

with evaluated with in place of . Recalling (95), it follows that the inverse matrix satisfies, on ,

in view of the ellipticity of the involved symbols. Then, the operators , , do not increase the -order of the resulting function with respect to that of their arguments, , which is the same of , uniformly with respect to . The proof of (110) then follows by Theorem , the proof of Lemma and formula () in [31]; see also [26, 32]. Indeed, by the mentioned results,

since on . It is then enough to sum all the expansions of , , and sort the terms by decreasing exponents of (as in the proof of Theorem in [31]) to obtain (109) with the usual expression

so that, in particular,

for any , . We can then integrate and its asymptotic expansions with respect to and find

Recall that and , for all . Moreover, for , the factors , , and are identically equal to (see the Appendix). Then, the coefficient of the leading term in (115) is given by

with evaluated in . We say that

To confirm this, first note that , , for any belonging to the support of the integrand, see the Appendix. Moreover, the integrand is uniformly bounded by the summable function , and its support is included in the set . Then, recalling (95) and setting ,

The second integral is always , since implies

The first integral can be estimated as follows. Since

by the properties of (see the appendix) we find

since . By (95), we similarly have , so that

If , , , the integral in is convergent for and . In this case, contributes an term to the expansion of , which is of lower order than the term, which is one of the remainders appearing in (77). On the other hand, if , the integral in is divergent, and itself is , since, trivially

Finally, if , is , by

and again contributes a term of lower order than the remainder . Similar conclusions can be obtained for the subsequent terms of the expansion of .

The proof is complete, combining the contributions of the remainders like with the other terms in the expansion of , and remembering that

Remark 34. The same conclusions concerning the behaviour of in the final step of the proof of Proposition 28 could have been obtained studying the Taylor expansion of the extension of , , to the interval , similarly to [32].

Proof of Theorem 1. The statement for follows by the arguments in Section 3 and Propositions 23, 25 and 28, summing up the contribution of the local symbol on the exit chart to the contributions of the remaining local symbols, which gives the desired multiple of the integral of on the cosphere bundle as coefficient of the leading term . The remainder has then order equal to the maximum between and , as claimed. The proof for is the same, by exchanging step by step the role of and .

Appendix

For the sake of completeness, here we illustrate some details of the proof of Theorem 1, which we skipped in the previous sections. They concern, in particular, formula (41), which expresses the relation between and the oscillatory integrals examined in Section 4. We mainly focus on the aspects which are specific for the manifolds with ends.

We also show more precisely how the constants are involved in the solution of (92) via the fixed point theorem, completing the proof of Proposition 31.

A. Solution of Cauchy Problems and SG Fourier Integral Operators

Using the so-called “geometric optics method”, specialised to che pseudodifferential calculus we use (see [22–24, 28, 29, 33]), the Cauchy problem (33) can be solved modulo by means of an operator family , defined for in a suitable interval , : induces continuous maps

First of all, we recall that the partition of unity and the family of functions of Definition 8 can be chosen so that and are SG-symbols of order on , extendable to symbols of the same class defined on (see [5]).

Remark A.1. (1) The complete symbol of depends, in general, on the choice of the admissible atlas, of and of . Anyway, if is another complete symbol of , for an admissible cutoff function supported in .

(2) The solution of (33) in the SG-classical case and the properties of and in (37) were investigated in [28] (see also [33, Section 4]). In particular, it turns out that , . According to ([23, page 101]), for every SG phase functions of the type involved in the definition of we also have, for all ,

with a constant not depending on . The function turns out to be a (SG-) diffeomorphism, smoothly depending on the parameters and (see [22]).

Before proving Theorem 16, we state a technical lemma, whose proof is immediate and henceforth omitted.

Lemma A.2. Let be an open set and define for arbitrary . Assume such that , and . Then, for any diffeomorphism , smoothly depending on , , and such that for all   with a constant independent of ,

for any multi-index and .

We remark that, since a manifold with ends is, in particular, a SG-manifold, the charts , and the functions , can be chosen such that

(i)for a fixed , each coordinate open set , , contains an open subset such that ;(ii)the supports of and , , satisfy hypotheses as the supports of and in Lemma A.2 (see, e.g., Section 3 of [5] for the construction of functions with the required properties).

