Abstract
We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator \(Q=(1+|x|^2)(1-\varDelta )\) on \(\mathbb {R}^d\).
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1 Introduction
Let (X, g) be a d-dimensional asymptotically Euclidean manifold. More explicitly, X belongs to a class of compact manifolds with boundary, whose interior is equipped with a Riemannian metric g which assumes a specific form close to the boundary \(\partial X\) (see Definition 29 in Section A.1 of “Appendix”). The elements of such class are also known as scattering manifolds, asymptotically conic manifolds, or manifolds with large conic ends. A typical example is the unit ball \({\mathbb {B}}^d\), equipped with a scattering metric.
On X, we consider a self-adjoint positive operator P, elliptic in the \({{\,\mathrm{SG}\,}}\)-calculus of order (m, n) with \(m, n\in (0,\infty )\).Footnote 1 By the compact embedding of weighted Sobolev spaces, the resolvent is compact, and hence, the spectrum of P consists of a sequence of eigenvalues
The goal of this article is to study the Weyl law of P, that is, the asymptotics of its counting function,
Hörmander [18] proved, for a positive elliptic self-adjoint classical pseudodifferential operator of order \(m > 0\) on a compact manifold, the Weyl law
It was pointed out that, in general, this is the sharp remainder estimate, since the exponent of \(\lambda \) in the remainder term cannot be improved for the Laplacian on the sphere. It was subsequently shown by Duistermaat and Guillemin [15] that, under a geometric assumption, there appears an additional term \(\gamma ' \lambda ^{(d-1)/m}\) and the remainder term becomes \(o(\lambda ^{(d-1)/m})\).
In the case of \({{\,\mathrm{SG}\,}}\)-operators on manifolds with cylindrical ends (see Definition 41 and the relationship with asymptotically Euclidean manifolds at the end of Section A.4 of “Appendix”), the leading order of the Weyl asymptotics was found by Maniccia and Panarese [22]. Battisti and Coriasco [3] improved the remainder estimate to \(O(\lambda ^{d/\max \{m,n\} - {\epsilon }})\) for some \({\epsilon }> 0\). For \(m \not = n\), Coriasco and Maniccia [10] proved the general sharp remainder estimate. We recall that \(\mathbb {R}^d\) is the simplest example of manifold with one cylindrical end.
In Theorem 1, we prove the analogue of Hörmander’s result for \(m = n\). This provides a more precise remainder term compared to the earlier result given in [3]. If the geodesic flow at infinity generated by the corner component \(p_ {\psi e} \) of the principal symbol of P is sufficiently generic, we have an even more refined estimate, parallel to the Duistermaat–Guillemin theorem, described in Theorem 2.
Theorem 1
Let \(P \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m,m}_\mathrm {cl}(X)\), \(m>0\), be a self-adjoint, positive, elliptic \({{\,\mathrm{SG}\,}}\)-classical pseudodifferential operator on an asymptotically Euclidean manifold X, and \(N(\lambda )\) its associated counting function. Then, the corresponding Weyl asymptotics reads as
If \(X^o\) is a manifold with cylindrical ends, then the coefficients \(\gamma _j\), \(j=1,2\), are given by
where \({{\,\mathrm{TR}\,}}\) and \(\widehat{{{\,\mathrm{TR}\,}}}_{x,\xi }\) are suitable trace operators on the algebra of \({{\,\mathrm{SG}\,}}\)-operators on X.
Theorem 2
Let \(P \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m,m}_\mathrm {cl}(X)\) and \(N(\lambda )\) be as in Theorem 1. Denote by \(p_ {\psi e} \) the corner component of the principal symbol of P. If the set of periodic orbits of the Hamiltonian flow of \(\mathsf {X}_{f}\), \(f=(p_ {\psi e} )^\frac{1}{m}\), has measure zero on \({\mathcal {W}}^{\psi e}\), then we have the estimate
with the coefficients \(\gamma _2\) and \(\gamma _1\) given in Theorem 1, and
if \(X^o\) is a manifold with cylindrical ends.
Remark 3
The trace operators \({{\,\mathrm{TR}\,}}\) and \(\widehat{{{\,\mathrm{TR}\,}}}_{x,\xi }\) appearing in Theorems 1 and 2 were introduced in [3], see also Section A.3. in “Appendix”. The coefficient \(\gamma _0\) can be calculated as the Laurent coefficient of order \(-2\) at \(s = d-1\) of \(\zeta (s)\), the spectral \(\zeta \)-function associated with P.
Remark 4
To our best knowledge, this is the first result of a logarithmic Weyl law with the remainder being one order lower than the leading term. (We refer, for example, to [2] for a discussion of other settings with logarithmic Weyl laws.)
Remark 5
In view of the analysis at the end of Section A.4 in “Appendix”, we can apply our results to \({{\,\mathrm{SG}\,}}\)-operators on the manifold \(X={\mathbb {B}}^d\), equipped with an arbitrary scattering metric g, that is, to \({{\,\mathrm{SG}\,}}\)-operators on the manifold with one cylindrical end \(\mathbb {R}^d\), identified with \(({\mathbb {B}}^d)^o\) by radial compactification.
Next, we apply our results to the model operator P associated with the symbol \(p(x,\xi )=\left\langle x\right\rangle \!\cdot \!\left\langle \xi \right\rangle \), \(\left\langle z\right\rangle =\sqrt{1+|z|^2}\), \(z\in \mathbb {R}^d\), that is, \(P=\left\langle \cdot \right\rangle \sqrt{1-\varDelta }\). In particular, we observe that the condition on the underlying Hamiltonian flow in Theorem 2 is not satisfied and compute explicitly the coefficients \(\gamma _1\) and \(\gamma _2\).
Theorem 6
Let \(P = \left\langle \cdot \right\rangle \left\langle D\right\rangle \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{1,1}_\mathrm {cl}(\mathbb {R}^d)\). Then,
Here, the coefficients are
where \(\displaystyle \gamma =\lim _{n\rightarrow +\infty }\left( \sum _{k=1}^n\frac{1}{k}-\log n\right) \) is the Euler–Mascheroni constant and
is the digamma function.
This implies that the Weyl asymptotics of the operator
is given by
with the same coefficients given in Theorem 6.
Remark 7
It could be conjectured that many operators satisfying the assumptions of Theorem 1 also satisfy the additional geometric requirement which allows to obtain the refined Weyl formula (2) (cf. [14] for a proof of such fact in a different setting). However, we remark that it is still an open problem to construct explicitly an operator fulfilling the hypotheses of Theorem 2.
Remark 8
Operators like Q arise, for instance, as local representations of Schrödinger-type operators of the form \(H=-\varDelta _{\mathfrak {h}}+V\) on manifolds with ends, for appropriate choices of the metric \({\mathfrak {h}}\) and potential V (see Section A.4 in “Appendix” for a description of this class of manifolds adopted, e.g. in [3, 22]). We just sketch an example of construction of such an operator (see [8, Example 5.21] for the details). Consider the cylinder \(C=\{(u,v,z)\in \mathbb {R}^3:u^2+v^2=1, \, z>1\}={\mathbb {S}}^1\times (1,+\infty )\subset \mathbb {R}^3\) as the model of an end. Pulling back to the metric \({\mathfrak {h}}\) on C the metric \({\mathfrak {h}}^\prime \) on \(\mathbb {R}^3\) given by \(\mathfrak {h^\prime }=4^{-1}\mathrm {diag}(z^2\left\langle z\right\rangle ^{-n},z^2\left\langle z\right\rangle ^{-n}, 4\left\langle z\right\rangle ^{-n})\), \(n>0\), it turns out that, in suitable local coordinates \(x=(x_1,x_2)\in \mathbb {R}^2\) on C, the Laplace–Beltrami operator has the form
with \(\varDelta \) the standard Laplacian. Choosing then, in local coordinates, \(V(x)=\left\langle x\right\rangle ^n\), we find
It is straightforward to see that \(H\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{2,n}\) and it is elliptic (see Sect. 2), as claimed.
The proofs of Theorems 1 and 2 are broken into two parts. First, we will establish a connection between the wave trace near \(t=0\) and the zeta function, to calculate the coefficients of the wave trace. Then, we use a parametrix construction to relate the wave trace to the counting function.
The paper is organized as follows. In Sect. 2, we fix most of the notation used throughout the paper and recall the basic elements of the calculus of \({{\,\mathrm{SG}\,}}\)-classical pseudodifferential operators, the associated wavefront set, and the computation of the parametrix of Cauchy problems for \({{\,\mathrm{SG}\,}}\)-hyperbolic operators of order (1, 1). In particular, we quickly recall the invariance properties of the \({{\,\mathrm{SG}\,}}\)-calculus. In Sect. 3, we consider the wave trace of a \({{\,\mathrm{SG}\,}}\)-classical operator P of order (1, 1). Section 4 is devoted to study the relation between the wave trace and the spectral \(\zeta \)-function of P. In Sect. 5, we prove our main Theorems 1 and 2, while in Sect. 6, we examine the example given by the model operator \(P=\left\langle \cdot \right\rangle \left\langle D\right\rangle \) and prove Theorem 6. For the convenience of the reader, we conclude with an “Appendix”, including a few facts concerning asymptotically Euclidean manifolds and manifolds with cylindrical ends, including a comparison of the two notions at the end of Section A.4. We also give a short summary of the various trace operators and \({{\,\mathrm{SG}\,}}\)-Fourier integral operators.
2 SG-Calculus
The Fourier transform \({\mathcal {F}} : \mathcal {S}(\mathbb {R}^d) \rightarrow \mathcal {S}(\mathbb {R}^d)\) is defined by
and extends by duality to a bounded linear operator \({\mathcal {F}} : \mathcal {S}'(\mathbb {R}^d) \rightarrow \mathcal {S}'(\mathbb {R}^d)\).
The set of pseudodifferential operators \(A = a^w(x,D) = {{\,\mathrm{Op}\,}}^w(a) : \mathcal {S}(\mathbb {R}^d) \rightarrow \mathcal {S}'(\mathbb {R}^d)\) on \(\mathbb {R}^d\) with Weyl symbol \(a \in \mathcal {S}'(\mathbb {R}^{2d})\) can be defined through the Weyl quantizationFootnote 2
A smooth function \(a \in \mathcal {C}^\infty (\mathbb {R}^d\times \mathbb {R}^d)\) is a SG-symbol of order \((m_\psi ,m_e) \in \mathbb {R}^2\), and we write \(a \in {{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^{2d})\), if for all multiindices \(\alpha ,\beta \in \mathbb {N}_0^d\), there exists \(C_{\alpha \beta }>0\) such that, for all \(x,\xi \in \mathbb {R}^d\),
The space \({{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^{2d})\) becomes a Fréchet space with the seminorms being the best constants in (5). The space of all \({{\,\mathrm{SG}\,}}\)-pseudodifferential operators of order \((m_\psi ,m_e)\) is denoted by
The corresponding calculus was established in the 70s by Cordes and Parenti (see, for example, [4, 32]). The letter “G” in the notation, after the usual initial “S” for “symbol space”, stands for “global”. This calculus of symbols of product type, globally defined on \(\mathbb {R}^d\), was also considered by Shubin (see [38]). Actually, the \({{\,\mathrm{SG}\,}}\)-calculus on \(\mathbb {R}^d\) is a special case of the Weyl calculus (see [19, Sections 18.4-18.6]), associated with the slowly varying Riemannian metric on \(\mathbb {R}^{2d}\) given by
(see, for example, [26, p. 71]; see also [8, Section 2.3] for more general \({{\,\mathrm{SG}\,}}\)-classes of symbols and operators on \(\mathbb {R}^d\) as elements of the Weyl calculus).
We list below some basic properties of SG-symbols and operators. (We refer to, for example, [4] and [30, Section 3.1] for an overview of the SG-calculus.) Some more information is provided in “Appendix”, for the convenience of the reader.
-
1.
