Resolvents of Bochner Laplacians in the semiclassical limit

L. Charles

Abstract

We introduce a new class of pseudodifferential operators, called Heisenberg semiclassical pseudodifferential operators, to study the space of sections of a power of a line bundle on a compact manifold, in the limit where the power is large. This class contains the Bochner Laplacian associated to a connection of the line bundle, and when the curvature is nondegenerate, its resolvent and some associated spectral projections, including generalised Bergman kernels.

1 Introduction

The spectral analysis of the Bochner Laplacians acting on sections of a line bundle with a large curvature, has many applications ranging from complex geometry to mathematical physics: holomorphic Morse inequalities and Bergman kernels [20], dynamical systems [13], geometric quantization [2] or large magnetic field limit of Schrödinger operators [23], [22], [17] to quote just a few references. In this paper we introduce an algebra of pseudodifferential operators, shaped to study the bottom of the spectra of these Laplacian at small scale.

To understand how the scales matter, let us state two Weyl laws corresponding to two different regimes. Let $(M^{n},g)$ be a closed Riemannian manifold and $L\rightarrow M$ a Hermitian line bundle with a connection $\nabla$ . For any integer $k$ , the Bochner Laplacian $\Delta_{k}=\tfrac{1}{2}\nabla^{*}\nabla$ acting on sections of $L^{k}$ is an elliptic differential operator. Hence $\Delta_{k}$ with domain ${\mathcal{C}}^{\infty}(M,L^{k})$ , is essentially self-adjoint. Its spectrum is a subset of ${\mathbb{R}}_{\geqslant 0}$ and consists only of eigenvalues with finite multiplicity.

When $k$ is large, the structure of this spectrum depends essentially on the curvature $\frac{1}{i}{\omega}$ of $\nabla$ . Since $\nabla$ is assumed to preserve the Hermitian structure, ${\omega}$ is a real closed $2$ -form of $M$ . For the first regime we will need the twisted symplectic form $\Omega$ of $T^{*}M$ defined by

\displaystyle\Omega=\sum d\xi_{i}\wedge dx_{i}+p^{*}{\omega}

(1)

where $p$ is the projection $T^{*}M\rightarrow M$ . For the second one, we will assume that ${\omega}$ and $g$ are compatible in the sense that ${\omega}(x,y)=g(jx,y)$ for an (almost) complex structure $j$ . The Weyl laws at the energy scales $k^{2}$ and $k$ are respectively:

For any $\lambda>0$ , the number $N_{k}(\lambda)$ of eigenvalues of $k^{-2}\Delta_{k}$ smaller than $\lambda$ and counted with multiplicities satisfies

\displaystyle N_{k}(\lambda)\sim\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\operatorname{vol}\{\xi\in T^{*}M,\;\tfrac{1}{2}|\xi|_{x}^{2}\leqslant\lambda\}\quad\text{ as $k\rightarrow\infty$}

(2)

with the volume computed with respect to the Liouville form $\frac{1}{n!}\Omega^{n}$ .

if ${\omega}$ and $g$ are compatible, then for any $M>0$ there exists $C>0$ such that for any $k$ ,

\displaystyle\operatorname{spec}(k^{-1}\Delta_{k})\cap]-\infty,M]\subset(\tfrac{1}{2}+{\mathbb{N}})+Ck^{-\frac{1}{2}}]-1,1[,

(3)

and for any $m\in{\mathbb{N}}$ ,

\displaystyle\sharp\Bigl{(}\operatorname{spec}(k^{-1}\Delta_{k})\cap\bigl{(}\tfrac{1}{2}+m+]-\tfrac{1}{4},\tfrac{1}{4}[\bigr{)}\Bigr{)}\sim\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n/2}{m+n-1\choose m}

(4)

The estimate (2) does not appear in the literature, but it is actually a small variation of the Weyl law of semiclassical pseudodifferential operators on a compact manifold with semiclassical parameter $h=k^{-1}$ , as we will explain below. The cluster structure (3) and the estimate (4) of the number of eigenvalues in each cluster have been proved in [13]. This eigenvalue number is actually given by a Riemann-Roch formula when $k$ is sufficiently large [6], [7].

These results rely on a good description of the resolvents $(k^{-\epsilon}\Delta_{k}-z)^{-1}$ , which allows to study the $f(k^{-\epsilon}\Delta_{k})$ for $f$ a smooth compactly supported function, where $\epsilon=2$ or $1$ according to the regime. In the first case, $f(k^{-2}\Delta_{k})$ and $(k^{-2}\Delta_{k}-z)^{-1}$ are semiclassical twisted pseudodifferential operators, a class of operators which was introduced in [4, Chapitre 4] and used recently in [14]. The local theory of these operators is exactly the same as the one of the standard semiclassical pseudodifferential operators with $h=k^{-1}$ , whereas the global theory involves the twisted symplectic form (1). Similar operators have been studied as well on ${\mathbb{R}}^{n}$ under the name of magnetic pseudodifferential operators, cf. [19] for an introduction and many references.

For the second regime, the theory is much less developed. The proof of (3) is based on an approximation of the resolvent of $k^{-1}\Delta_{k}$ , which is obtained by gluing local resolvents of some model operators deduced from $\Delta_{k}$ by freezing the coordinates in an appropriate way. The main difficulty in this construction is that the number of terms we have to glue increases with $k$ . Still these approximations have been used successfully to prove (3) and later to describe the spectral projections onto the various clusters as generalizations of Bergman kernels [6], [7], [18].

In this paper, we introduce a new class of pseudodifferential operators, and we prove that it contains the resolvent of $k^{-1}\Delta_{k}$ when ${\omega}$ is nondegenerate, as well as the spectral projector associated to the cluster (3). This operator class is in a sense a semiclassical version of the class of Heisenberg pseudodifferential operators, which has been developed for the study of the $\overline{\partial}_{b}$ -complex [1], [12]. For this reason, we call our operators semiclassical Heisenberg pseudodifferential operators.

Just as the usual Heisenberg operators have a symbol of type $(\frac{1}{2},\frac{1}{2})$ , the semiclassical ones have an exotic symbol of type $\frac{1}{2}$ : at each derivative, a power of $h^{\frac{1}{2}}$ is lost. Recall that the semiclassical operators with a symbol of type $\frac{1}{2}$ form a limit class: they are closed under product, the usual operator norm estimates hold, but the standard expansions in the symbolic calculus do not hold because all the terms in these expansions have the same magnitude, cf. for instance [10, Proposition 7.7, Theorem 7.9 and Theorem 7.11]. As we will see in the sequel [3] of this paper, the semiclassical Heisenberg operators are closed under product. However the composition of their principal symbols, which are functions on $T^{*}M$ , is not the usual product. Instead it is a fiberwise product, whose restriction to each fiber $T_{x}^{*}M$ depends on the curvature ${\omega}_{x}$ : when ${\omega}_{x}$ is nondegenerate, it is essentially the Weyl product whereas when ${\omega}_{x}=0$ , this is the usual function product. So in general the principal symbol composition is not commutative.

Our definition of the Heisenberg operators is based on the usual semiclassical pseudodifferential operators, from which we deduce easily many of their properties, except what regards their composition. In this paper, we only address the Heisenberg composition of differential operators with pseudodifferential ones, because this suffices to show that the resolvent and cluster spectral projectors of $k^{-1}\Delta_{k}$ are Heisenberg operators. We have also included an exposition of the theory of twisted pseudodifferential operators, because this is not really standard material, and this helps to understand the specificity of the Heisenberg pseudodifferential operators. In the remainder of the introduction, we state our main result.

Twisted pseudodifferential operators

As the usual Laplace-Beltrami operator, $\Delta_{k}$ has the following local expression in a coordinate chart $(U,x_{i})$

k^{-2}\Delta_{k}=-\tfrac{1}{2\sqrt{g}}\sum_{j,\ell=1}^{n}\pi_{j}g^{j\ell}\sqrt{g}\,\pi_{\ell}

where for any $j=1$ , $\pi_{j}$ is the dynamical moment $\pi_{j}:=(ik)^{-1}\nabla^{L^{k}}_{j}$ with $\nabla^{L^{k}}_{j}$ the covariant derivative of $L^{k}$ with respect to $\partial_{x_{j}}$ .

