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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 909782, 21 pages
On the Spectral Asymptotics of Operators on Manifolds with Ends
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.
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.
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  and Parenti , see also Melrose  and Shubin , 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  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  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 , Maniccia and Panarese , Melrose , 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 . 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  and Guillemin , see also the book by Kumano-go . On the other hand, for operators globally defined on , see Boggiatto et al. , Helffer , Hörmander , Mohammed , Nicola , and Shubin . Many other situations have been considered, see the cited book by Ivrii. On manifolds with ends, Christiansen and Zworski  studied the Laplace-Beltrami operator associated with a scattering metric, while Maniccia and Panarese  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 , 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 , 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 , 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 , 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.  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 , see also .
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 . To treat the integrals we proceed similarly to Grigis and Sjöstrand , Helffer and Robert , see also Tamura .
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 ). 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 .
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 .
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  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 .
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., ). 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 .
Lemma 15. is an operator with kernel .
Clearly, we then have
By the analysis in [22–24, 28] (see also ), 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  (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 , .
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
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 .
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