\(\displaystyle {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}(\mathbb {R}^d)=\bigcup _{(m_\psi ,m_e)\in \mathbb {R}^2}{{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^d)\) is a graded *-algebra; its elements are linear continuous operators from \(\mathcal {S}(\mathbb {R}^d)\) to itself, extendable to linear continuous operators from \(\mathcal {S}'(\mathbb {R}^d)\) to itself;
-
2.
the differential operators of the form
$$\begin{aligned} \sum _{|\alpha | \le m_e, |\beta | \le m_\psi } a_{\alpha ,\beta } x^\alpha D^\beta , \quad m_e,m_\psi \in \mathbb {N}_0, \end{aligned}$$(6)are \({{\,\mathrm{SG}\,}}\) operators of order \((m_\psi ,m_e)\);
-
3.
if \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{0,0}(\mathbb {R}^d)\), then A extends to a bounded linear operator
$$\begin{aligned} A : L^2(\mathbb {R}^d) \rightarrow L^2(\mathbb {R}^d); \end{aligned}$$ -
4.
there is an associated scale of \({{\,\mathrm{SG}\,}}\)-Sobolev spaces (also known as Sobolev–Kato spaces), defined by
$$\begin{aligned} H^{s_\psi ,s_e}(\mathbb {R}^d) = \{u \in \mathcal {S}'(\mathbb {R}^d) :\Vert \left\langle \cdot \right\rangle ^{s_e} \left\langle D\right\rangle ^{s_\psi } u\Vert _{L^2(\mathbb {R}^d)} < \infty \}, \end{aligned}$$and for all \(m_\psi ,m_e,s_\psi ,s_e\in \mathbb {R}\) the operator \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^d)\) extends to a bounded linear operator
$$\begin{aligned} A : H^{s_\psi ,s_e}(\mathbb {R}^d) \rightarrow H^{s_\psi -m_\psi ,s_e-m_e}(\mathbb {R}^d); \end{aligned}$$ -
5.
the inclusions \(H^{s_\psi ,s_e}(\mathbb {R}^d)\subset H^{r_\psi ,r_e}(\mathbb {R}^d)\), \(s_\psi \ge r_\psi \), \(s_e\ge r_e\), are continuous, compact when the order components inequalities are both strict; moreover, the scale of the \({{\,\mathrm{SG}\,}}\)-Sobolev spaces is global, in the sense that
$$\begin{aligned} \bigcup _{s_\psi ,s_e} H^{s_\psi ,s_e}(\mathbb {R}^d) = \mathcal {S}'(\mathbb {R}^d), \quad \bigcap _{s_\psi ,s_e} H^{s_\psi ,s_e}(\mathbb {R}^d) = \mathcal {S}(\mathbb {R}^d); \end{aligned}$$ -
6.
an operator \(A = {{\,\mathrm{Op}\,}}^w(a) \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^d)\) is elliptic if its symbol a is invertible for \(|x|+|\xi | \ge R>0\), and \(\chi (|x|+|\xi |)[a(x,\xi )]^{-1}\) is a symbol in \({{\,\mathrm{SG}\,}}^{-m_\psi ,-m_e}(\mathbb {R}^{2d})\), where \(\chi \in \mathcal {C}^\infty (\mathbb {R})\) with \(\chi (t) = 1\) for \(t > 2R\) and \(\chi (t) = 0\) for \(t < R\);
-
7.
if \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^d)\) is an elliptic operator, then there is a parametrix \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{-m_\psi ,-m_e}(\mathbb {R}^d)\) such that
$$\begin{aligned} AB - {{\,\mathrm{I}\,}}\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{-\infty ,-\infty }(\mathbb {R}^d), \quad BA - {{\,\mathrm{I}\,}}\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{-\infty ,-\infty }(\mathbb {R}^d). \end{aligned}$$
2.1 SG-Classical Symbols
We first introduce two classes of \({{\,\mathrm{SG}\,}}\)-symbols which are homogeneous in the large with respect either to the variable or the covariable. For any \(\rho >0\), \(x_0\in \mathbb {R}^d\), we let \(B_\rho (x_0)=\{x\in \mathbb {R}^d:|x-x_0|<\rho \}\) and we fix a cut-off function \(\omega \in \mathcal {C}_c^\infty (\mathbb {R}^d)\) with \(\omega \equiv 1\) on the ball \(B_\frac{1}{2}(0)\). For proofs, we refer to [30, Section 3.2].
-
1.
A symbol \(a = a(x, \xi )\) belongs to the class \({{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_{\mathrm {cl}(\xi )}(\mathbb {R}^{2d})\) if there exist functions \(a_{m_\psi -i, \cdot } (x, \xi )\), \(i=0,1,\ldots \), homogeneous of degree \(m_\psi -i\) with respect to the variable \(\xi \), smooth with respect to the variable x, such that,
$$\begin{aligned} a(x, \xi ) - \sum _{i=0}^{M-1} (1-\omega (\xi )) \, a_{m_\psi -i, \cdot } (x, \xi )\in {{\,\mathrm{SG}\,}}^{m_\psi -M, m_e}(\mathbb {R}^{2d}), \quad M=1,2, \ldots \end{aligned}$$ -
2.
A symbol a belongs to the class \({{\,\mathrm{SG}\,}}_{\mathrm {cl}(x)}^{m_\psi ,m_e}(\mathbb {R}^{2d})\) if \(a \circ R \in {{\,\mathrm{SG}\,}}^{m_e,m_\psi }_{\mathrm {cl}(\xi )}(\mathbb {R}^{2d})\), where \(R(x,\xi ) = (\xi , x)\). This means that \(a(x,\xi )\) has an asymptotic expansion into homogeneous terms in x.
Definition 9
A symbol a is called \({{\,\mathrm{SG}\,}}\)-classical, and we write \(a \in {{\,\mathrm{SG}\,}}_{\mathrm {cl}(x,\xi )}^{m_\psi ,m_e}(\mathbb {R}^{2d})={{\,\mathrm{SG}\,}}_{\mathrm {cl}}^{m_\psi ,m_e}(\mathbb {R}^{2d})\), if the following two conditions hold true:
-
(i)
there exist functions \(a_{m_\psi -j, \cdot } (x, \xi )\), homogeneous of degree \(m_\psi -j\) with respect to \(\xi \) and smooth in x, such that \((1-\omega (\xi )) a_{m_\psi -j, \cdot } (x, \xi )\in {{\,\mathrm{SG}\,}}_{\mathrm {cl}(x)}^{m_\psi -j, m_e}(\mathbb {R}^{2d})\) and
$$\begin{aligned} a(x, \xi )- \sum _{j=0}^{M-1} (1-\omega (\xi )) \, a_{m_\psi -j, \cdot }(x, \xi ) \in {{\,\mathrm{SG}\,}}^{m_\psi -M, m_e}_{\mathrm {cl}(x)}(\mathbb {R}^{2d}), \quad M=1,2,\ldots ; \end{aligned}$$ -
(ii)
there exist functions \(a_{\cdot , m_e-k}(x, \xi )\), homogeneous of degree \(m_e-k\) with respect to the x and smooth in \(\xi \), such that \((1-\omega (x))a_{\cdot , m_e-k}(x, \xi )\in {{\,\mathrm{SG}\,}}_{\mathrm {cl}(\xi )}^{m_\psi , m_e-k}(\mathbb {R}^{2d})\) and
$$\begin{aligned} a(x, \xi ) - \sum _{k=0}^{M-1} (1-\omega (x)) \, a_{\cdot , m_e-k} (x,\xi )\in {{\,\mathrm{SG}\,}}^{m_\psi , m_e-M}_{\mathrm {cl}(\xi )}(\mathbb {R}^{2d}), \quad M=1,2,\ldots \end{aligned}$$
Note that the definition of \({{\,\mathrm{SG}\,}}\)-classical symbol implies a condition of compatibility for the terms of the expansions with respect to x and \(\xi \). In fact, defining \(\sigma ^\psi _{m_\psi -j}\) and \(\sigma ^e_{m_e-i}\) on \({{\,\mathrm{SG}\,}}_{\mathrm {cl}(\xi )}^{m_\psi ,m_e}\) and \({{\,\mathrm{SG}\,}}_{\mathrm {cl}(x)}^{m_\psi ,m_e}\), respectively, as
it possible to prove that (cf. [30, (3.2.7)])
for all \(j,k \in \mathbb {N}_0\).
Moreover, the composition of two \({{\,\mathrm{SG}\,}}\)-classical operators is still classical. For \(A={{\,\mathrm{Op}\,}}{a}\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^d)\), the triple
where
is called the principal symbol of A. This definition keeps the usual multiplicative behaviour; that is, for any \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^d)\), \(B\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{r_\psi ,r_e}(\mathbb {R}^d)\), \((m_\psi ,m_e),(r_\psi ,r_e)\in \mathbb {R}^2\), the principal symbol of AB is given by
where the product is taken componentwise. Proposition 10 allows to express the ellipticity of \({{\,\mathrm{SG}\,}}\)-classical operators in terms of their principal symbol.
Fixing a cut-off function \(\omega \in \mathcal {C}_c^\infty (\mathbb {R}^d)\) as above, we define the principal part of a to be
2.2 SG-Wavefront Sets
We denote by \({\mathcal {W}}\) the disjoint union
which may be viewed as the boundary of the (double) radial compactification of the phase space \(T^*\mathbb {R}^d\simeq \mathbb {R}^{d}\times \mathbb {R}^d\) (see, for example, [4] and “Appendix”). Therefore, it is natural to define smooth functions on \({\mathcal {W}}\) as follows:
By restriction, the principal symbol can be defined as a map \(\sigma : {{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^{2d})\ni a\mapsto \sigma (a) \in \mathcal {C}^\infty ({\mathcal {W}})\).
Proposition 10
An operator \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^d)\) is elliptic if and only if \(\sigma (A)(x,\xi ) \not = 0\) for all \((x,\xi ) \in {\mathcal {W}}\).
For \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^d)\) we define the following sets (see [9, 27]):
-
1.
the elliptic set
$$\begin{aligned} {{\,\mathrm{ell_{{{\,\mathrm{SG}\,}}}}\,}}(A) = \{ (x,\xi ) \in {\mathcal {W}}:\sigma (A)(x,\xi ) \not = 0\}, \end{aligned}$$ -
2.
the characteristic set
$$\begin{aligned} \Sigma _{{{\,\mathrm{SG}\,}}}(A) = {\mathcal {W}}{\setminus } {{\,\mathrm{ell_{{{\,\mathrm{SG}\,}}}}\,}}(A), \end{aligned}$$ -
3.
the operator \({{\,\mathrm{SG}\,}}\)-wavefront set \({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '(A) \subset {\mathcal {W}}\), via its complement: \((x,\xi ) \notin {{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '(A)\) if there exists \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{0,0}_\mathrm {cl}(\mathbb {R}^d)\) such that \((x,\xi ) \in {{\,\mathrm{ell_{{{\,\mathrm{SG}\,}}}}\,}}(B)\) and \(AB \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{-\infty ,-\infty }(\mathbb {R}^d)\). More concisely,
$$\begin{aligned} {{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '(A) = \bigcap _{\begin{array}{c} B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\\ AB \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{-\infty ,-\infty } \end{array}} \Sigma _{{{\,\mathrm{SG}\,}}}(B). \end{aligned}$$
The \({{\,\mathrm{SG}\,}}\)-wavefront set of a distribution \(u \in \mathcal {S}'(\mathbb {R}^d)\) is defined as
see [4, 9, 27]. Following the concept of wavefront space by Cordes (see [4, Sect. 2.3]) and the approach in [9], we will decompose the \({{\,\mathrm{SG}\,}}\)-wavefront set of \(u\in \mathcal {S}'(\mathbb {R}^d)\) into its components in \({\mathcal {W}}\), namely,
Then, we have that
where \({{\,\mathrm{WF}\,}}_\mathrm {cl}(u)\) is the classical Hörmander’s wavefront set.
The \({{\,\mathrm{SG}\,}}\)-wavefront set is well behaved with respect to the Fourier transform (see, for example, [9, Lemma 2.4]):
2.3 Complex Powers
As in the case of closed manifolds, it is possible to define complex powers of \({{\,\mathrm{SG}\,}}\)-pseudodifferential operators. We will only review the crucial properties of complex powers for a positive elliptic self-adjoint operator \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^d)\), \(m_\psi ,m_e>0\). For the definition and proofs of the following properties, we refer to [3, Proposition 2.8, Theorems 3.1 and 4.2] (cf. also [23, 35]).
-
(i)
\(A^z A^s= A^{z+s}\) for all \(z, s \in \mathbb {C}\).