Let $s\in{\mathcal{C}}^{\infty}(U,L)$ be such that $|s|=1$ and let $\beta\in\Omega^{1}(U,{\mathbb{R}})$ be the corresponding connection 1-form, $\nabla s=-i\beta\otimes s$ . Identifying ${\mathcal{C}}^{\infty}(U,L^{k})$ with ${\mathcal{C}}^{\infty}(U)$ through the frame $s^{k}$ ,

\pi_{j}=(ik)^{-1}\partial_{x_{j}}-\beta_{j}

where $\beta_{j}(x)=\beta(x)(\partial_{x_{j}})$ . Under these local identifications, the $\pi_{j}$ ’s and consequently $k^{-2}\Delta_{k}$ are semiclassical differential operators with $h=k^{-1}$ , their symbols being respectively $\xi_{j}-\beta_{j}$ and $\frac{1}{2}|\xi-\beta(x)|^{2}$ .

These symbols become independent of $s$ if we pull-back them by the momentum shift $T_{\beta}:T^{*}U\rightarrow T^{*}U$ , $(x,\xi)\rightarrow(x,\xi+{\beta}(x))$ . The same method will be used to define the symbol of a twisted pseudodifferential operator, cf. (8). Similarly, the shift $T^{*}_{\beta}$ is used in classical mechanic to write the motion equations of a particle in a magnetic field in an invariant way: $T_{\beta}$ sends the Hamiltonian $\frac{1}{2}|\xi-\beta|^{2}$ with the standard symplectic form $\sum d\xi_{i}\wedge dx_{i}$ to the usual kinetic energy $\frac{1}{2}|\xi|^{2}$ with the twisted symplectic form $\Omega$ .

Let us briefly introduce the twisted pseudodifferential operators of $L$ , the complete definition will be given in Section 2. Let us start with the residual class. An operator family

\displaystyle P=(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}),\;k\in{\mathbb{N}})

(5)

belongs to $k^{-\infty}\Psi^{-\infty}(L)$ if for each $k$ , the Schwartz kernel of $P_{k}$ is smooth, its pointwise norm is in $\mathcal{O}(k^{-\infty})$ uniformly on $M$ and the same holds for its successive derivatives. For any local data $\delta=(U,s,\rho)$ consisting of an open set $U$ of $M$ , a frame $s\in{\mathcal{C}}^{\infty}(U,L)$ such that $|s|=1$ and a function $\rho\in{\mathcal{C}}^{\infty}_{0}(U)$ , we define the local form of a family $P$ as in (5) as:

\displaystyle P^{\delta}_{k}:{\mathcal{C}}^{\infty}(U)\rightarrow{\mathcal{C}}^{\infty}(U),\qquad(P^{\delta}_{k}f)s^{k}=\rho P_{k}(\rho fs^{k}),\qquad k\in{\mathbb{N}}

(6)

A twisted pseudodifferential operator of order $m$ is a family $P$ as in (5) such that for any $\rho_{1}$ , $\rho_{2}\in{\mathcal{C}}^{\infty}(M)$ with disjoint supports, $(\rho_{1}P_{k}\rho_{2})$ belongs to $k^{-\infty}\Psi^{-\infty}(L)$ and for any local data $\delta=(U,s,\rho)$ , $P_{k}^{\delta}=Q^{\delta}_{k^{-1}}$ where $(Q_{h}^{\delta}:{\mathcal{C}}^{\infty}(U)\rightarrow{\mathcal{C}}^{\infty}(U),\;h\in(0,1])$ is a semiclassical pseudodifferential operator of order $m$ . So in terms of coordinates $(x_{i})$ on $U$ the Schwartz kernel of $P_{k}^{\delta}$ has the form

\displaystyle P_{k}^{\delta}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int e^{ik\xi\cdot(x-y)}a(k^{-1},\tfrac{1}{2}(x+y),\xi)\;d\xi

(7)

for some semiclassical polyhomogeneous symbol $(a(h,\cdot)\in{\mathcal{C}}^{\infty}(U\times{\mathbb{R}}^{n}),\;h\in(0,1])$ of order $m$ . The (principal) symbol of $P$ is the function ${\sigma}\in{\mathcal{C}}^{\infty}(T^{*}M)$ such that for any local data $\delta$ as above

\displaystyle a(h,x,\xi+\beta(x))=\rho(x){\sigma}(x,\xi)+\mathcal{O}(h),

(8)

where $\beta\in\Omega^{1}(U,{\mathbb{R}})$ is the connection one-form of $s$ .

Examples of twisted (pseudo)differential operators of respective order $0$ , $1$ and $2$ are the multiplications by the functions of ${\mathcal{C}}^{\infty}(M)$ , the covariant derivatives $(ik)^{-1}\nabla^{L^{k}}_{X}$ where $X\in{\mathcal{C}}^{\infty}(M,TM)$ , the symbol being $(x,\xi)\rightarrow\langle\xi,X(x)\rangle$ , and the normalised Laplacian $k^{-2}\Delta_{k}$ , its symbol is $(x,\xi)\rightarrow\frac{1}{2}|\xi|_{x}^{2}$ .

It is not difficult to adapt the standard results [8], [27] on the resolvents of elliptic operators and their functional calculus in this setting. Denoting by $\Psi_{\operatorname{tsc}}^{m}(L)$ the space of twisted pseudodifferential operators of order $m$ , we have

•

for any $z\in{\mathbb{C}}\setminus{\mathbb{R}}_{\geqslant 0}$ , the resolvent $(z-k^{-2}\Delta_{k})^{-1}$ belongs to $\Psi_{\operatorname{tsc}}^{-2}(L)$ and its symbol is $(z-\frac{1}{2}|\xi|^{2})^{-1}$ .
•

for any $f\in{\mathcal{C}}^{\infty}_{0}({\mathbb{R}})$ , $f(k^{-2}\Delta_{k})$ belongs to $\Psi_{\operatorname{tsc}}^{-\infty}(L)$ and its symbol is $f(\frac{1}{2}|\xi|^{2})$ .

In particular $\operatorname{tr}f(k^{-2}\Delta_{k})=\bigl{(}\frac{k}{2\pi}\bigr{)}^{n}\int_{T^{*}M}f(\frac{1}{2}|\xi|^{2})\frac{1}{n!}\Omega^{n}+\mathcal{O}(k^{-1+n})$ . The Weyl law (2) follows.

Heisenberg pseudodifferential operators

A (semiclassical) Heisenberg pseudodifferential operator of $(L,\nabla)$ of order $m$ is by definition a family $P$ of operators of the form (5) such that for any $\rho_{1}$ , $\rho_{2}\in{\mathcal{C}}^{\infty}(M)$ with disjoint supports, $(\rho_{1}P_{k}\rho_{2})$ belongs to $k^{-\infty}\Psi^{-\infty}(L)$ and for any local data $\delta=(U,s,\rho)$ as above with a coordinate set $(x_{i})$ on $U$ , the Schwartz kernel of $P_{k}^{\delta}$ has the form

\displaystyle e^{ik{\beta}\bigl{(}\tfrac{x+y}{2}\bigr{)}\cdot(x-y)}\Bigl{(}\frac{\sqrt{k}}{2\pi}\Bigr{)}^{n}\int_{{\mathbb{R}}^{n}}e^{i\sqrt{k}\;\xi\cdot(x-y)}a(k^{-\frac{1}{2}},\tfrac{1}{2}(x+y),\xi)\;d\xi

(9)

where

-

${\beta}=\sum\beta_{i}(x)dx_{i}$ is the connection one-form of $s$ defined as above and viewed as the ${\mathbb{R}}^{n}$ -valued function $x\rightarrow({\beta}_{1}(x),\ldots,{\beta}_{n}(x))$
-

$(a(h,\cdot)\in{\mathcal{C}}^{\infty}(U\times{\mathbb{R}}^{n}),\;h\in(0,1])$ is a semiclassical polyhomogeneous symbols of order $m$ , so in particular $\partial_{x}^{{\alpha}}\partial_{\xi}^{\beta}a=\mathcal{O}_{{\alpha},{\beta}}(\langle\xi\rangle^{m-|\beta|})$ and $a\sim\sum_{\ell=0}^{\infty}\hbar^{\ell}a_{\ell}(x,\xi)$ with polyhomogeneous coefficients $a_{\ell}$ of order $m-\ell$ .