In fact, this is relevant only for .

Proof of Theorem 16. We will write when and when the functions are smooth, nonnegative, supported in , satisfy and are SG-symbols of order on . Obviously, implies . To simplify notation, in the following computations we will not distinguish between the functions , , and so forth, and their local representations.

obviously satisfies (A.3). To prove (A.2), choose functions supported in such that . Then and, for all , (see [1], Section 4.4; cf. also [11]), so that

That the first term in the sum (A.6) is smoothing comes from the SG symbolic calculus in and the observations above, since . The same property holds for each in the second term, provided , small enough. In fact, by Theorems 7 and 8 of [22], is a SG FIO with the same phase function and amplitude such that

with suitable SG-symbols defined in terms of and . By Remark A.1 and Lemma A.2, for small enough. The proof that satisfies (A.2) is completed once we set . The last part of the theorem can be proved as in [25], Proposition 12.3, since, setting , it is easy to see , so that , with smooth dependence on , as claimed.

B. Trace Formula and Asymptotics for

Proof of Lemma 15. Consider first the finite sum

and reduce to the local situation (cf. Schrohe [5]), via the -compatible partition of unity subordinate to the atlas , by

Then, by and the fact that is supported and at most of polynomial growth in , it turns out that we can extend and to elements of . By an argument similar to the proof of Proposition in [13] (or by direct estimates of the involved seminorms, as in [25]), in when , with kernel of . This proves that is an operator with kernel .

The proof of Theorem 19 is essentially the one in [25], while the proof of Lemma B.1 comes from [13]. We include both of them here for convenience of the reader.

Proof of Theorem 19. Setting and integrating (42) in , we obtain

Now, observe that

where is the Heaviside function. Bringing the series under the integral sign, we can write

since . In view of the monotonicity of and next Lemma B.1 (cf. Lemma of [13]), for

We can then conclude that , , since , and this, together with (B.4) and (B.6), completes the proof.

Lemma B.1. Under the hypotheses of Theorem 19, there exists a constant such that for any and any

Proof. Let and such that for all . Then, trivially,

Let us now prove that

Indeed, this is clear for and , suitably large, in view of hypothesis (iii). For , choose a constant so large that . This shows that, for all ,

For arbitrary there exists such that . We write

By (B.11), the last sum can be estimated by

as claimed.

C. The Solution of the Equation

We know that , , and that . Moreover, is chosen so large that, in particular, on , the absolute value of the -derivative of is less than , uniformly with respect to , , . We want to show that once is fixed, the choice of such a suitably large allows to make a contraction on the compact set , uniformly with respect to , , provided , . This gives the existence and unicity of such that is the unique stationary point of , with respect to , which belongs to the support of for .

First of all, the presence of the factors and in the expression of implies and

Since , clearly . With an arbitrarily chosen , take such that implies and , which is possible, in view of (3) and of the fact that is bounded on . Fix and . Then, on ,

Since , for all , , we have proved that for any choice of , as above, has a unique fixed point in , solution of .

By well-known corollaries of the fixed point theorem for strict contractions on compact subsets of metric spaces, we of course have that depends smoothly on and . Moreover, since for all , , obviously and

pointwise for any . Moreover, by the choices of , , and ,

These imply, for any , , such that ,

Of course, by the choice of , for , ,

Acknowledgments

The authors wish to thank U. Battisti, L. Rodino, and E. Schrohe for useful discussions and hints. Thanks are also due to N. Batavia. The first author was partially supported by the PRIN Project “Operatori Pseudo-Differenziali ed Analisi Tempo-Frequenza” (Director of the national project: G. Zampieri; local supervisor at Università di Torino: L. Rodino). The first author also gratefully acknowledges the support by the Institut für Analysis, Fakultät für Mathematik und Physik, Gottfried Wilhelm Leibniz Universität Hannover, during his stay as Visiting Scientist in the Academic Year 2011/2012, where this paper was partly developed and completed.

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