-
(ii)
\(A^k= \underbrace{A \circ \ldots \circ A}_{k \text { times}}\) for \(k \in \mathbb {N}_0\).
-
(iii)
If \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi , m_e}(\mathbb {R}^d)\), then \(A^z \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi {\text {Re}}z, m_e {\text {Re}}z}(\mathbb {R}^d)\).
-
(iv)
If A is a classical \({{\,\mathrm{SG}\,}}\)-operator, then \(A^z\) is classical and its principal symbol is given by
$$\begin{aligned} \sigma (A^z) = \sigma (A)^z. \end{aligned}$$ -
(v)
For \({\text {Re}}z < -d \cdot \min \{1/m_e, 1/m_\psi \}\), \(A^z\) is trace class.
For any \(A={{\,\mathrm{Op}\,}}(a)\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi , m_e}(\mathbb {R}^d)\) as above, the full symbolFootnote 3 of \(A^z\) will be denoted by
Let \(s \in \mathbb {C}\) with \({\text {Re}}(s) > \max \{d/m_e, d/m_\psi \}\). Using the property (v), it is possible to define \(\zeta (s)\) by
where \(K_{A^z}\) is the Schwartz kernel of \(A^z\). We note that the \(\zeta \)-function may be written as
with \((\lambda _j)_{j\in \mathbb {N}}\) the sequence of eigenvalues of A.
Theorem 11
(Battisti–Coriasco [3]) The function \(\zeta (s)\) is holomorphic for \({\text {Re}}(s)>d\cdot \max \{1/m_\psi , 1/m_e\}\). Moreover, it can be extended as a meromorphic function with possible poles at the points
Such poles can be of order two if and only if there exist integers j, k such that
2.4 Parametrix of SG-Hyperbolic Cauchy Problems
Let \(P \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1}(\mathbb {R}^d)\) be a self-adjoint positive elliptic operator. By the construction from [11, Theorem 1.2] (cf. also [5, 6, 9]), it is possible to calculate a suitable parametrix for the Cauchy problem associated with the wave equation, namely
The solution operator of (10) exists by the spectral theorem and is denoted by \(U(t) = \hbox {e}^{-itP} = [\mathcal {F}_{\lambda \rightarrow t}(dE)](t)\), where dE is the spectral measure of P. There exists a short-time parametrix \({\widetilde{U}}(t)\), which is given by a regular family of \({{\,\mathrm{SG}\,}}\)-Fourier integral operators of type I (cf. Section A.2 in “Appendix”), defined through the integral kernel
where \({\tilde{a}} \in \mathcal {C}^\infty ( (-{\epsilon },{\epsilon }), {{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0})\) with \({\tilde{a}}(0) - 1 \in {{\,\mathrm{SG}\,}}^{-\infty ,-\infty }\) and \(\phi \in \mathcal {C}^\infty ( (-{\epsilon },{\epsilon }), {{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1})\).
The parametrix \({\widetilde{U}}(t)\) solves the wave equation (10) in the sense that \({\tilde{u}}(t,x) = [{\widetilde{U}}(t) u_0](x)\) satisfies
By a Duhamel argument, \(U(t)-{\widetilde{U}}(t) \in \mathcal {C}^\infty ( (-{\epsilon }, {\epsilon }), {\mathcal {L}}(\mathcal {S}'(\mathbb {R}^d), \mathcal {S}(\mathbb {R}^d)))\), (cf. [10, Theorem 16], [14, p. 284]). Since the error term is regularizing, we obtain that
for \(a \in \mathcal {C}^\infty ( (-{\epsilon },{\epsilon }), {{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0})\) with \(a(0) = 1\) (cf. [7, Lemma 4.14]).
Let p be the principal part of the full Weyl-quantized symbol of P. The phase function \(\phi \) satisfies the eikonal equation
This implies that we have a Taylor expansion in t of the form
for t small enough.
For any \(f \in \mathcal {C}^\infty (\mathbb {R}^{2d})\), we define the Hamiltonian vector field by
and we denote its flow by \(t \mapsto \exp (t\mathsf {X}_f)\). For \(P\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1}(\mathbb {R}^d)\), we will collectively denote by \(\mathsf {X}_{\sigma (P)}\) the Hamiltonian vector fields on \({\mathcal {W}}^\bullet \) generated by \(\sigma ^\bullet (P)\), \(\bullet \in \{\psi ,e, {\psi e} \}\), and by \(t \mapsto \exp (t\mathsf {X}_{\sigma (P)})\) the three corresponding flows.
By the group property, \(U(t+s) = U(t) U(s)\), we can extend propagation of singularities results for small times to \(t \in \mathbb {R}\). In [9], the propagation of the \({{\,\mathrm{SG}\,}}\)-wavefront set under the action of \({{\,\mathrm{SG}\,}}\)-classical operators and operator families like U(t) has been studied. In particular, the following theorem was proved there, by means of \({{\,\mathrm{SG}\,}}\)-Fourier integral operators (see also [8] and the principal-type propagation result for the scattering wavefront set [27, Proposition 7]).
Theorem 12
Let \(u_0\in \mathcal {S}'(\mathbb {R}^d)\) and \(U(t) = \hbox {e}^{-itP}\). Then,
where \(\varPhi ^\bullet \) is the smooth family of canonical transformations on \({\mathcal {W}}^\bullet \) generated by \(\sigma ^\bullet (\phi )\) with \(\bullet \in \{\psi ,e, {\psi e} \}\).
Remark 13
In view of (15), Theorem 12 can also be stated in the following way: For any \(u_0\in \mathcal {S}'(\mathbb {R}^d)\) and \(t\in (-{\epsilon }/2,{\epsilon }/2)\), \({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} ^\bullet (U(t)u_0)\subset \exp (t\mathsf {X}_{\sigma ^\bullet (p)})({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} ^\bullet (u_0))\), where \(\bullet \in \{\psi ,e, {\psi e} \}\), and \(\mathsf {X}_f\) is the Hamiltonian vector field generated by f. In the sequel, we will express this fact in the compact form
2.5 SG-Operators on Manifolds
In the 80s, Schrohe [34] introduced the class of \({{\,\mathrm{SG}\,}}\)-manifolds, whose elements admit \(\mathcal {C}^\infty \) structures associated with finite atlases, where the changes of coordinates satisfy suitable estimates of \({{\,\mathrm{SG}\,}}\)-type. This class includes non-compact manifolds, namely, for instance, the Euclidean space \(\mathbb {R}^d\), the infinite-holed torus (see, for example, [36, Page 25]), and the manifolds with ends (see, for example, [36, Page 27]), including those considered in [3, 22] (see Section A.4 in “Appendix”). In particular, Schrohe showed that the concepts of Schwartz functions and distributions, as well as of weighted Sobolev space, have an invariant meaning on \({{\,\mathrm{SG}\,}}\)-manifolds, and the same holds true for \({{\,\mathrm{SG}\,}}\)-operators. In fact, the corresponding symbol and operator classes are preserved by the admissible changes of coordinates (cf. also [5, Section 4.4], for an alternative proof of this property). Maniccia and Panarese [22] considered a class of manifolds with ends and showed that a type of \({{\,\mathrm{SG}\,}}\)-classical operators can be defined there, with the principal symbol triple having an invariant meaning. In [3], this was employed, with reference to \({{\,\mathrm{SG}\,}}\)-classical operators locally described by the symbols recalled in Sect. 2.1, to study the \(\zeta \)-function of \({{\,\mathrm{SG}\,}}\)-operators on manifolds with ends satisfying suitable ellipticity properties, as well as to extend to such environment the concepts of Wodzicki residue and of the trace operators introduced by Nicola [31]. The latter appear in the statements of Theorems 1 and 2. The classical \({{\,\mathrm{SG}\,}}\)-operators have also been employed by Schulze (see, for example, [37]), with the terminology symbols with exit behaviour, in some steps of the construction of pseudodifferential calculi on singular manifolds.
Melrose [26, 27] introduced the so-called scattering calculus on asymptotically Euclidean manifolds. As it can be seen in the quoted references (see, for example, [26, Sect. 6.3]), up to a different choice of compactification with respect to the one we employ (cf. Section A.1 in “Appendix”), and an opposite sign for the e-order of the symbols (that is, the order with respect to the x variable), the operators belonging to the scattering calculus are locally represented by \({{\,\mathrm{SG}\,}}\) pseudodifferential operators. In particular, the principal symbol of the classical operators has an invariant meaning, and the same holds true for the wavefront set (about the latter, see also the comparison in [9, Section 6] between the scattering wavefront set and the \({{\,\mathrm{SG}\,}}\)-wavefront set on \(\mathbb {R}^d\) recalled in Sect. 2.2).
In view of the above observations about the invariance property of the \({{\,\mathrm{SG}\,}}\)-calculus on asymptotically Euclidean manifolds, in the sequel we will mostly work and prove our results for the locally defined operators, that is, on \(\mathbb {R}^d\), with the global results following by a partition of unity and local coordinates argument.
3 Wave Trace
We fix a positive elliptic operator \(P \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1}(\mathbb {R}^{d})\) with \( {\psi e} \)-principal symbol \(p_ {\psi e} = \sigma ^ {\psi e} (P)\). By the compactness of the embedding of \({{\,\mathrm{SG}\,}}\)-Sobolev spaces, we have that the resolvent \((\lambda - P)^{-1}\) is compact for \(\lambda > 0\), and hence, there exists an orthonormal basis \(\{\psi _j\}\) of \(L^2\) consisting of eigenfunctions of P with eigenvalues \(\lambda _j\) with the property that
Therefore, the spectral measure is given by \(dE(\lambda ) = \sum _{j=1}^\infty \delta _{\lambda _j}(\lambda )\left\langle \cdot , \psi _j\right\rangle \psi _j\), where \(\delta _\mu \) is the delta distribution centred at \(\mu \), and we have that
The wave trace w(t) is (formally) defined as
As usual, w(t) is well defined as a distribution by means of integration by parts and the fact that \(P^{-N}\) is trace class for \(N > d\) (cf. Schrohe [35, Theorem 2.4]).
Theorem 12 directly implies Lemma 14.
Lemma 14
Choose \(t_0 \in \mathbb {R}\). Let \(\varGamma \subset {\mathcal {W}}\) be an open subset and such that \(\left[ \exp (t\mathsf {X}_{\sigma (P)}) (\varGamma )\right] \cap \varGamma = \emptyset \), for all \(t\in (t_0-\delta ,t_0+\delta )\) and \(\delta > 0\) small. Then, for all \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}(\mathbb {R}^{d})\) with \({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '(B) \subset \varGamma \), and all \(t\in (t_0-\delta ,t_0+\delta )\), we have that \(B U(t) B \in {\mathcal {L}}(\mathcal {S}'(\mathbb {R}^d), \mathcal {S}(\mathbb {R}^d))\).
We will show that the improvement of the Weyl law is only related to the corner component
The structure of the singularities of w(t) is more involved. This comes from the fact that the boundary at infinity is not a manifold or equivalently the flow is not homogeneous. In contrast to the case of a closed manifold, the distribution w(t) will not be a conormal distribution near 0, but it turns out that it is a log-polyhomogeneous distribution.
Let \({\epsilon }> 0\) as in Sect. 2.4 and choose a function \(\chi \in \mathcal {S}(\mathbb {R})\) with \({{\,\mathrm{supp}\,}}{\hat{\chi }} \subset (-{\epsilon }, {\epsilon })\) and \({\hat{\chi }} = 1\) on \((-{\epsilon }/2,{\epsilon }/2)\).
Proposition 15
Let \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\) and denote by \(N_B(\lambda ) = {{\,\mathrm{Tr}\,}}(E_\lambda B B^*)\) the microlocalized counting function. There exist coefficients \(w_{jk} \in \mathbb {R}\) with \(k \in \mathbb {N}_0\) and \(j\in \{0,1\}\) independent of \(\chi \) such that
as \(\lambda \rightarrow \infty \).
Remark 16
Note that \([\mathcal {F}(N_B')](t) = {{\,\mathrm{Tr}\,}}(U(t) B B^*)\).