As we see, the Schwartz kernel (9) is the product of an oscillatory factor depending on the frame $s$ with the Schwartz kernel of a semiclassical operator where the semiclassical parameter is $k^{-\frac{1}{2}}$ . We will prove that this formula is consistent with change of frame and that we can define a (principal) symbol ${\sigma}\in{\mathcal{C}}^{\infty}(T^{*}M)$ such that for any local data as above,

a(h,x,\xi)=\rho(x){\sigma}(x,\xi)+\mathcal{O}(h).

Let us denote by $\Psi_{\operatorname{Heis}}^{m}(L,\nabla)$ the space of Heisenberg pseudodifferential operators of order $m$ , and by $\Psi_{\operatorname{Heis}}^{-\infty}(L,\nabla)$ the intersection $\cap_{m}\Psi_{\operatorname{Heis}}^{m}(L,\nabla)$ .

To state our main result, we need to introduce some symbols $R_{d,z}$ and $\pi_{d,E}$ . Recall that for any tempered distribution $a\in\mathcal{S}^{\prime}({\mathbb{R}}_{s}^{d}\times{\mathbb{R}}_{\varsigma}^{d})$ , the Weyl quantization of $a$ is the operator $a^{w}:\mathcal{S}({\mathbb{R}}^{d})\rightarrow\mathcal{S}^{\prime}({\mathbb{R}}^{d})$ with Schwartz kernel at $(s,t)$ :

(2\pi)^{-d}\int e^{i\varsigma\cdot(s-t)}a(\tfrac{1}{2}(s+t),\varsigma)\;d\varsigma.

The (quantum) harmonic oscillator is $H^{w}$ with $H(s,\varsigma)=\tfrac{1}{2}\sum_{i=1}^{d}(s_{i}^{2}+\varsigma_{i}^{2})$ . As an operator of $L^{2}({\mathbb{R}}^{d})$ with domain the Schwartz space, $H^{w}$ is essentially self-adjoint with spectrum $\operatorname{sp}H^{w}=\frac{d}{2}+{\mathbb{N}}$ . Then for any $z\in{\mathbb{C}}\setminus\operatorname{sp}H^{w}$ and $E\in\operatorname{sp}H^{w}$ , $R_{d,z}$ and $\pi_{d,E}$ are the tempered distributions such that

\displaystyle(H^{w}-z)^{-1}=R_{d,z}^{w},\qquad 1_{\{E\}}(H^{w})=\pi^{w}_{d,E}

(10)

By Weyl calculus, $R_{d,z}$ belongs to the symbol class $S^{-2}({\mathbb{R}}^{2d})$ and $R_{d,z}=(H-z)^{-1}$ modulo $S^{-3}({\mathbb{R}}^{2d})$ . Moreover $\pi_{d,E}$ belongs to the Schwartz space, being the Weyl symbol of an orthogonal projector onto a finite dimensional subspace of $\mathcal{S}({\mathbb{R}}^{d})$ . The analytic Fredholm theory can be developed in this setting and it says that the function $z\rightarrow R_{d,z}$ with values in ${\mathcal{C}}^{\infty}({\mathbb{R}}^{2d})$ , or better the symbol space $S^{-2}({\mathbb{R}}^{2d})$ , is meromorphic on ${\mathbb{C}}$ with simple poles at $\frac{d}{2}+{\mathbb{N}}$ whose residues are the $\pi_{d,E}$ .

Theorem 1.1.

Assume ${\omega}$ and $g$ are compatible so that $n=2d$ with $d\in{\mathbb{N}}$ . Then

1.
For any $z\in{\mathbb{C}}\setminus(\tfrac{d}{2}+{\mathbb{N}})$ , there exists $Q(z)\in\Psi_{\operatorname{Heis}}^{-2}(L,\nabla)$ such that
1. -
  
  $(k^{-1}\Delta_{k}-z)Q_{k}(z)\equiv\operatorname{id}$ and $Q_{k}(z)(k^{-1}\Delta_{k}-z)\equiv\operatorname{id}$ mod $k^{-\infty}\Psi^{-\infty}(L)$ .
2. -
  
  $(k^{-1}\Delta_{k}-z)Q_{k}(z)=Q_{k}(z)(k^{-1}\Delta_{k}-z)=\operatorname{id}$ when $k$ is large.
3. -
  
  the symbol of $Q(z)$ restricted to $T_{x}^{*}M$ is the Weyl symbol $R_{d,z}$ of the resolvent of the harmonic oscillator with symbol $\tfrac{1}{2}|\xi|^{2}$ , cf. (10).
2.

For any $E\in\frac{d}{2}+{\mathbb{N}}$ , the spectral projector family

$(1_{[E-1/2,E+1/2]}(k^{-1}\Delta_{k}),k\in{\mathbb{N}})$

belongs to $\Psi_{\operatorname{Heis}}^{-\infty}(L,\nabla)$ . The restriction of its symbol to $T_{x}^{*}M$ is the Weyl symbol $\pi_{d,E}$ of the spectral projector on the $E$ -eigenspace of the harmonic oscillator with symbol $\tfrac{1}{2}|\xi|^{2}$ , cf. (10)

In this statement, we view $H$ , $R_{d,z}$ and $\pi_{d,E}$ as functions on $T^{*}_{x}M$ as follows: choose an orthosymplectic basis $(e_{i},f_{i})$ of $T_{x}M$ , that is $(e_{i},f_{i})$ is an orthonormal basis and for any $i,j$ , ${\omega}(x)(e_{i},e_{j})=0={\omega}(x)(f_{i},f_{j})$ , ${\omega}(x)(e_{i},f_{j})=\delta_{ij}$ . Let $(s_{i},{\varsigma}_{i})$ be the associated coordinates of $T^{*}_{x}M$ , so $s_{i}(\xi):=\xi(e_{i})$ and ${\varsigma}_{i}(\xi):=\xi(f_{i})$ for any $\xi\in T_{x}^{*}M$ . Then any function $f:{\mathbb{R}}^{2d}\rightarrow{\mathbb{C}}$ identifies with the function of $T_{x}^{*}M$

\displaystyle\xi\in T^{*}_{x}M\rightarrow f(s(\xi),\varsigma(\xi)).

(11)

In particular $H(s(\xi),\varsigma(\xi))=\frac{1}{2}|\xi|^{2}$ because $(e_{i},f_{i})$ is orthonormal. The fact that $R_{d,z}(s(\xi),\varsigma(\xi))$ and $\pi_{d,E}(s(\xi),\varsigma(\xi))$ are independent of the choice of the basis $(e_{i},f_{i})$ follows from the symplectic invariance of the Weyl quantization.

It follows as well from the symplectic invariance of Weyl quantization and the $O(n)$ invariance of $H$ that $R_{d,z}$ and $\pi_{d,E}$ are radial functions. A computation from Mehler formula leads to [9]

\displaystyle R_{d,z}=\int_{0}^{1}(1-\tfrac{1}{2}s)^{\frac{d}{2}-z-1}(1+\tfrac{1}{2}s)^{\frac{d}{2}+z-1}e^{-sH}ds,\quad\text{ if }\operatorname{Re}z<d

(12)

We can also compute $\pi_{d,E}$ in terms of Laguerre polynomial [26] :

\displaystyle\pi_{d,E}(\xi)=2^{d}(-1)^{m}e^{-|\xi|^{2}}L_{m}^{d-1}(2|\xi|^{2}),\qquad\text{ where }m=E-\tfrac{d}{2}

(13)

and $L_{m}^{\alpha}(x)=\frac{1}{m!}e^{x}x^{-{\alpha}}\partial_{x}^{m}(e^{-x}x^{m+{\alpha}}).$

Remark 1.2.

1. The proof of the first part of Theorem 1.1 is an adaptation of the standard parametrix construction of an elliptic pseudodifferential operator, the main change being in the symbolic calculus: if $(P_{k})$ belongs to $\Psi_{\operatorname{Heis}}^{m}(L,\nabla)$ and has symbol ${\sigma}$ , then $(k^{-1}\Delta_{k}P_{k})$ belongs to $\Psi_{\operatorname{Heis}}^{m-2}(L,\nabla)$ and its symbol restricted to $T_{x}^{*}M$ is the Weyl product of $\tfrac{1}{2}|\xi|_{x}^{2}$ and ${\sigma}(x,\cdot)$ . By Weyl product, we mean the product of symbols in Weyl quantization, and the identification of functions of $T_{x}^{*}M$ with symbols is done through (11). This explains how the symbols $R_{d,z}$ and $\pi_{d,E}$ appear.

2. A remarkable fact is that the proof of the second assertion of Theorem 1.1 is a direct application of Cauchy formula for the spectral projector of an operator with compact resolvent. This part is much simpler than the proof that $f(k^{-2}\Delta_{k})\in\Psi_{\operatorname{tsc}}^{-\infty}(L)$ , even with the modern approach through Helffer-Sjöstrand formula.

3. The Schwartz kernel of the cluster spectral projectors was described in [18] and [7] as a generalisation of the Bergman kernel. The advantage of considering these projectors as Heisenberg pseudodifferential operators is merely that it connects them directly to the Laplacian and its resolvent. Moreover, in [3], we will explain how can use the Heisenberg calculus instead of the algebra $\mathcal{L}(A)$ of [6] to compute the dimension of each cluster and develop the theory of Toeplitz operators as it was done in [7]. ∎

Outline of the paper

In Section 2, we introduce notations and basic analytical tools to address the large $k$ limit of the space of sections of the $k$ -th power of $L$ , including the theory of semiclassical twisted pseudodifferential operators with their Sobolev spaces. The study of Heisenberg pseudodifferential operators starts in Section 3, from their Schwartz kernel asymptotic to their mapping properties. In Section 4, we introduce the symbol product, which is then used in Section 5 for the composition of differential operators with pseudodifferential operators. This is applied to resolvents and spectral projections in Section 6. In Section 7, we explain how we can add auxiliary bundles to the theory, which provides some important examples.

Acknowledgment

I would like to thank Clotilde Fermanian Kammerer, Colin Guillarmou and Thibault Lefeuvre for useful discussions.

2 Twisted pseudodifferential operators

Symbols

We will use the class of semiclassical polyhomogeneous symbols introduced in [11, Section E.1.2], cf. also [8, Section 6.1]. Let $V$ be an open set of ${\mathbb{R}}^{p}$ and $m\in{\mathbb{R}}$ . For any $\xi\in{\mathbb{R}}^{n}$ , let $|\xi|$ and $\langle\xi\rangle$ be the Euclidean norm and Japanese bracket, so $|\xi|^{2}=\sum\xi_{i}^{2}$ , $\langle\xi\rangle^{2}=1+|\xi|^{2}$ . Let $S^{m}(V,{\mathbb{R}}^{n})$ , $S^{m}_{\operatorname{ph}}(V,{\mathbb{R}}^{n})$ and $S^{m}_{\operatorname{sc}}(V,{\mathbb{R}}^{n})$ be the spaces of symbols (resp. polyhomogeneous symbols, semiclassical polyhomogeneous symbols) of order $m$ . By definition

•

$S^{m}(V,{\mathbb{R}}^{n})$ consists of the families $(a(h,\cdot),\;h\in(0,1])$ of ${\mathcal{C}}^{\infty}(V\times{\mathbb{R}}^{n})$ such that for any compact set $K$ of $V$ , ${\alpha}\in{\mathbb{N}}^{p},{\beta}\in{\mathbb{N}}^{n}$ , there exists $C>0$ such that

|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(h,x,\xi)|\leqslant C\langle\xi\rangle^{m-|{\beta}|},\qquad\forall x\in K,\;\xi\in{\mathbb{R}}^{n},\;h\in(0,1]

•

$b\in S_{\operatorname{ph}}^{m}(V,{\mathbb{R}}^{n})$ if $b\in S^{m}(V,{\mathbb{R}}^{n})$ , $b$ is independent on $h$ and for every $N$ , $b=\sum_{j=0}^{N-1}b_{j}\mod S^{m-N}(V,{\mathbb{R}}^{n})$ with coefficients $b_{j}\in{\mathcal{C}}^{\infty}(V\times{\mathbb{R}}^{n})$ such that $b_{j}(x,t\xi)=t^{m-j}b_{j}(x,\xi)$ when $|\xi|\geqslant 1$ and $t>0$ .
•

$a\in S^{m}_{\operatorname{sc}}(V,{\mathbb{R}}^{n})$ if $a\in S^{m}(V,{\mathbb{R}}^{n})$ and $a-\sum_{\ell=0}^{N-1}h^{\ell}a_{\ell}\in h^{N}S^{m-N}(V,{\mathbb{R}}^{n})$ for some coefficients $a_{\ell}\in S_{\operatorname{ph}}^{m-\ell}(V,{\mathbb{R}}^{n})$ .

More generally these definitions make sense for a real vector bundle $E\rightarrow N$ instead of the product $V\times{\mathbb{R}}^{n}$ . We denote by $S_{*}^{m}(N,E)$ the corresponding spaces and set $S^{\infty}_{*}(N,E)=\bigcup_{m}S^{m}_{*}(N,E)$ for $*=\emptyset,\operatorname{ph},\operatorname{sc}$ . An easy remark is that for any section $u$ of $E$ , the translation $T_{u}$ of ${\mathcal{C}}^{\infty}(E,{\mathbb{C}})$ given by $T_{u}f(x,v)=f(x,v-u(x))$ preserves $S_{*}^{m}(N,E)$ . When $V$ is reduced to a point, we set $S^{m}_{*}({\mathbb{R}}^{n}):=S^{m}_{*}(\{\cdot\},{\mathbb{R}}^{n})$ .

Negligible families

We say that a family $(f_{h},h\in(0,1])$ of ${\mathcal{C}}^{\infty}(N)$ is negligible, and we write $f_{h}=\mathcal{O}_{\infty}(h^{\infty})$ , if all its ${\mathcal{C}}^{\infty}$ -seminorms are in $\mathcal{O}(h^{\infty})$ . This definition is meaningful if $(f_{h})$ is only defined for $h\in D$ where $D$ is any subset of $(0,1]$ whose closure contains $0$ .

Let $L\rightarrow M$ be a Hermitian line bundle and $A\rightarrow M$ a complex vector bundle with rank $r$ . A family $(t_{k}\in{\mathcal{C}}^{\infty}(M,L^{k}\otimes A),\;k\in{\mathbb{N}})$ is said to be negligible if for any open set $U$ of $M$ , any $s\in{\mathcal{C}}^{\infty}(U,L)$ with pointwise norm $|s|=1$ and any frame $(a_{i}\in{\mathcal{C}}^{\infty}(U,A),i=1,\ldots,r)$ , $t_{k}=\sum f_{i,k^{-1}}s^{k}\otimes a_{i}$ on $U$ where each coefficient $f_{i,h}\in\mathcal{O}_{\infty}(h^{\infty})$ . We denote by $\mathcal{O}_{\infty}(k^{-\infty})$ the space of negligible families.

Let $P$ be a family of operators

\displaystyle P=(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}),\;k\in{\mathbb{N}}),

(14)

The Schwartz kernel of each $P_{k}$ is a section of $(L^{k}\boxtimes\overline{L}^{k})\otimes({\mathbb{C}}_{M}\boxtimes|\Lambda|(M))$ , where we denote by $\boxtimes$ the external tensor product of vector bundles, by ${\mathbb{C}}_{M}$ the trivial line bundle over $M$ and by $|\Lambda|(M)$ the density bundle. Since $L^{k}\boxtimes\overline{L}^{k}=(L\boxtimes\overline{L})^{k}$ , the previous definition of a negligible family applies to the family $(P_{k})$ of Schwartz kernels.