Proof
From Sect. 2.4, through the calculus of \({{\,\mathrm{SG}\,}}\) FIOs (see [5] and Section A.2 in “Appendix”), we obtain that there is a parametrix \({\widetilde{U}}(t)\) for U(t) and we have the local representation of the kernel
for \(t \in (-{\epsilon },{\epsilon })\). The amplitude satisfies \(\sigma (a(0)) = \sigma (BB^*)\).
Set
By the previous remark, we have that \(\mathcal {T}_B(t)\) is the Fourier transform of \((N_B' * \chi )(\lambda )\). We will now calculate the inverse Fourier transform of \(\mathcal {T}_B\).
Using the Taylor expansion of the phase function, we have that
where \(\psi \) is smooth in t. Formally, we can write the trace as
We proceed as in Hörmander [20, pp. 254–256], and we set
Note that ellipticity implies that \({\tilde{A}}_B(t,\lambda ) < \infty \).
Set \(X = {{\mathbb {B}}^d}\) and \({}^{\mathrm {sc}}\,\overline{T}^* X = {{\mathbb {B}}^d}\times {{\mathbb {B}}^d}\) with boundary defining functions \(\rho _X\) and \(\rho _\Xi \) as explained in Section A.1 of “Appendix”, and let
Under the compactification \(\iota _2 = \iota \times \iota : \mathbb {R}^d\times \mathbb {R}^d \rightarrow {{\mathbb {B}}^d}\times {{\mathbb {B}}^d}= {}^{\mathrm {sc}}\,\overline{T}^* X\), we have that
In the language of Melrose [25], \((\iota _2)_* u \in {\mathcal {A}}^{{\mathcal {K}}}_{{\text {phg}}}({}^{\mathrm {sc}}\,\overline{T}^* X, {}^b\Omega )\) with index set \({\mathcal {K}}\) given by \(K(\{\rho _X = 0\}) = K(\{\rho _\Xi = 0\}) = (-d + \mathbb {N}_0) \times \{0\}\).
It follows from (17) that
The function \({\tilde{\psi }}(t) = (\iota \circ (-\psi ) \circ \iota _2^{-1})(t) : {}^{\mathrm {sc}}\,\overline{T}^* X \rightarrow {\mathbb {B}}^1 =:{\overline{\mathbb {R}}}\) is a b-fibration with exponent matrix (1, 1) since the symbol \(\psi \) is of order (1, 1). Hence, the Push–Forward Theorem (cf. Melrose [25] and Grieser and Gruber [17]) implies that
where \(\psi _\# {\mathcal {K}} = -d + \mathbb {N}_0 \times \{0,1\}\). Recall that the bundle \({}^b\Omega \) is generated by \(\rho ^{-1}d\rho \) and \(dy_j\) near the boundary \(\{\rho = 0\}\).
Pulling this distribution back to \(\mathbb {R}\) and pairing with \(\delta _\lambda \), we obtain the asymptotics
where the coefficients \(a_{jk}\) are smooth and compactly supported in t. Defining \(A_B(\lambda ) = \hbox {e}^{iD_tD_\lambda }{\tilde{A}}_B(t,\lambda )|_{t=0}\), we find
The above calculation implies that \({\tilde{A}}_B\) and \(A_B\) have the asymptotics
We conclude that
We note that the coefficients are determined by derivatives of \({\tilde{A}}_B(t,\lambda )\) at \(t = 0\) and since \({\hat{\chi }} = 1\) near \(t=0\), the specific choice of \(\chi \) does not change the coefficients. \(\square \)
4 Relation with the Spectral \(\zeta \)-Function
As in the case of pseudodifferential operators on closed manifolds (cf. Duistermaat and Guillemin [15, Corollary 2.2]), the wave trace at \(t = 0\) is related to the spectral \(\zeta \)-function. This relation extends to the \({{\,\mathrm{SG}\,}}\) setting.
Recall that for a positive self-adjoint elliptic operator \(P \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{1,1}_\mathrm {cl}(\mathbb {R}^d)\), the function \(\zeta (s)\) is defined for \({\text {Re}}s > d\) by
In addition, we consider the microlocalized version of \(\zeta (s)\), defined by
for \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\). Of course, \(\zeta _{{{\,\mathrm{I}\,}}}(s) = \zeta (s)\).
By Theorem 11, \(\zeta (s)\) admits a meromorphic continuation to \(\mathbb {C}\) with poles of maximal order two at \(d-k\), \(k\in \mathbb {N}_0\). This result extends to \(\zeta _B(s)\), and we characterize the Laurent coefficients in terms of the wave trace expansion at \(t = 0\).
Proposition 17
The function \(\zeta _B(s)\) extends meromorphically to \(\mathbb {C}\) and has at most poles of order two at the points \(d - k\), \(k\in \mathbb {N}_0\). We have the expansion
where f is holomorphic near \(s = d-k\) and
where the \(w_{jk}\), \(k\in \mathbb {N}_0\), \(j=0,1\), are the coefficients appearing in the asymptotic expansion (16) of \(N_B(\lambda )\).
Proof
The meromorphic continuation and the possible location of the poles follow from similar arguments as in [3, Theorem 3.2] (see also the proof of Proposition 19). Hence, we only have to show that the poles are related to \(N_B(\lambda )\).
Let \({\epsilon }\in (0, \lambda _1)\) be sufficiently small. Choose an excision function \(\chi \in \mathcal {C}^\infty (\mathbb {R})\) such that \(\chi (\lambda ) = 0\) for \(\lambda < {\epsilon }\) and \(\chi (\lambda ) = 1\) for \(\lambda \ge \lambda _1\). Set \(\chi _s(\lambda ) = \chi (\lambda ) \lambda ^{-s}\). Then, using Remark 16,
Let \(\rho \in \mathcal {S}(\mathbb {R})\) such that \(\rho \) is positive, \({\hat{\rho }}(0) = 1\), \({\hat{\rho }} \in \mathcal {C}_c^\infty (\mathbb {R})\), and \(\rho \) is even. By an argument similar to the one in [15, Corollary 2.2], we have that
is entire in s and polynomially bounded for \({\text {Re}}s > C\).
Now, we can insert the asymptotic expansion of \(N_B' * \rho \) to calculate the residues of \(\zeta _B(s)\). Taking the derivative of (16), we see that the asymptotic expansion of \(N_B' * \rho \) is given by
for any \(N \in \mathbb {N}_0\) and \(A_{j,k}\) are given by (18).
Let \(k\in \mathbb {N}_0\) be arbitrary. If \(f \in \mathcal {C}^\infty (\mathbb {R})\) with \(f(\lambda ) = O(\lambda ^{d-k-1}\log \lambda )\) as \(\lambda \rightarrow \infty \), then \(\int f(\lambda ) \chi (\lambda ) \lambda ^{-s} \mathrm{d}\lambda \) is bounded and holomorphic in s for \({\text {Re}}s > d - k\). Let
By partial integration, we obtain
where \(\psi (s) = \int \lambda ^{d-k-s} \chi '(\lambda ) \mathrm{d}\lambda \) is holomorphic and \(\psi (d-k) = 1\). Therefore, we have
Hence, the integral near \(s = d-k\) is given by
where f is holomorphic in a neighbourhood of \(s = d-k\). This shows that \(A_{k,2}\) and \(A_{k,1}\) as defined by (18) are the Laurent coefficients of \(\zeta (s)\) at \(s = d-k\). \(\square \)
The main advantage in employing the \(\zeta \)-function is that the coefficients are easier to calculate than for the wave trace.
Proposition 18
Let \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\) with principal \( {\psi e} \)-symbol \(b_ {\psi e} \). The function \(\zeta _B(s)\) has a pole of order two at \(s = d\) with leading Laurent coefficient
Proof
This follows from the same arguments as in [3, Theorems 3.2 and 4.2] (cf. the proof of Proposition 19), with the modification that the full symbol is \(a(z) = p(z) \# b\), where p(z) denotes the full symbol of \(P^z\). The principal \( {\psi e} \)-symbol of \(A(z) = P^z B\) is given by \(a_{z,z}(x,\xi ;z) = [p_ {\psi e} (x,\xi )]^z \cdot b_ {\psi e} (x,\xi )\). \(\square \)
For the three-term asymptotics, we compute the third coefficient more explicitly.
Proposition 19
Let \(p(s) = p(x,\xi ; s)\) be the full symbol of \(P^s\). The leading Laurent coefficient of \(\zeta (s)\) at \(s = d - 1\) is given by
Remark 20
The equality \(A_{2,1} = {{\,\mathrm{TR}\,}}(P^{-(d-1)})\) also holds on any manifold with cylindrical ends.
Proof
The second equality follows directly from Proposition 40 in “Appendix”.
As in [3], we split the zeta function into four parts
where, for \({\text {Re}}s>d\),
and
Let us recall the main aspects of the proof of the properties of the four terms \(\zeta _j(s)\), \(j=1,\ldots ,4\), shown in [3].
-
1.
\(\zeta _1(s)\) is holomorphic, since we integrate \(p(-s)\), a holomorphic function in s and smooth with respect to \((x,\xi )\), on a bounded set with respect to \((x,\xi )\).
-
2.
Let us first assume \({\text {Re}}s > d\). Using the expansion of \(p(-s)\) with \(M\ge 1\) terms homogeneous with respect to \(\xi \), switching to polar coordinates in \(\xi \) and integrating the radial part, one can write
$$\begin{aligned} \zeta _2(s)&=(2\pi )^{-d} \sum _{j=0}^{M-1}\frac{1}{s-(d-j)} \int _{|x|\le 1}\int _{{\mathbb {S}}^{d-1}} p_{-s-j, \cdot } (x, \omega ;-s) \mathrm{d}\omega \mathrm{d}x \\&\quad +(2\pi )^{-d}\iint _{\Omega _2} r_{-s-M, \cdot }(x, \xi ; -s) \mathrm{d}\xi \mathrm{d}x. \end{aligned}$$Notice that the last integral is convergent and provides a holomorphic function in s. Arguing similarly to the case of operators on smooth, compact manifolds, \(\zeta _2(s)\) turns out to be holomorphic for \({\text {Re}}(s)>d\), extendable as a meromorphic function to the whole complex plane with, at most, simple poles at the points \(s^1_j=d-j\), \(j=0,1,2,\ldots \)
-
3.
Using now the expansion of \(p(-s)\) with respect to x, exchanging the role of variable and covariable with respect to the previous point, again first assuming \({\text {Re}}s > d\) and choosing \(M\ge 1\), one can write
$$\begin{aligned} \zeta _3(s)&= (2\pi )^{-d} \sum _{k=0}^{M-1}\frac{1}{s-(d-k)}\int _{{\mathbb {S}}^{d-1}} \int _{|\xi |\le 1} p_{\cdot , -s-k}(\theta , \xi ;-s)\mathrm{d}\xi \mathrm{d}\theta \\&\quad +(2\pi )^{-d}\iint _{\Omega _3} t_{\cdot , -s-M}(x,\xi ;-s) \mathrm{d}\xi \mathrm{d}x. \end{aligned}$$Arguing as in point 2, \(\zeta _3(s)\) turns out to be holomorphic for \({\text {Re}}s>d\), extendable as a meromorphic function to the whole complex plane with, at most, simple poles at the points \(s^2_k= d-k\), \(k=0,1,2,\ldots \)
-
4.