We denote by $k^{-\infty}\Psi^{-\infty}(L)$ the space consisting of operator families of the form (14) such that each $P_{k}$ is smoothing with a Schwartz kernel family in $\mathcal{O}_{\infty}(k^{-\infty})$ . As we will see, $k^{-\infty}\Psi^{-\infty}(L)$ is both the residual space of twisted pseudodifferential operators and of Heisenberg pseudodifferential operators.

Semiclassical pseudodifferential operators

Let $\Psi^{m}_{\operatorname{sc}}(M)$ be the space of semiclassical pseudodifferential operators of order $m$ acting on smooth functions of $M$ . By definition $P\in\Psi^{m}_{\operatorname{sc}}(M)$ is a family of operators $(P_{h}:{\mathcal{C}}^{\infty}(M)\rightarrow{\mathcal{C}}^{\infty}(M),h\in(0,1])$ with a Schwartz kernel $K_{h}(x,y)$ satisfying for any $\rho\in{\mathcal{C}}^{\infty}(M^{2})$ ,

1.

if $\operatorname{supp}\rho\cap\operatorname{diag}M=\emptyset$ , then $\rho K_{h}$ is smooth and negligible.
2.

if $\operatorname{supp}\rho\subset U^{2}$ where $(U,x_{i})$ is a coordinate chart of $M$ , then on $U^{2}$

$\displaystyle(\rho K_{h})(x,y)=(2\pi h)^{-n}\int e^{ih^{-1}\xi\cdot(x-y)}a(h,x,y,\xi)\;d\xi$ (15)

with $a\in S_{\operatorname{sc}}^{m}(U^{2},{\mathbb{R}}^{n})$ .

Here and in the sequel, when the Schwartz kernel is written in a coordinate chart, we implicitly use the density $|dx_{1}\ldots dx_{n}|$ . The principal symbol of $P$ is the function ${\sigma}\in S_{\operatorname{ph}}^{m}(M,T^{*}M)$ such that $a(h,x,x,\xi)=\rho(x){\sigma}(x,\xi)+\mathcal{O}(h)$ on $U$ .

Twisted pseudodifferential operators

Let $L\rightarrow M$ be a Hermitian line bundle.

Definition 2.1.

A semiclassical twisted pseudodifferential operator $P$ of $L$ is a family having the form (14) such that for any $\rho\in{\mathcal{C}}^{\infty}(M^{2})$ ,

1.

if $\operatorname{supp}\rho\cap\operatorname{diag}M=\emptyset$ , then $\rho P_{k}$ is smooth and negligible.

if $\operatorname{supp}\rho\subset U^{2}$ where $(U,x_{i})$ is a coordinate chart of $M$ and $s\in{\mathcal{C}}^{\infty}(U,L)$ is such that $|s|=1$ , then on $U^{2}$

(\rho P_{k})(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int e^{ik\xi\cdot(x-y)}a(k^{-1},x,y,\xi)\;d\xi\;s^{k}(x)\otimes\overline{s}^{k}(y)

with $a\in S_{\operatorname{sc}}^{m}(U^{2},{\mathbb{R}}^{n})$ .

To understand the dependence of the oscillatory integral with respect to the choice of the frame $s$ , consider a new frame $t=e^{i\varphi}s$ where $\varphi\in{\mathcal{C}}^{\infty}(U,{\mathbb{R}})$ . Then $\varphi(x)-\varphi(y)=\sum_{j}\psi_{j}(x,y)(x_{j}-y_{j})$ where $\psi_{j}\in{\mathcal{C}}^{\infty}(U^{2},{\mathbb{R}})$ is such that $\psi_{j}(x,x)=\partial_{x_{j}}\varphi(x)$ . Using that $s(x)\otimes\overline{s}(y)=\exp(-i\psi(x,y)\cdot(x-y))t(x)\otimes\overline{t}(y))$ and changing the variable $\xi$ into $\xi+\psi(x,y)$ , we obtain

\displaystyle\begin{split}&\int e^{ik\xi\cdot(x-y)}a(k^{-1},x,y,\xi)\;d\xi\;s^{k}(x)\otimes\overline{s}^{k}(y)\\ &=\int e^{ik\xi\cdot(x-y)}a(k^{-1},x,y,\xi+\psi(x,y))\;d\xi\;t^{k}(x)\otimes\overline{t}^{k}(y)\end{split}

(16)

So multiplying $s$ by $e^{i\varphi}$ amounts to change the amplitude $a$ to $b$ such that $b(h,x,y,\xi)=a(h,x,y,\xi+\psi(x,y))$ . This relation writes on the diagonal

b(h,x,x,\xi)=a(h,x,x,\xi+d\varphi(x)).

Let $\nabla$ be a connection on $L$ preserving the metric. Then $\nabla s=\frac{1}{i}\beta_{s}\otimes s$ , where $\beta_{s}$ is a real one-form of $U$ . Observe that $\nabla t=\frac{1}{i}\beta_{t}\otimes t$ where $\beta_{t}=\beta_{s}-d\varphi$ . So we can define the principal symbol as follows.

Definition 2.2.

The principal symbol ${\sigma}_{\nabla}(P)$ of $P\in\Psi_{\operatorname{tsc}}^{m}(L)$ is the element of $S^{m}_{\operatorname{ph}}(M,T^{*}M)$ such that for any local data $(\rho,U,s,a)$ as in Definition 2.1, we have

a(h,x,x,\xi+\beta_{s}(x))=\rho(x){\sigma}_{\nabla}(P)(x,\xi)+\mathcal{O}(h)

If $\nabla^{\prime}$ is another connection of $L$ preserving the metric, then $\nabla^{\prime}=\nabla+\frac{1}{i}{\alpha}$ with ${\alpha}\in\Omega^{1}(M,{\mathbb{R}})$ and ${\sigma}_{\nabla^{\prime}}(P)(x,\xi)={\sigma}_{\nabla}(P)(x,\xi+{\alpha}(x))$ . It is easy to extend the basic properties of pseudodifferential operators to our setting:

-

If $P\in\Psi_{\operatorname{tsc}}^{m}(L)$ , then ${\sigma}_{\nabla}(P)=0$ if and only if $P\in k^{-1}\Psi_{\operatorname{tsc}}^{m-1}(L)$ .
-

$\bigcap_{m}k^{-m}\Psi_{\operatorname{tsc}}^{-m}(L)=k^{-\infty}\Psi^{-\infty}(L)$ .
-
if $P\in\Psi_{\operatorname{tsc}}^{m}(L)$ and $Q\in\Psi_{\operatorname{tsc}}^{p}(L)$ , then
1. i)
  
  $(P_{k}Q_{k})$ belongs to $\Psi_{\operatorname{tsc}}^{m+p}(L)$ and its principal symbol is the product of the principal symbols of $P$ and $Q$ .
2. ii)
  
  $ik[P_{k},Q_{k}]$ belongs to $\Psi_{\operatorname{tsc}}^{m+p-1}(L)$ and its symbol is the Poisson bracket for the twisted symplectic form (1) where $\frac{1}{i}{\omega}$ is the curvature of $\nabla$ .

It is possible to define the twisted pseudodifferential operators without using local frames. Recall that the Schwartz kernel of an operator acting on ${\mathcal{C}}^{\infty}(M,L^{k})$ is a section of $(L\boxtimes\overline{L})^{k}\otimes({\mathbb{C}}_{M}\boxtimes|\Lambda|^{1}(M))$ . Introduce an open neighborhood $V$ of the diagonal of $M^{2}$ and a section $F\in{\mathcal{C}}^{\infty}(V,L\boxtimes\overline{L})$ such that $|F|=1$ on $V$ and $F(x,x)=1$ for any $x\in M$ , in the sense that $F(x,x)=u\otimes\overline{u}$ for any $u\in L_{x}$ with norm $1$ . We claim that $P\in\Psi_{\operatorname{tsc}}^{m}(L)$ if and only if its Schwartz kernel has the form

\displaystyle F^{k}(x,y)\phi(x,y)K_{k^{-1}}(x,y)+\mathcal{O}_{\infty}(k^{-\infty})

(17)

where $\phi\in{\mathcal{C}}^{\infty}_{0}(V)$ is equal to $1$ on a neighborhood of the diagonal and $(K_{h},\;h\in(0,1])$ is the Schwartz kernel family of a semiclassical pseudodifferential operator $Q\in\Psi_{\operatorname{sc}}^{m}(M)$ . If furthermore $\nabla$ is a connection of $L$ such that the corresponding covariant derivative of $F$ is zero on the diagonal, then $\sigma_{\nabla}(P)={\sigma}(Q)$ .

These facts follow from a computation similar to (16), by writing $F(x,y)=\exp(i\varphi(x,y))s(x)\otimes\overline{s}(y)$ where $s\in{\mathcal{C}}^{\infty}(U,L)$ is such that $|s|=1$ on $U$ and $\varphi\in{\mathcal{C}}^{\infty}(U^{2},{\mathbb{R}})$ satisfies $\varphi(x,x)=0$ .

Semiclassical Sobolev norms

Let $m\in{\mathbb{R}}$ . Denote by $H^{m}(M,L^{k})$ the Sobolev space of sections of $L^{k}$ of order $m$ . Let us give three equivalent definitions of the semiclassical Sobolev norms of a section $u$ of $L^{k}$ . First the norm of $H^{0}(M,L^{k})=L^{2}(M,L^{k})$ is defined by

\displaystyle\|u\|_{L^{2}(M,L^{k})}^{2}=\int_{M}|u(x)|^{2}d\mu(x)

(18)

where $\mu$ is a volume element of $M$ independent of $k$ .

1.

only for integral exponent $m\in{\mathbb{N}}$ : choose a connection $\nabla$ of $L$ , vector fields $(X_{i})_{i=1}^{N}$ of $M$ which generates $T_{x}M$ at each $x$ , and set

$\|u\|_{m}:=\sum_{|{\alpha}|\leqslant m}k^{-|{\alpha}|}\|\nabla_{X}^{\alpha}u\|_{L^{2}(M,L^{k})}$

where for any ${\alpha}\in{\mathbb{N}}^{N}$ , $\nabla_{X}^{\alpha}=\nabla_{X_{1}}^{{\alpha}(1)}\ldots\nabla_{X_{N}}^{{\alpha}(N)}$ .

based on local semi-norms: for any chart $(U,\chi)$ of $M$ , frame $s\in{\mathcal{C}}^{\infty}(U,L)$ such that $|s|=1$ and $\rho\in{\mathcal{C}}^{\infty}_{o}(U)$ we set

\|u\|_{m,U,\chi,s,\rho}=\|\langle k^{-1}\xi\rangle^{m}\hat{v}(\xi)\|_{L^{2}({\mathbb{R}}^{n})}\qquad\text{where }\rho u=(\chi^{*}v)s^{k}

and $\hat{v}$ is the Fourier transform of $v$ . Choose a finite family $(U_{i},\chi_{i},s_{i},\rho_{i})$ of local data such that $M$ is covered by the $\{\rho_{i}=1\}$ and set $\|u\|_{m}:=\sum_{i}\|u\|_{m,U_{i},\chi_{i},s_{i},\rho_{i}}$ .

3.

based on twisted pseudodifferential operators: choose $E\in\Psi_{\operatorname{tsc}}^{m}(L)$ which is elliptic and invertible for any $k$ , and set $\|u\|_{m}=\|Eu\|_{L^{2}(M,L^{k})}$ .

The ellipticity condition is as usual that the principal symbol satisfies for some $C>0$ , $|{\sigma}_{\nabla}(L)(x,\xi)|\geqslant C^{-1}|\xi|^{m}$ when $|\xi|\geqslant C$ . It does not depend on the choice of $\nabla$ .

We claim that all these norms are equivalent with constants uniform in $k$ . Furthermore for any twisted pseudodifferential operator $P\in\Psi_{\operatorname{tsc}}^{p}(L)$ and any $m\in{\mathbb{R}}$ , there exists $C$ such that for any $k$ ,

\displaystyle\|P_{k}u\|_{m}\leqslant C\|u\|_{m+p},\qquad\forall\;u\in{\mathcal{C}}^{\infty}(M,L^{k}).

(19)

3 Heisenberg semiclassical operator

In the introduction, we defined the Heisenberg pseudodifferential operators by expressing locally their Schwartz kernels as oscillatory integrals. Here we will start with a global definition which has the advantage that we can deduce some basic properties of these operators directly from the ones of the semiclassical pseudodifferential operators.

Let $L\rightarrow M$ be a Hermitian line bundle with a connection $\nabla$ preserving the metric. The line bundle $L\boxtimes\overline{L}$ inherits from $L$ a Hermitian metric and a connection. Its restriction to the diagonal is the flat trivial bundle with a natural trivialisation obtained by sending $u\otimes\overline{v}\in L_{x}\otimes\overline{L}_{x}$ to the scalar product of $u$ and $v$ . In the sequel we will use a particular extension of this trivialisation.

Lemma 3.1.

There exist a tubular neighborhood $V$ of the diagonal of $M^{2}$ and $F\in{\mathcal{C}}^{\infty}(V,L\boxtimes\overline{L})$ such that $|F|=1$ on $V$ and

F(x,x)=1,\quad\nabla F(x,x)=0,\quad\nabla_{Y}\nabla_{Y}F(x,x)=0\qquad\forall x\in M

for any vector field $Y$ of $M^{2}$ having the form $Y(x,y)=(X(x),-X(y))$ with $X\in{\mathcal{C}}^{\infty}(M,TM)$ . If $(V^{\prime},F^{\prime})$ satisfies the same conditions, then $F=F^{\prime}\exp(i\psi)$ where $\psi\in{\mathcal{C}}^{\infty}(V\cap V^{\prime},{\mathbb{R}})$ vanishes to third order along the diagonal

Proof.

Consider more generally a closed submanifold $N$ of $M$ , a flat section $E$ of $L|_{N}$ , and a subbundle $\mathcal{D}$ of $TM|_{N}$ such that $\mathcal{D}\oplus TN=TM|_{N}$ . Then we can extend $E$ to a neighborhood of $N$ in such a way that it satisfies on $N$ : $\nabla E=0$ and $\nabla_{X}\nabla_{X}E=0$ for any vector field $X$ of $M$ such that $X|_{N}$ is a section of $\mathcal{D}$ . To see this, introduce a coordinate chart $(U,x_{i},y_{j})$ of $M$ and a unitary frame $s:U\rightarrow L$ such that $N\cap U=\{x_{1}=\ldots=x_{k}=0\}$ , $(\partial_{x_{1}},\ldots,\partial_{x_{k}})$ is a frame of $\mathcal{D}$ and $s$ extends $E$ . Then the section we are looking for is $e^{i\varphi}s$ with

\varphi=\sum_{i=1}^{k}\beta_{i}(0,y)x_{i}+\tfrac{1}{2}\sum_{i,j=1}^{k}(\partial_{x_{j}}\beta_{i})(0,y)x_{i}x_{j}+\mathcal{O}(|x|^{3})

where the $\beta_{i}$ ’s are the functions in ${\mathcal{C}}^{\infty}(U)$ such that $\nabla_{\partial_{x_{i}}}s=\frac{1}{i}\beta_{i}\,s$ . Applying this to $M^{2}$ , $L\boxtimes\overline{L}$ and $\operatorname{diag}M$ instead of $M$ , $L$ , $N$ concludes the proof. ∎

Definition 3.2.

A semiclassical Heisenberg pseudodifferential operator of order $m\in{\mathbb{R}}$ is a family of operators $(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}),\;k\in{\mathbb{N}})$ whose Schwartz kernels have the form

\displaystyle F^{k}(x,y)\phi(x,y)K_{k^{-\frac{1}{2}}}(x,y)+\mathcal{O}_{\infty}(k^{-\infty})

(20)

where $(V,F)$ satisfies the conditions of Lemma 3.1, $\phi\in{\mathcal{C}}^{\infty}_{0}(V)$ is equal to $1$ on a neighborhood of the diagonal and $(K_{h},\;h\in(0,1])$ is the Schwartz kernel family of a semiclassical pseudodifferential operator $(Q_{h})\in\Psi_{\operatorname{sc}}^{m}(M)$ .

The principal symbol ${\sigma}(P)$ of $(P_{k})$ is defined as the principal symbol of $(Q_{h})$ .