To treat the last term, both the expansions with respect to x and with respect to \(\xi \) are needed. We assume that \({\text {Re}}s > d\) and choose \(M \ge 1\). We argue as in point 2 to obtain
$$\begin{aligned} \zeta _4(s)&= (2\pi )^{-d}\sum _{j=0}^{M-1} \frac{1}{s-(d-j)}\int _{|x|\ge 1} \int _{{\mathbb {S}}^{d-1}}p_{-s-j, \cdot }(x, \omega ;-s)\mathrm{d}\omega \mathrm{d}x\\&\quad + (2\pi )^{-d}\iint _{\Omega _4} r_{-s-M, \cdot }(x, \xi ;-s)\mathrm{d}\xi \mathrm{d}x. \end{aligned}$$Now, we introduce the expansion with respect to x, switching to polar coordinates and integrating the x-radial variable in the homogeneous terms, for both integrals
$$\begin{aligned}&\int _{|x|\ge 1}\int _{{\mathbb {S}}^{d-1}}p_{-s-j, \cdot }(x, \omega ;-s)\mathrm{d}\omega \mathrm{d}x\\&\quad = \sum _{k=0}^{M-1} \frac{1}{s-(d-k)} \int _{\mathbb {S}^{d-1}}\int _{{\mathbb {S}}^{d-1}} p_{-s-j, -s-k}(\theta , \omega ;-s) \mathrm{d}\theta \mathrm{d}\omega \\&\qquad + \int _{|x|\ge 1} \int _{{\mathbb {S}}^{d-1}} t_{-s-j, -s-M}(x, \omega ;-s) \mathrm{d}x \mathrm{d}\omega \end{aligned}$$and
$$\begin{aligned}&\iint _{\Omega _4} r_{-s-M, \cdot }(x, \xi ;-s)\mathrm{d}\xi \mathrm{d}x\\&\quad = \sum _{k=0}^{M-1} \frac{1}{s-(d-k)}\int _{{\mathbb {S}}^{d-1}}\int _{|\xi |\ge 1} r_{-s-M, -s-k}(\theta ,\xi ; -s) \mathrm{d}\xi \mathrm{d}\theta \\&\qquad + \iint _{\Omega _4}r_{-s-M, -s-M}(x, \xi ; -s)\mathrm{d}x \mathrm{d}\xi . \end{aligned}$$We end up with
$$\begin{aligned} \zeta _4(s)&= \sum _{k=0}^{M-1} \sum _{j=0}^{M-1} \frac{1}{s-(d-j)} \frac{1}{s-(d-k)} I_{j}^{k}(s)\\&\quad + \sum _{j=0}^{M-1} \frac{1}{s-(d-j)} R_j^M(s) +\sum _{k=0}^{M-1} \frac{1}{s-(d-k)} R_M^k(s) + R^M_M(s), \end{aligned}$$
where
and \(R^j_M\), \(R^M_k\), \(R_M^M\) are holomorphic in s for \({\text {Re}}s > M + d\), \(j,k=0,\ldots , M-1\). It follows that \(\zeta _4(s)\) is holomorphic for \({\text {Re}}(s)>d\) and can be extended as a meromorphic function to the whole complex plane with, at most, poles at the points \(s^1_j=d-j\), \(s^2_k= d-k\) with \(j,k \in \mathbb {N}_0\). Clearly, such poles can be of order two if and only if \(j=k\) (cf. Theorem 11).
In view of the properties of \(\zeta (s)\) recalled above, the limit
proves the desired claim. \(\square \)
5 Proof of the Main Theorems
We choose a positive function \(\rho \in \mathcal {S}(\mathbb {R})\) such that \({\hat{\rho }}(0) = 1\), \({{\,\mathrm{supp}\,}}{\hat{\rho }} \subset [-1,1]\), and \(\rho \) is even. For \(T > 0\), we set \(\rho _T(\lambda ) :=T \rho (T\lambda )\), which implies that \({\hat{\rho _T}}(t) = {\hat{\rho }}(t/T)\). Let \(\nu > 0\) be arbitrary. Then, it is possible to prove the next Tauberian theorem by following the proof in [33, Appendix B].
Theorem 21
(Tauberian theorem) Let \(N : \mathbb {R}\rightarrow \mathbb {R}\) such that N is monotonically non-decreasing, \(N(\lambda ) = 0\) for \(\lambda \le 0\), and is polynomially bounded as \(\lambda \rightarrow +\infty \). If
for \(C_1 > 0\), then
Proof of Theorem 1
The first part of Theorem 1 follows directly from the Tauberian theorem and Proposition 15, due to the identity
and using a partition of unity to locally represent \(U(t) = \hbox {e}^{-itP}\) as a smooth family of \({{\,\mathrm{SG}\,}}\)-Fourier integral operators, as discussed in Sect. 2.4. To calculate the coefficients \(w_{j,0}\), we use Proposition 17, to see that
From the definition of the traces, recalled in Definition 39 in “Appendix” (see also [3, p. 247]), we have that
which gives the claimed coefficients. \(\square \)
To prove Theorem 2 it suffices to prove that
where \((N*\rho )(\lambda )\) is obtained through Propositions 15 and 17.
We define the microlocal return time function \(\varPi : {\mathcal {W}}\rightarrow \mathbb {R}_+ \cup \{\infty \}\) by
and \(\varPi (x,\xi ) = \infty \) if no such t exists. For a set \(\varGamma \in {\mathcal {W}}\), we set \(\varPi _\varGamma = \inf _{z \in \varGamma } \varPi (z)\).
We will need a microlocalized version of the Poisson relation.
Proposition 22
Let \(\varGamma \subset {\mathcal {W}}\) and \({\hat{\chi }} \in \mathcal {C}_c^\infty (\mathbb {R})\) with \({{\,\mathrm{supp}\,}}{\hat{\chi }} \subset (0,\varPi _\varGamma )\). For all \(B \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\) with \({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '(B) \subset \varGamma \), we have that
In particular, \((\chi * N_B')(\lambda ) \in O(\lambda ^{-\infty }).\)
The proof is a standard argument (cf. Wunsch [39]) and is only sketched here.
Proof of Proposition 22
For \(t_0 \in {{\,\mathrm{supp}\,}}{\hat{\chi }}\) and \((x,\xi ) \in \varGamma \), we choose a conic neighbourhood U of \((x,\xi )\) such that
for all \(t \in (t_0-{\epsilon },t_0+{\epsilon })\) with \({\epsilon }> 0\) sufficiently small. The existence of this neighbourhood is guaranteed by the conditions on \(\varGamma \) and \({{\,\mathrm{supp}\,}}{\hat{\chi }}\). Choose \({{\tilde{B}}} \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\) with \({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '({{\tilde{B}}}) \subset U\). Lemma 14 implies that for any \(k \in \mathbb {N}\),
hence \({\tilde{B}} U(t) {\tilde{B}}\) and all its derivatives are trace class. We obtain the claim by using a partition of unity. \(\square \)
We also define the modified return time
where \({\epsilon }\) is given as in (12), and set \({\tilde{\varPi }}_\varGamma =\inf _{z\in \varGamma }{\tilde{\varPi }}(z)\). The main tool to prove Theorem 2 is Proposition 23.
Proposition 23
It holds true that
Proof of Theorem 2
The claim follows immediately by Proposition 23, since the assumptions imply that \(\varPi (x,\xi )^{-1} = 0\) almost everywhere on \({\mathcal {W}}^{\psi e}\). From Proposition 19, we obtain the coefficient \(\gamma _0\). \(\square \)
Proof of Proposition 23
Consider an open covering \(\{\varGamma ^\bullet _j\}\) of \({\mathcal {W}}\) with \(\bullet \in \{\psi ,e, {\psi e} \}\) and \(j \in \{1,\cdots , n_\bullet \}\) such that \(\varGamma _j^\psi \subset {\mathcal {W}}^\psi \) and \(\varGamma _j^e \subset {\mathcal {W}}^e\) do not intersect \({\mathcal {W}}^{\psi e}\), and \(\varGamma _j^ {\psi e} \cap {\mathcal {W}}^ {\psi e} \not =\emptyset \).
We consider a partition of unity on the level of operators such that
where \(A^\psi _j \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,-\infty }\), \(A^e_j \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{-\infty ,0}\), \(A^ {\psi e} _j \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{0,0}\), and \(R \in {\mathcal {L}}(\mathcal {S}',\mathcal {S})\). Furthermore, we assume that \({{\,\mathrm{WF}\,}}_{{{\,\mathrm{SG}\,}}} '(A^\bullet _j) \subset \varGamma _j^\bullet \).
Inserting the partition of unity into the counting function yields
where as before \(N^\bullet _j(\lambda ) = {{\,\mathrm{Tr}\,}}E_\lambda A^\bullet _j (A^\bullet _j)^* = \sum _{\lambda _k < \lambda } \Vert (A^\bullet _j)^* \psi _k\Vert ^2\). Here, \(\psi _k\) are the eigenfunctions of P with eigenvalue \(\lambda _k\).
Since \(A^e_j\) and \(A^ {\psi e} _j\) have wavefront set near the boundary of X, we can choose local coordinates such that \(A^e_j\) and \(A^ {\psi e} _j\) become SG-operators on \(\mathbb {R}^d\) of order \((-\infty ,0)\) and (0, 0), respectively. As in the proof of Theorem 1, the parametrix of the wave equation then is locally a smooth family of \({{\,\mathrm{SG}\,}}\)-Fourier integral operators.
The asymptotics for \(N^\psi _j\) and \(N^e_j\) are standard and follow from similar arguments as in Hörmander [18], exchanging the roles of variable and covariable for \(N^e_j\). We can also adapt the proof of Proposition 15. By observing that for \(N^e_j(\lambda )\) and \(N^\psi _j(\lambda )\), the amplitude is supported only near one boundary face, we obtain
Using the standard Tauberian theorem [33, Theorem B.2.1] yields
The operator \(E_\lambda R\) is regularising; thus, its trace is uniformly bounded. We arrive at
It remains to estimate the terms \(N^ {\psi e} _j(\lambda ) - (N^ {\psi e} _j*\rho )(\lambda )\). For this, let
For \(1/T < {\epsilon }\), we have by Proposition 15 that
This implies that the derivative is given by
where \(w_{1,0}\) is given by Proposition 18. Namely,
Together with Proposition 22, this implies that
for \(1/T < {{\tilde{\varPi }}}_j\).
Applying the Tauberian theorem to \(N^ {\psi e} _j * \rho _T\) yields
for \(\lambda \ge {{\tilde{\varPi }}}_j\). Taking the \(\limsup \) and summing over all j gives
The right-hand side is an upper Riemann sum; therefore, we obtain the claim by shrinking the partition of unity. \(\square \)
6 An Example: The Model Operator \(P=\left\langle \cdot \right\rangle \left\langle D\right\rangle \)
In this section, we will consider the case of the operator \(P = \left\langle \cdot \right\rangle \left\langle D\right\rangle \) on \(\mathbb {R}^d\). First, we compute the full symbol of P near the corner:
It follows that \(p_ {\psi e} (x,\xi ) =\sigma ^ {\psi e} (P)(x,\xi )= |x||\xi |\), \(p_\psi (x,\xi ) = |\xi | \left\langle x\right\rangle \), and \(p_e(x,\xi ) = |x|\left\langle \xi \right\rangle \).
We have to investigate the flow of the principal symbol \(p_ {\psi e} \) in the corner. The Hamiltonian vector field on \(\mathbb {R}^{2d}\) is given by
First, we show that the angle between x and \(\xi \) is invariant under the flow. This follows from
Hence, the quantity
is preserved by the flow. The Hamiltonian flow \(\varPhi ^ {\psi e} (t) : {\mathcal {W}}^ {\psi e} \rightarrow {\mathcal {W}}^ {\psi e} \) is given by the angular part.
Lemma 24
The differential equation for \(\omega = x/\left|x\right|\) and \(\theta = \xi /\left|\xi \right|\) describing the Hamiltonian flow \(\varPhi ^ {\psi e} (t) : {\mathcal {W}}^ {\psi e} \rightarrow {\mathcal {W}}^ {\psi e} \) is given by
Proof
We observe that
The calculation of \({\partial }_t |x|\) is straightforward:
This implies
as claimed. The second equation follows likewise. \(\square \)
Proposition 25
The return time function \(\varPi : {\mathcal {W}}^ {\psi e} \rightarrow \mathbb {R}\) is given by
Proof
The system of differential equations (20) decomposes into d decoupled systems of the form
where
We note that the eigenvalues of the matrix A are given by \(\lambda _\pm = \pm i\sqrt{1 - c^2}\). Thus, we have that the fundamental solution to the differential equation (20) for \((\omega , \theta )\) is given by
for some unitary matrix \(S = S(c)\). The claim follows by choosing the minimal \(t > 0\) with \(t\sqrt{1-c^2} \in 2\pi \mathbb {Z}\) and noting that \(c = \left\langle \omega (0),\theta (0)\right\rangle =\left\langle \omega _0,\theta _0\right\rangle \) for \(\omega _0, \theta _0 \in \mathbb {S}^{d-1}\). \(\square \)
Remark 26
Proposition 25 shows that Theorem 2 cannot be applied to P. Nevertheless, we calculate \(\gamma _0\) in Proposition 28.
Proof of Theorem 6
By the Weyl law, Theorem 1, we have that
So it remains to calculate the corresponding Laurent coefficients of \(\zeta (s)\). By Proposition 40, we have
Again by Proposition 40,
First, we note that \(\widehat{{{\,\mathrm{TR}\,}}}_\psi (P^{-d}) = \widehat{{{\,\mathrm{TR}\,}}}_e(P^{-d})\) and the last term we already calculated for \(\gamma _2\). We recall that \(p_ {\psi e} = 1\) on \(\mathbb {S}^{d-1} \times \mathbb {S}^{d-1}\). Thus, we have for \(\widehat{{{\,\mathrm{TR}\,}}}_\theta (P^{-d})\) that
This implies
Hence, we only have to calculate \(\widehat{{{\,\mathrm{TR}\,}}}_e(P^{-d})\):
Using polar coordinates, we see that
Now, we perform a change of variables \(r=t^{-\frac{1}{2}}\Leftrightarrow t=r^{-2}>0\), so that
For \({\text {Re}}z > 0\), we have that (cf. [16, #8.36])
where \(\varPsi (z)\) is the digamma function, defined by (3). By elementary computations, we obtain
Hence, we have that
Summing up, we have obtained
The proof is complete. \(\square \)
Remark 27
Using the properties of the function \(\varPsi \), we can make (22) more explicit. Indeed (see, for example, [16, #8.366, page 945]), we find:
In particular, we have that
Even though the coefficient \(\gamma _0\) does not appear in the Weyl law of P, for the sake of completeness we show that it is not hard to compute in this case.
Proposition 28
For the operator \(P = \left\langle \cdot \right\rangle \left\langle D\right\rangle \), we have that \(\gamma _0 = 0\).
Proof
To calculate \(\gamma _0\), we switch to Weyl quantization. Letting \(p^w(x,D) = P\), the symbol p has an asymptotic expansion
where \(p_{0,0}(x,\xi ) = \frac{\left\langle x,\xi \right\rangle }{|x||\xi |}\) is the subprincipal symbol of p. Hence, for the symbol of \(P^{d-1} = [a^w(x,D)]^{d-1}\) we find an asymptotic expansion
This implies for the symbol \(p^{-d+1}\) of the inverse of \([p^w(x,D)]^{d-1}\) that
Therefore, by Propositions 17 and 19, we have that
\(\square \)
Notes
The formula involving integrals only holds true for \(a \in \mathcal {S}(\mathbb {R}^{2d})\), but the quantization can be extended to any \(a \in \mathcal {S}'(\mathbb {R}^{2d})\), using the Fourier transform, the pullback by linear transformations, and the Schwartz kernel theorem.
For the definition of the zeta function, it does not matter which quantization we choose.
Note that Melrose uses the stereographic projection to compactify \(\mathbb {R}^d\) to \(\mathbb {S}^d_+\), but it was shown in [7, Remark 1.3] that these two compactifications are equivalent. Moreover, the space \({{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}\) is denoted by \(\varPsi _{\text {sc}}\).
As explained in [3], it suffices to assume \(\Lambda \)-ellipticity and some sectoriality conditions on the spectrum.
In the coordinates above, the radial compactification map is given by \((r,\theta ) \mapsto (1/r, \theta )\).
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Acknowledgements
We would like to thank R. Schulz for many helpful discussions and various remarks on the manuscript. We are grateful to the anonymous Referees, for their comments, suggestions and constructive criticism.
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Appendix A. Classical Operators on Asymptotically Euclidean Manifolds
Appendix A. Classical Operators on Asymptotically Euclidean Manifolds
1.1 A.1. Scattering Geometry
We refer the reader to, for example, [7, 28] for a detailed study of scattering geometry and recall here only a few definitions and notation.
Definition 29
An asymptotically Euclidean manifold (X, g) is a compact manifold with boundary X, whose interior is equipped with a Riemannian metric g that is supposed to take the form, in a tubular neighbourhood of the boundary,
where \(\rho _X\) is a boundary defining function and \(g_\partial \in \mathcal {C}^\infty (X,{{\,\mathrm{Sym}\,}}^2 T^*X)\) restricts to a metric on \(\partial X\).
We set \({{\mathbb {B}}^d}=\{x\in \mathbb {R}^d\,:\,|x|\le 1\}\) and denote \(\partial {{\mathbb {B}}^d}={{\mathbb {S}}^{d-1}}\), \(({{\mathbb {B}}^d})^o=\{x\in \mathbb {R}^d\,:\,|x|<1\}\), and \(\mathbb {R}_+=(0,\infty )\). Pick any diffeomorphism \(\iota :\mathbb {R}^d\rightarrow ({{\mathbb {B}}^d})^o\) that for \(|x|>3\) is given by
Then, its inverse is given, for \(|y|\ge \frac{2}{3}\), by
The map \(\iota \) is called the radial compactification map. The associated polar coordinates equip \(\mathbb {R}^d\) with a differential structure “at infinity”. Indeed, introducing polar coordinates \((r,\varphi )\in \mathbb {R}^d\) we see that \(\iota \) is simply given (for large r) by
Denote by \(x\mapsto [x]\) any smooth function \(\mathbb {R}^d\rightarrow \mathbb {R}_+\) that coincides with |x| for \(|x|>3\). Then, the map \({{\mathbb {B}}^d}\rightarrow [0,\infty ) \) given by \(y\mapsto \frac{1}{[\iota ^{-1}(y)]}=: \rho _{{{\mathbb {B}}^d}}\) is a boundary defining function for \({{\mathbb {B}}^d}\). Notice that, for \(|y|>2/3\), the map \(y\mapsto \rho _{{{\mathbb {B}}^d}}\) is simply given by \(y\mapsto (1-|y|)\). In a collar neighbourhood of the boundary, \(0 \le \rho _{{{\mathbb {B}}^d}} < 1/3\), the metric induced by these coordinates from the standard Euclidean metric on \(\mathbb {R}^d\) is given by
where \(g_{\mathbb {S}^{d-1}}\) is the (lifted) standard metric on the \((d-1)\)-sphere.
For any compact manifold with boundary X with boundary defining function \(\rho _X\), we define the space of scattering vector fields \({}^{\mathrm {sc}}\,\mathcal {V}(x) :=\rho _X\, {}^{b}\mathcal {V}(X)\), where \( {}^{b}\mathcal {V}(X)\) is the space of tangential vector fields. There is a natural vector bundle, \({}^{\mathrm {sc}}\,TX\) such that the sections of \({}^{\mathrm {sc}}\,TX\) are exactly the scattering vector fields. The dual bundle is the scattering cotangent bundle, \({}^{\mathrm {sc}}\,T^* X\). Using the fibrewise radial compactification, we obtain a manifold with corners \({}^{\mathrm {sc}}\,\overline{T}^* X\) with boundary defining functions \(\rho _X\) and \(\rho _\Xi \).
The new-formed fibre boundary may be identified with a rescaling of the cosphere bundle, called \({}^{\mathrm {sc}}\,S^*X\). Since X is a compact manifold with boundary, \({}^{\mathrm {sc}}\,\overline{T}^*X\) is a compact manifold with corners. The boundary \({\mathcal {W}}\) of \({}^{\mathrm {sc}}\,\overline{T}^*X\) splits into three components:
It can be shown (cf. [7, Section 1], in particular Example 1.14) that, under the above identification of \(\mathbb {R}^d\) with the interior of \({{\mathbb {B}}^d}\), the \({{\,\mathrm{SG}\,}}\)-classical symbols spaces \({{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(\mathbb {R}^d)\) become \(\rho _X^{-m_e}\rho _{\Xi }^{-m_\psi }\mathcal {C}^\infty ({{\mathbb {B}}^d}\times {{\mathbb {B}}^d})\).
Let X be an asymptotically Euclidean manifold, the \({{\,\mathrm{SG}\,}}\)-classical symbols of order \(m_\psi ,m_e \in \mathbb {R}\) are given by
Using local coordinates, one can define \({{\,\mathrm{SG}\,}}\)-operators, which is denoted by \({{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}(X)\), on general asymptotically Euclidean manifolds, see Melrose [27, Definition 3].Footnote 4
Let \(a \in \rho _X^{-m_e} \rho _\Xi ^{-m_\psi } \mathcal {C}^\infty ({}^{\mathrm {sc}}\,\overline{T}^* X)\), and set \(f = \rho _X^{m_e} \rho _\Xi ^{m_\psi } a\). The symbols \(\sigma ^\bullet (a)\) of a at a point \(p = (\rho _X, x, \rho _\Xi , \xi ) \in \mathcal {W}^\bullet \) are defined by
The principal symbol \(\sigma (a) \in \mathcal {C}^\infty (\mathcal {W})\) is the triple
A symbol \(a \in \rho _X^{-m_e} \rho _\Xi ^{-m_\psi } \mathcal {C}^\infty ({}^{\mathrm {sc}}\,\overline{T}^* X)\) is called elliptic at \(p \in \mathcal {W}\) if \(\sigma (a)(p) \not = 0\) (cf. [26, Section 6.4]).
Melrose and Zworski [29] defined, for \(a \in \rho _X^{-m_e}\rho _\Xi ^{-m_\psi } \mathcal {C}^\infty ({}^{\mathrm {sc}}\,\overline{T}^*X)\), the Hamiltonian vector field
which generalizes the usual Hamiltonian vector field to the compactified scattering cotangent bundle of asymptotically Euclidean manifolds.
For \(a \in \rho _X^{-1}\rho _\Xi ^{-1} \mathcal {C}^\infty ({}^{\mathrm {sc}}\,\overline{T}^*X)\), the Hamiltonian vector field is tangential to the boundary, and hence, its flow \(\exp (t\, {}^{\mathrm {sc}}\mathsf {X}_f)\) can be restricted to a map
that preserves the components \({\mathcal {W}}^e\), \({\mathcal {W}}^\psi \), and \({\mathcal {W}}^{\psi e}\). Note that the flow \(t\mapsto \exp (t\, {}^{\mathrm {sc}}\mathsf {X}_a)|_{\mathcal {W}}\) depends only on the principal symbol \(\sigma (a)\) of a.
1.2 A.2. The Calculus of SG-Fourier Integral Operators
The calculus of Fourier integral operators defined by means of general \({{\,\mathrm{SG}\,}}\)-symbols was initially studied in [5] and subsequently applied to the analysis of the corresponding hyperbolic problems in [6]. Their \({{\,\mathrm{SG}\,}}\)-classical counterparts have been considered in [11]. The theory was expanded along the years, and such operator class has been studied and employed also by other authors. Some of the most recent developments in this field have been obtained in the series of papers [7, 12, 13] (see also [8] and the references quoted there). In this section, we recall some basic elements of the calculus of \({{\,\mathrm{SG}\,}}\) Fourier integral operators on \(\mathbb {R}^d\) that are involved in the proof of the main results of this paper, relying on materials appeared, for example, in [1] (see [8] for more general classes of Fourier integral operators of \({{\,\mathrm{SG}\,}}\) type). Here, we write \(A\asymp B\) when \(A\lesssim B\) and \(B\lesssim A\), where \(A\lesssim B\) means that \(A\le c\cdot B\), for a suitable constant \(c>0\). We will also write FIO for Fourier integral operator.
Definition 30
A real-valued function \(\varphi \in \mathcal {C}^\infty (\mathbb {R}^{2d})\) belongs to the class \({\mathcal {P}}\) of \({{\,\mathrm{SG}\,}}\) phase functions if it satisfies the following conditions:
-
1.
\(\varphi \in {{\,\mathrm{SG}\,}}^{1,1}(\mathbb {R}^{2d})\);
-
2.
\(\left\langle \varphi '_x(x,\xi )\right\rangle \asymp \left\langle \xi \right\rangle \) as \(|(x,\xi )|\rightarrow \infty \);
-
3.
\(\left\langle \varphi '_\xi (x,\xi )\right\rangle \asymp \left\langle x\right\rangle \) as \(|(x,\xi )|\rightarrow \infty \).