We denote by $\Psi^{m}_{\operatorname{Heis}}(L,\nabla)$ the space of semiclassical Heisenberg pseudodifferential operators of order $m$ . For any $P\in\Psi^{m}_{\operatorname{Heis}}(L,\nabla)$ , for any fixed $k$ , $P_{k}$ is a pseudo-differential operator of order $m$ , so $P_{k}$ act on ${\mathcal{C}}^{\infty}(M,L^{k})$ and on $\mathcal{C}^{-\infty}(M,L^{k})$ . The definition clearly does not depend on the choice of the cutoff function $\phi$ . It neither doesn’t depend on the choice of $F$ as will be explained below. To compare with the twisted pseudodifferential operators, observe first that the section $F$ in (17) satisfies a weaker condition than in Definition 3.2 and second in (17), the Schwartz kernel of $Q$ is evaluated at $h=k^{-1}$ , whereas in (20) we have $h=k^{-1/2}$ .

By defining globally the Heisenberg pseudodifferential operators in terms of scalar pseudodifferential operators as in Definition 3.2 instead of the local oscillatory integrals (9), we avoid the usual discussions on the coordinate changes and the principal symbol and we deduce easily the following three facts:

-

If $P\in\Psi_{\operatorname{Heis}}^{m}(L,\nabla)$ , then ${\sigma}(P)=0$ if and only if $P\in k^{-\frac{1}{2}}\Psi_{\operatorname{Heis}}^{m-1}(L,\nabla)$ .
-

$\bigcap_{m}k^{-\frac{m}{2}}\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)=k^{-\infty}\Psi^{-\infty}(L)$ .
-

If $P\in\Psi_{\operatorname{Heis}}^{m}(L)$ and $\rho\in{\mathcal{C}}^{\infty}(M^{2})$ is such that $\operatorname{supp}\rho\cap\operatorname{diag}M=\emptyset$ , then the kernel $(x,y)\rightarrow\rho(x,y)P_{k}(x,y)$ is smooth and negligible.

Unfortunately, the definition 3.2 does not allow to deduce the composition properties of the Heisenberg operators from the one of the semiclassical pseudodifferential operators.

By Lemma 3.1, $F$ is uniquely defined modulo a factor $e^{i\psi}$ with $\psi\in{\mathcal{C}}^{\infty}(U^{2})$ vanishing to third order along the diagonal. Write

\psi=\sum_{|{\alpha}|=3}\psi_{{\alpha}}(x,y)(x-y)^{{\alpha}}

with smooth coefficients $\psi_{\alpha}$ . For any symbol $a\in S^{\infty}(U^{2},{\mathbb{R}}^{n})$ , let $I(a)$ be the oscillatory integral

I(a)(h,x,y)=\int e^{ih^{-1}\xi\cdot(x-y)}a(h,x,y,\xi)\;d\xi.

Lemma 3.3.

For all $a\in S^{m}(U^{2},{\mathbb{R}}^{n})$ , $e^{ih^{-2}\psi(x,y)}I(a)(h,x,y)=I(b)(h,x,y)$ with $b\in S^{m}(U^{2},{\mathbb{R}}^{n})$ having the asymptotic expansion

\displaystyle b=\sum_{\ell=0}^{\infty}\frac{h^{\ell}}{\ell!}L^{\ell}(a),\qquad\text{ with }\quad L=\sum_{|{\alpha}|=3}\psi_{{\alpha}}(x,y)\partial_{\xi}^{\alpha}.

(21)

In particular, if $a\in S^{m}_{\operatorname{sc}}(U^{2},{\mathbb{R}}^{n})$ , then $b\in S^{m}_{\operatorname{sc}}(U^{2},{\mathbb{R}}^{n})$ .

This proves that Definition 3.2 does not depend on the choice of $F$ . Moreover, since $b=a+\mathcal{O}(h)$ , the principal symbol of $(Q_{k})$ is also independent of $F$ .

Proof.

By integration by part, $(x_{i}-y_{i})I(a)=ihI(\partial_{\xi_{i}}a)$ , so $ih^{-2}\psi I(a)=hI(L(a))$ with $L$ given by (21). By Taylor formula, it comes that

e^{ih^{-2}\psi}I(a)=\sum_{\ell=0}^{N}\frac{h^{\ell}}{\ell!}I(L^{\ell}(a))+h^{N+1}r_{N}I(L^{N+1}(a))

with

r_{N}(h,x,y)=\frac{1}{N!}\int_{0}^{1}e^{ith^{-2}\psi(x,y)}(1-t)^{N}\;dt.

Observe that $r_{N}$ is smooth and $h^{2|{\alpha}|}\partial^{\alpha}_{x,y}r_{N}=\mathcal{O}(1)$ . Furthermore, since $I(a)$ is a genuine integral for $m<-n$ , by derivating under the integral sign, when $k\in{\mathbb{N}}$ satisfies $k+m<-n$ , $I(a)\in\mathcal{C}^{k}$ and for any $|{\alpha}|=k$ , $h^{|{\alpha}|}\partial^{\alpha}_{x,y}I(a)=O(1)$ . Since $L^{N+1}(a)$ is a symbol of order $m-3(N+1)$ , it comes that for any $k$ , when $N$ is sufficiently large, $r_{N}I(L^{N+1}(a))$ is of class $\mathcal{C}^{k}$ and for any $|{\alpha}|=k$ , $h^{2|{\alpha}|}\partial^{\alpha}_{x,y}(r_{N}I(L^{N+1}(a)))=\mathcal{O}(1)$ .

So for any $b\in S^{m}(U^{2},{\mathbb{R}}^{n})$ having the asymptotic expansion (21), we have that $e^{ih^{-2}\psi}I(a)=I(b)+\rho$ with $\rho\in h^{\infty}{\mathcal{C}}^{\infty}(U^{2})$ , and we can absorb $\rho$ in $I(b)$ by modifying $b$ by a summand in $h^{\infty}S^{-\infty}(U^{2},{\mathbb{R}}^{n})$ . ∎

Let us explain how we recover the local expression (9) of the introduction. Let $(U,x_{i})$ be a local chart of $M$ and $s\in{\mathcal{C}}^{\infty}(U,L)$ such that $|s|=1$ on $U$ . Let $\beta\in\Omega^{1}(U,{\mathbb{R}})$ be the connection form, $\nabla s=\frac{1}{i}\beta\otimes s$ . Then we easily check that the section $F\in{\mathcal{C}}^{\infty}(U^{2},L\boxtimes\overline{L})$ given by

\displaystyle F(x,y)=e^{i\beta\bigl{(}\tfrac{x+y}{2}\bigr{)}\cdot(x-y)}s(x)\otimes\overline{s}(y)

(22)

satisfies the condition of Lemma 3.1. Consequently, the Schwartz kernel of an operator in $\Psi^{m}_{\operatorname{Heis}}(L)$ has the form $K_{k}s^{k}\boxtimes\overline{s}^{k}$ on $U^{2}$ with

\displaystyle K_{k}(x,y)=e^{ik{\beta}\bigl{(}\tfrac{x+y}{2}\bigr{)}\cdot(x-y)}\Bigl{(}\frac{\sqrt{k}}{2\pi}\Bigr{)}^{n}\int e^{i\sqrt{k}\;\xi\cdot(x-y)}a(k^{-\frac{1}{2}},x,y,\xi)\;d\xi

(23)

with $a\in S^{m}_{\operatorname{sc}}(U^{2},{\mathbb{R}}^{n})$ . Of course, we can assume that $a$ does not depend on $y$ (resp. $x$ ) or that it is on the Weyl form $a(h,x,y,\xi)=b(h,\frac{1}{2}(x+y),\xi)$ with $b\in S^{m}_{\operatorname{sc}}(U,{\mathbb{R}}^{n})$ . In this last case, we recover exactly the expression (9).