Functions of class \({\mathcal {P}}\) are those used in the construction of the \({{\,\mathrm{SG}\,}}\) FIOs calculus. The \({{\,\mathrm{SG}\,}}\) FIOs of type I and type II, \({{\,\mathrm{Op}\,}}_\varphi (a)\) and \({{\,\mathrm{Op}\,}}^*_\varphi (b)\), are defined as
and
respectively, with \(\varphi \in {\mathcal {P}}\) and \(a,b\in {{\,\mathrm{SG}\,}}^{m_\psi ,m_e}\), \(u \in \mathcal {S}\). Operators of type I and type II with the same phase function and symbol are formal \(L^2\)-adjoint of each other. Both operators of type I and type II are linear and continuous on \(\mathcal {S}\), extendable to linear continuous operators on \(\mathcal {S}'\).
Theorem 31 about composition between \({{\,\mathrm{SG}\,}}\) pseudodifferential operators and \({{\,\mathrm{SG}\,}}\) FIOs was originally proved in [5], see also [8, 11].
Theorem 31
Let \(\varphi \in {\mathcal {P}}\) and assume \(p\in {{\,\mathrm{SG}\,}}^{s_\psi ,s_e}(\mathbb {R}^{2d})\), \(a,b\in {{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^{2d})\). Then,
for some \(c_j\in {{\,\mathrm{SG}\,}}^{m_\psi +s_\psi ,m_e+s_e}(\mathbb {R}^{2d})\), \(r_j\in {{\,\mathrm{SG}\,}}^{-\infty ,-\infty }(\mathbb {R}^{2d})\), \(j=1,\ldots ,4\).
To obtain the composition of \({{\,\mathrm{SG}\,}}\) FIOs of type I and type II, some more hypotheses are needed, leading to the definition of the class \({\mathcal {P}}_r\) of regular \({{\,\mathrm{SG}\,}}\) phase functions.
Definition 32
Let \(r>0\). A function \(\varphi \in {\mathcal {P}}\) belongs to the class \({\mathcal {P}}_r\) if it satisfies, for all \((x,\xi )\in \mathbb {R}^{2d}\),
Theorem 33 shows that the composition of \({{\,\mathrm{SG}\,}}\) FIOs of type I and type II with the same regular \({{\,\mathrm{SG}\,}}\) phase functions is a \({{\,\mathrm{SG}\,}}\) pseudodifferential operator, see [5] for a detailed proof.
Theorem 33
Let \(\varphi \in {\mathcal {P}}_r\) and assume \(a\in {{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(\mathbb {R}^{2d})\), \(b\in {{\,\mathrm{SG}\,}}^{s_\psi ,s_e}(\mathbb {R}^{2d})\). Then,
for some \(c_j\in {{\,\mathrm{SG}\,}}^{m_\psi +s_\psi ,t_\psi +t_e}(\mathbb {R}^{2d})\), \(r_j\in {{\,\mathrm{SG}\,}}^{-\infty ,-\infty }(\mathbb {R}^{2d})\), \(j=5,6\).
Furthermore, asymptotic formulae can be given for \(c_j\), \(j=1,\ldots ,6\), in terms of \(\varphi \), p, a and b, see [5].
Remark 34
In particular, we have the following first-order expansion of the symbol of \(c_1\), coming from [5]:
Remark 35
All the results in this section have classical counterparts, that is, when all the starting symbols and phase functions are \({{\,\mathrm{SG}\,}}\)-classical, the resulting objects are \({{\,\mathrm{SG}\,}}\)-classical as well, see [11].
Given a symbol \(p \in {{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1}\), let us consider the eikonal equation
By the theory developed in [6, 11], Proposition 36 holds true.
Proposition 36
For some small enough \(T_0\in (0,T]\), Eq. (26) admits a unique solution \(\varphi \in C^1([0,T_0],{{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1}(\mathbb {R}^{2d}))\). Moreover, \(\varphi (t,x,\xi )\in {\mathcal {P}}_r\) for all \(0\le t\le T_0\).
Remark 37
Using the standard procedure for solving hyperbolic evolution equations modulo regular terms (cf. [6, 10, 11]), it is possible to construct a short-time parametrix for \(\hbox {e}^{-itP}\), where \(P \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{1,1}\).
1.3 A.3. Trace Operators on the SG-Algebra
Various trace functionals on \({{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}(\mathbb {R}^d)\) were introduced and studied by Nicola in [31], including a notion of Wodzicki residue. In [3], such concepts have been extended to \({{\,\mathrm{SG}\,}}\)-classical operators on the class of manifolds with ends described in Section A.4, see also [23, 24]. Here, we recall the definitions of such functionals in terms of the symbolic structure, as well as their relation with the spectral \(\zeta \)-function.
Let \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(\mathbb {R}^d)\) be elliptic, self-adjoint, and positive.Footnote 5 We set
where, respectively,
A further functional \(\widehat{{{\,\mathrm{TR}\,}}}_{\theta }(A)\) was defined in [3, (3.6)], called angular term:
Remark 38
In general, it is rather cumbersome to evaluate the angular term defined in (27). In the case \(m_\psi =m_e=-d\), the computation is easier. Indeed, by [3, Proposition 1.10],
One can also define the traces as the residues of the spectral zeta function (cf. [3, Section 3 and Section 4]). Notice the opposite sign convention for \(\zeta (s) = {{\,\mathrm{Tr}\,}}A^{-s}\) in [3].
Definition 39
The trace operators \({{\,\mathrm{TR}\,}}\) and \(\widehat{{{\,\mathrm{TR}\,}}}_{x,\xi }\) are given by
The following proposition was proved in [3, Theorem 3.3 and Theorem 3.4] with arguments similar to the one employed in the proof of Proposition 19. It gives the relation between the functionals \(\text {Tr}_{\psi , e}(A)\), \(\widehat{\text {Tr}}_\psi (A)\), \(\widehat{\text {Tr}}_e(A)\), \(\widehat{{{\,\mathrm{TR}\,}}}_{\theta }(A)\), and the spectral \(\zeta \)-function of A.
Proposition 40
Let A be as above. Then,
The functional \({{\,\mathrm{TR}\,}}\), the Wodzicki residue for the \({{\,\mathrm{SG}\,}}\)-classical operators setting (cf., e.g. [3, 31] and the references therein), can be extended to all \({{\,\mathrm{SG}\,}}\)-classical operators with integer order in a standard way, cf. [21]. It is also possible to prove that \({{\,\mathrm{TR}\,}}\) is a trace on the algebra \({\mathscr {A}}={{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}/ {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{-\infty ,-\infty }\), see again [21].
1.4 A.4. Manifolds with Cylindrical Ends
We briefly recall the definition of a class of manifolds with cylindrical ends given in [3], together with the concepts, in such environment, of rapidly decreasing function, temperate distribution, \({{\,\mathrm{SG}\,}}\)-calculus, and weighted Sobolev space. In [3], such notions have been illustrated with slight modifications with respect to their original definition in [22]. This class of manifolds was introduced by describing explicitly the admissible atlases, in the spirit of the definition of \({{\,\mathrm{SG}\,}}\)-manifolds given by Schrohe [34]. Without loss of generality, to keep notation simpler, we focus on the manifolds with a single cylindrical end.
Definition 41
A manifold with a cylindrical end of dimension d is a triple (M, Y, [f]), where \(M= {\mathscr {M}} \amalg _C {\mathscr {C}}\) is a d-dimensional smooth manifold and
-
1.
\({\mathscr {M}}\) is a smooth manifold, given by \({\mathscr {M}}=(M_0{\setminus } D)\cup C\) with a d-dimensional smooth compact manifold without boundary \(M_0\), D a closed disc of \(M_0\), and \(C\subset D\) a collar neighbourhood of \(\partial D\) in \(M_0\);
-
2.
\({\mathscr {C}}\) is a smooth manifold with boundary \(\partial {\mathscr {C}}=Y\), with Y diffeomorphic to \(\partial D\);
-
3.
\(f: [\delta _f, \infty ) \times {\mathbb {S}}^{d-1} \rightarrow {\mathscr {C}}\), \(\delta _f>0\), is a diffeomorphism, \(f(\{\delta _f \}\times {\mathbb {S}}^{d-1})=Y\), and \(f([\delta _f,\delta _f+\varepsilon _f) \times {\mathbb {S}}^{d-1})\), \(\varepsilon _f>0\), is diffeomorphic to C;
-
4.
the symbol \(\amalg _C\) means that we are gluing \({\mathscr {M}}\) and \({\mathscr {C}}\), through the identification of C and \(f([\delta _f,\delta _f+\varepsilon _f) \times {\mathbb {S}}^{d-1})\);
-
5.
the symbol [f] represents an equivalence class in the set of functions
$$\begin{aligned} \{ g: [\delta _g, \infty ) \times {\mathbb {S}}^{d-1} \rightarrow {\mathscr {C}} :&g \text { is a diffeomorphism, } \\&g(\{\delta _g\}\times {\mathbb {S}}^{d-1})=Y \text{ and } \\&g([\delta _g, \delta _g+\varepsilon _g) \times {\mathbb {S}}^{d-1}), \hbox {} \varepsilon _g>0 \hbox {, is diffeomorphic to~} C\} \end{aligned}$$where \(f \sim g\) if and only if there exists a diffeomorphism \(\Theta \in \text {Diff}({\mathbb {S}}^{d-1})\) such that
$$\begin{aligned} (g^{-1} \circ f)(\rho , \omega )= (\rho , \Theta (\omega )) \end{aligned}$$(32)for all \(\rho \ge \max \{\delta _f, \delta _g\}\) and \(\omega \in {\mathbb {S}}^{d-1}\).
We use the following notation:
-
\(U_{\delta _f}= \{x \in \mathbb {R}^d :|x|> \delta _f\}\);
-
\( {\mathscr {C}}_\tau = f([\tau , \infty ) \times {\mathbb {S}}^{d-1})\), where \(\tau \ge \delta _f\). The equivalence condition (32) implies that \({\mathscr {C}}_\tau \) is well defined;
-
\(\displaystyle \pi : \mathbb {R}^d{\setminus }\{0\}\rightarrow (0, \infty ) \times {\mathbb {S}}^{d-1}: x \mapsto \pi (x)= \Big (|x|, \frac{x}{|x|}\Big )\);
-
\(f_\pi = f\circ \pi : \overline{U_{\delta _f}} \rightarrow {\mathscr {C}}\) is a parameterization of the end. Let us notice that, setting \(F=g^{-1}_\pi \circ f_\pi \), the equivalence condition (32) implies
$$\begin{aligned} F(x)= |x| \; \Theta \Big (\frac{x}{|x|}\Big ). \end{aligned}$$(33)We also denote the restriction of \(f_\pi \) mapping \(U_{\delta _f}\) onto \(\dot{{\mathscr {C}}}={\mathscr {C}}{\setminus } Y\) by \({\dot{f}}_\pi \).
The couple \((\dot{{\mathscr {C}}}, {\dot{f}}_\pi ^{-1})\) is called the exit chart. If \({\mathscr {A}}=\{(\Omega _j, \psi _j)\}_{j=1}^N\) is such that the subset \(\{(\Omega _j, \psi _j)\}_{j=1}^{N-1}\) is a finite atlas for \({\mathscr {M}}\) and \((\Omega _N, \psi _N)=(\dot{{\mathscr {C}}}, {\dot{f}}_\pi ^{-1})\), then M, with the atlas \({\mathscr {A}}\), is a \({{\,\mathrm{SG}\,}}\)-manifold (see [34]). An atlas \({\mathscr {A}}\) of such kind is called admissible. From now on, we restrict the choice of atlases on M to the class of admissible ones. We introduce the following spaces, endowed with their natural topologies:
M is also tacitly assumed to be endowed with a volume form \(\mathrm{d}\mu \) (for instance, induced by a Riemannian metric), so that the spaces \(L^p(M)\), \(p\in [1,\infty ]\), can be defined as well.
Definition 42
The set \({{\,\mathrm{SG}\,}}^{m_\psi , m_e}(U_{\delta _f})\) consists of all the symbols \(a \in \mathcal {C}^\infty (U_{\delta _f})\) which fulfill (5) for \((x,\xi ) \in U_{\delta _f} \times \mathbb {R}^d\) only. Moreover, the symbol a belongs to the subset \({{\,\mathrm{SG}\,}}_{\mathrm {cl}}^{m_\psi , m_e}(U_{\delta _f})\) if it admits expansions in asymptotic sums of homogeneous symbols with respect to x and \(\xi \) as in Definition 9, where the remainders are now given by \({{\,\mathrm{SG}\,}}\)-symbols of the required order on \(U_{\delta _f}\).
Note that since \(U_{\delta _f}\) is conical, the definition of homogeneous and classical symbol on \(U_{\delta _f}\) makes sense. Moreover, the elements of the asymptotic expansions of the classical symbols can be extended by homogeneity to smooth functions on \(\mathbb {R}^d{\setminus }\{0\}\), which will be denoted by the same symbols. It is a fact that, given an admissible atlas \(\{(\Omega _j, \psi _j)\}_{j=1}^N\) on M, there exists a partition of unity \(\{\varphi _j\}\) and a set of smooth functions \(\{\chi _j\}\) which are compatible with the \({{\,\mathrm{SG}\,}}\)-structure of M, that is (see [34]):
-
\(\mathrm {supp}\,\varphi _j\subset \Omega _j\), \(\mathrm {supp}\,\chi _j\subset \Omega _j\), \(\chi _j\,\varphi _j=\varphi _j\), \(j=1,\ldots ,N\);
-
\(|\partial ^\alpha (\varphi _N\circ {\dot{f}}_\pi )(x)|\le C_\alpha |x|^{-|\alpha |}\) and \(|\partial ^\alpha (\chi _N\circ {\dot{f}}_\pi )(x)|\le C_\alpha |x|^{-|\alpha |}\) for all \(x\in U_{\delta _f}\).
Moreover, \(\varphi _N\) and \(\chi _N\) can be chosen so that \(\varphi _N\circ {\dot{f}}_\pi \) and \(\chi _N\circ {\dot{f}}_\pi \) are homogeneous of degree 0 on \(U_\delta \). We denote by \(u^*\) the composition of \(u:\psi _j(\Omega _j)\subset \mathbb {R}^d\rightarrow \mathbb {C}\) with the coordinate patches \(\psi _j\), and by \(v_*\) the composition of \(v:\Omega _i\subset M\rightarrow \mathbb {C}\) with \(\psi _j^{-1}\), \(j=1,\ldots ,N\). We now recall the definition of \({{\,\mathrm{SG}\,}}\)-pseudodifferential operator on M.
Definition 43
Let M be a manifold with a cylindrical end. A linear operator \(A:{\mathscr {S}}(M)\rightarrow {\mathscr {S}}'(M) \) is a \({{\,\mathrm{SG}\,}}\)-pseudodifferential operator of order \((m_\psi , m_e)\) on M, and we write \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(M)\) if, for any admissible atlas \(\{(\Omega _j, \psi _j)\}_{j=1}^N\) on M with exit chart \((\Omega _N,\psi _N)\):
-
1.
for all \(j=1, \ldots , N-1\) and any \(\varphi _j,\chi _j\in \mathcal {C}^\infty _c(\Omega _j)\), there exist symbols \(a_j \in S^{m_\psi }(\psi _j(\Omega _j))\) such that
$$\begin{aligned}&(\chi _j A \varphi _j \,u^*)_*(x)\\&\quad = (2\pi )^{-d}\iint \hbox {e}^{i(x-y)\xi }\chi _j(x)\,a_j(x,\xi )\, \varphi _j(y)\,u(y) \mathrm{d}y \mathrm{d}\xi , u \in \mathcal {C}^\infty (\psi _j(\Omega _j)); \end{aligned}$$ -
2.
for any \(\varphi _N,\chi _N\) of the type described above, there exists a symbol \(a_N \in {{\,\mathrm{SG}\,}}^{m_\psi , m_e}(U_{\delta _f})\) such that
$$\begin{aligned}&(\chi _N A \varphi _N\,u^*)_*(x)\\&\quad = (2\pi )^{-d}\iint \hbox {e}^{i(x-y) \xi }\chi _N(x)\,a_N(x,\xi )\,\varphi _N(y)\, u(y) \mathrm{d}y \mathrm{d}\xi , u \in {\mathscr {S}}_0(U_{\delta _f}); \end{aligned}$$ -
3.
\(K_A\), the Schwartz kernel of A, is such that
$$\begin{aligned}&K_A \in \mathcal {C}^\infty \big ((M \times M) {\setminus } \varDelta \big ) \bigcap {\mathscr {S}}\big ((\dot{{\mathscr {C}}} \times \dot{{\mathscr {C}}}){\setminus } W\big ) \end{aligned}$$where \(\varDelta \) is the diagonal of \(M \times M\) and \(W= ({\dot{f}}_\pi \times {\dot{f}}_\pi )(V)\) with any conical neighbourhood V of the diagonal of \(U_{\delta _f} \times U_{\delta _f}\).
The most important local symbol of A is \(a_N\), which we will also denote \(a^f\), to remind its dependence on the exit chart. Our definition of \({{\,\mathrm{SG}\,}}\)-classical operator on M differs slightly from the one in [22].
Definition 44
Let \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(M)\). A is a \({{\,\mathrm{SG}\,}}\)-classical operator on M, and we write \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\), if \(a^f(x,\xi ) \in {{\,\mathrm{SG}\,}}_{\mathrm {cl}}^{m_\psi , m_e}(U_{\delta _f})\) and the operator A, restricted to the manifold \({\mathscr {M}}\), is classical in the usual sense.
Remark 45
Since M is a \({{\,\mathrm{SG}\,}}\)-manifold, the concepts of \({{\,\mathrm{SG}\,}}\) symbols and operators, as well as of the Schwartz spaces of functions and distributions, are invariant with respect to the choice of the atlas in the class of admissible ones. Given the special structure of the exit chart, the same is true also for the classical \({{\,\mathrm{SG}\,}}\) symbols and operators, see [22].
The principal homogeneous symbol \(a_{m_\psi , \cdot }\) of a \({{\,\mathrm{SG}\,}}\)-classical operator \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\) is of course well defined as a smooth function on \(T^*M{\setminus } 0\). In order to give an invariant definition of principal symbol with respect to x of an operator \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\), the subbundle \(T_Y^*M=\{(x,\xi ) \in T^*M:x \in Y, \, \xi \in T_x^*M\}\) was introduced. The notion of ellipticity can be extended to operators on M as well.
Definition 46
Let \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\), and let us fix an exit map \(f_\pi \). We can define local objects \(a_{m_\psi -j, m_e-k}, a_{\cdot , m_e-k} \) as
Definition 47
An operator \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\) is elliptic if the principal part of \(a^f \in SG^{m_\psi , m_e}(U_{\delta _f})\) satisfies the \({{\,\mathrm{SG}\,}}\)-ellipticity conditions on \(U_{\delta _f}\times \mathbb {R}^d\) and the operator A, restricted to the manifold \({\mathscr {M}}\), is elliptic in the usual sense.
Proposition 48
The properties \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(M)\) and \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\) as well as the notions of ellipticity do not depend on the (admissible) atlas. Moreover, the local functions \(a_{\cdot , m_e}\) and \(a_{m_\psi , m_e}\) give rise to invariantly defined elements of \(\mathcal {C}^\infty (T_Y^*M)\) and \(\mathcal {C}^\infty (T_Y^*M{\setminus } 0)\), respectively.
Then, with any \(A\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{m_\psi ,m_e}_\mathrm {cl}(M)\), it is associated an invariantly defined principal symbol in three components \(\sigma (A)=(a_{m_\psi ,.},a_{.,m_e},a_{m_\psi , m_e})\). Finally, through local symbols given by \(p_j(x,\xi )=\left\langle \xi \right\rangle ^{s_\psi }\), \(j=1,\ldots ,N-1\), and \(p^f(x,\xi )=\left\langle \xi \right\rangle ^{s_\psi }\left\langle x\right\rangle ^{s_e}\), \(s_\psi , s_e\in \mathbb {R}\), we get a \({{\,\mathrm{SG}\,}}\)-elliptic operator \(\varPi _{s_\psi ,s_e}\in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}^{s_\psi ,s_e}_\mathrm {cl}(M)\) and introduce the (invariantly defined) weighted Sobolev spaces \(H^{s_\psi ,s_e}(M)\) as
The properties of the spaces \(H^{s_\psi ,s_e}(\mathbb {R}^d)\) extend to \(H^{s_\psi ,s_e}(M)\) without any change, as well as the continuous action on them of the \({{\,\mathrm{SG}\,}}\)-operators.
Proposition 49
([3])n The zeta function \(\zeta (s) = {{\,\mathrm{Tr}\,}}A^{-s}\) and the trace operators \({{\,\mathrm{TR}\,}}(A)\) and \(\widehat{{{\,\mathrm{TR}\,}}}_{x,\xi }(A)\) are well defined for any positive elliptic \(A \in {{\,\mathrm{Op}\,}}{{\,\mathrm{SG}\,}}_\mathrm {cl}^{m_\psi ,m_e}(M)\) on a manifold with cylindrical ends M.
The relationship between manifolds with cylindrical ends and asymptotically Euclidean manifolds is as follows. Let M be a manifold with cylindrical ends as in Definition 41. Then, we can use the radial compactification and choose a scattering metric on the compactification X that is compatible with the \({{\,\mathrm{SG}\,}}\)-structure (see the similar concept of \({\mathcal {S}}\)-manifolds in [4, Sects. 4.1, 4.2]). The boundary \(\partial X\) is then a disjoint union of components diffeomorphic to spheres.
An asymptotically Euclidean manifold X with \(\partial X\cong \mathbb {S}^{d-1}\) as the compactification of a manifold M with a cylindrical end. Step (I): decomposition of X. Step (II): construction of \(M_0\). Step (III): construction of M
On the other hand, if the boundary of an asymptotically Euclidean manifold (X, g), with boundary defining function \(\rho _X\), consists of a disjoint union of components diffeomorphic to spheres, then it is the compactification of a manifold with cylindrical ends with the same \({{\,\mathrm{SG}\,}}\)-structure (see Fig. 1). In fact, we may assume without loss of generality that the boundary has a single connected component, that is, \({\partial }X \cong \mathbb {S}^{d-1}\). Choose \(\delta > 0\) sufficiently small, and let \(U \subset X\) be a relatively open collar neighbourhood of \({\partial }X\), with coordinates \(\phi : [0,\delta ) \times \mathbb {S}^{d-1} \rightarrow U\), such that \(\rho _X(\phi (r,\theta )) = r\) for all \((r,\theta ) \in [0,\delta ) \times \mathbb {S}^{d-1}\) and \(\rho _X(x) \ge \delta \) for \(x \in X {\setminus } U\).
We set \(M = X^o\) and \({\mathscr {C}}=\phi ((0,\delta /2]\times \mathbb {S}^{d-1})\), \(C=\phi ((\delta /4,\delta /2]\times \mathbb {S}^{d-1})\), and define the local coordinates near the boundary
where \(\delta _f = 2/\delta \).
We find \(Y={\partial }{\mathscr {C}} = \{x \in M :\rho _X(x) = \delta /2\}\cong \mathbb {S}^{d-1}\cong {\partial }X\). Finally, setting \(D={{\mathbb {B}}^d}\), \(M_0=\overline{(X{\setminus }\overline{{\mathscr {C}}})}\amalg _{Y} D\), and \({\mathscr {M}} = (M_0 {\setminus } D)\cup C\), we obtain a decomposition of M as in Definition 41 and f satisfies condition (3) of Definition 41 with \(\varepsilon _f = 2 / \delta \). Then, (M, Y, [f]) is a manifold with cylindrical ends. As in [7, Example 1.14], we see that \({{\,\mathrm{SG}\,}}^{m_\psi ,m_e}(M) \cong \rho _X^{-m_e} \rho _\Xi ^{-m_\psi } \mathcal {C}^\infty ({}^{\mathrm {sc}}\,\overline{T}^*X)\) under radial compactification,Footnote 6 which proves the claim. In this sense, we may view manifolds with cylindrical ends as a proper subclass of asymptotically Euclidean manifolds.
To keep this exposition within a reasonable length, and avoid to deviate from our main focus, the detailed analysis of the extension of some of the existing results mentioned above, which we employ to prove the main theorems of this paper, to general asymptotically Euclidean manifolds will be illustrated elsewhere.
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Coriasco, S., Doll, M. Weyl Law on Asymptotically Euclidean Manifolds. Ann. Henri Poincaré 22, 447–486 (2021). https://doi.org/10.1007/s00023-020-00995-1
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DOI: https://doi.org/10.1007/s00023-020-00995-1