Another interesting expression is obtained by rescaling the variable $\xi$ by a square root of $k$ in (23) and absorbing the $\beta$ factor into the amplitude:

\displaystyle\begin{split}K_{k}(x,y)&=e^{ik{\beta}\bigl{(}\tfrac{x+y}{2}\bigr{)}\cdot(x-y)}\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int e^{ik\;\xi\cdot(x-y)}a(k^{-\frac{1}{2}},x,y,\sqrt{k}\;\xi)\;d\xi\\ &=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int e^{ik\;\xi\cdot(x-y)}a\bigl{(}k^{-\frac{1}{2}},x,y,\sqrt{k}\bigl{(}\xi-{\beta}\bigl{(}\tfrac{x+y}{2}\bigr{)}\bigr{)}\bigr{)}\;d\xi\end{split}

(24)

Assume that $a$ is on the Weyl form, $a(h,x,y,\xi)=b(h,\frac{1}{2}(x+y),\xi)$ , then we have that

\displaystyle K_{k}(x,y)=\Bigl{(}\frac{k}{2\pi}\Bigr{)}^{n}\int e^{ik\;\xi\cdot(x-y)}\tilde{b}\bigl{(}k^{-1},\tfrac{1}{2}(x+y),\xi\bigr{)}\;d\xi

(25)

where $\tilde{b}(h,x,\xi)=b(\sqrt{h},x,h^{-\frac{1}{2}}(\xi-{\beta}(x)))$ . So we recognise a semiclassical pseudodifferential operator at $k=h^{-1}$ with a Weyl symbol $\tilde{b}$ .

We call $\tilde{b}$ the effective symbol. As we will see, it satisfies some exotic estimates. Let us introduce the symbol semi-norms of $S^{m}(U,{\mathbb{R}}^{n})$ ,

\|a\|_{m,\ell,K}=\max_{|{\alpha}|+|{\beta}|\leqslant\ell}\sup_{x\in K,\xi\in{\mathbb{R}}^{n}}|\partial_{x}^{{\alpha}}\partial_{\xi}^{{\beta}}a(x,\xi)|\langle\xi\rangle^{-m+|{\beta}|}

where $K$ is a compact subset of $U$ .

Lemma 3.4.

For any $m\in{\mathbb{R}}$ , ${\alpha},{\beta}\in{\mathbb{N}}^{n}$ and compact subset $K$ of $U$ , there exists $C>0$ such that for any $a\in{\mathcal{C}}^{\infty}(U\times{\mathbb{R}}^{n})$ , the function $\tilde{a}(h,x,\xi)=a(x,h^{-\frac{1}{2}}(\xi-{\beta}(x))$ satisfies

|\partial_{x}^{{\alpha}}\partial_{\xi}^{{\beta}}\tilde{a}(h,x,\xi)|\leqslant C\|a\|_{m,\ell,K}h^{-\frac{1}{2}(m_{+}+\ell)}\langle\xi\rangle^{m-|{\beta}|}

for all $0<h\leqslant 1$ , $x\in K$ , $\xi\in{\mathbb{R}}^{n}$ with $\ell=|{\alpha}|+|{\beta}|$ , $m_{+}=\max(m,0)$ .

Proof.

For any $0<\epsilon\leqslant 1$ , we have $\langle\eta\rangle\leqslant\langle\epsilon^{-1}\eta\rangle\leqslant\epsilon^{-1}\langle\eta\rangle$ . Furthermore, if $x\in K$ , $C^{-1}\langle\xi\rangle\leqslant\langle\xi-{\beta}(x)\rangle\leqslant C\langle\xi\rangle$ . So for any $m\in{\mathbb{R}}$ ,

\displaystyle\langle\epsilon^{-1}(\xi-{\beta}(x))\rangle^{m}\leqslant C_{m}\epsilon^{-m_{+}}\langle\xi\rangle^{m}.

(26)

The derivatives of $\tilde{a}_{\epsilon}(x,\xi)=a(x,\epsilon^{-1}(\xi-{\beta}(x)))$ have the form $\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\tilde{a}_{\epsilon}=\tilde{b}_{\epsilon}$ with

\displaystyle b=\sum_{{\alpha}^{\prime},{\beta}^{\prime}}\epsilon^{-|{\beta}^{\prime}|}f_{{\alpha}^{\prime},{\beta}^{\prime}}\partial_{x}^{{\alpha}^{\prime}}\partial_{\xi}^{{\beta}^{\prime}}a

(27)

where the coefficients $f_{{\alpha}^{\prime},{\beta}^{\prime}}$ are in ${\mathcal{C}}^{\infty}(U)$ and don’t depend on $a$ , and we sum over the multi-indices satisfying ${\beta}\leqslant{\beta}^{\prime}$ and $|{\alpha}^{\prime}|+|{\beta}^{\prime}|\leqslant|{\alpha}|+|{\beta}|$ . So for $x\in K$ ,

	$\displaystyle\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\tilde{a}_{\epsilon}(x,\xi)\|$	$\displaystyle\leqslant C\sum_{{\alpha}^{\prime},{\beta}^{\prime}}\epsilon^{-\|{\beta}^{\prime}\|}\\|a\\|_{m,\ell,K}\langle\epsilon^{-1}(\xi-{\beta}(x))\rangle^{m-\|{\beta}^{\prime}\|}\quad\text{ by }\eqref{eq:e2}$
		$\displaystyle\leqslant C\sum_{{\alpha}^{\prime},{\beta}^{\prime}}\epsilon^{-\|{\beta}^{\prime}\|}\\|a\\|_{m,\ell,K}\langle\xi\rangle^{m-\|{\beta}^{\prime}\|}\epsilon^{-m_{+}}\quad\text{ by }\eqref{eq:e1}$
		$\displaystyle\leqslant C\epsilon^{-(\|{\alpha}\|+\|{\beta}\|)}\\|a\\|_{m,\ell,K}\langle\xi\rangle^{m-\|{\beta}\|}\epsilon^{-m_{+}}$

because $|{\beta}|\leqslant|{\beta}^{\prime}|\leqslant|{\alpha}|+|{\beta}|$ and we conclude by setting $\epsilon=h^{\frac{1}{2}}$ . ∎

So $\tilde{b}$ belongs to the class $S_{\delta}$ with exponent $\delta=1/2$ , that is at each derivative we loose a factor $h^{-\delta}$ . Recall that $\delta=1/2$ is the critical exponent: the space of pseudodifferential operators with symbol in $S_{\delta}$ is an algebra for $\delta\in[0,1/2]$ , but the standard asymptotic expansions of the symbolic calculus only hold for $\delta\in[0,1/2[$ , cf. for instance [10, Proposition 7.7]. As we will see in Section 5 and in [5], the Heisenberg pseudodifferential operators form an algebra and have an associated symbol calculus, but this can not be deduced form the usual composition rules of pseudodifferential operators. Nevertheless, Lemma 3.4 has some useful consequences, the first of them being the $L^{2}$ mapping property. Recall the definition (18) of the $L^{2}$ -norm with a volume element independent of $k$ .

Theorem 3.5.

For any $Q\in\Psi^{0}_{\operatorname{Heis}}(L)$ , there exists $C>0$ such that for any $k$ , $\|Q_{k}\|_{\mathcal{L}(L^{2}(M,L^{k}))}\leqslant C$ .

Proof.

Introduce a finite atlas $(U_{i},\phi_{i})$ of $M$ with functions $\varphi_{i},\psi_{i}\in{\mathcal{C}}^{\infty}_{0}(U_{i})$ such that $\sum\varphi_{i}=1$ and $\operatorname{supp}\varphi_{i}\subset\operatorname{int}\{\psi_{i}=1\}$ . Write

\displaystyle P=\sum\psi_{i}P\varphi_{i}+Q.

(28)

Since $\sum\psi_{i}(x)\varphi_{i}(y)=1$ when $x$ is close to $y$ , $Q$ is in $k^{-\infty}\Psi^{-\infty}$ . Identifying $U_{i}$ with $\phi_{i}(U_{i})$ , the Schwartz kernel of $\psi_{i}P\varphi_{i}$ has the form (25) with a symbol $\tilde{b}_{i}$ satisfying

|\partial^{\alpha}_{x}\partial^{\beta}_{\xi}\tilde{b}_{i}(h,x,\xi)|\leqslant h^{-\frac{1}{2}(|{\alpha}|+|{\beta}|)}C_{{\alpha},{\beta}}.

By [10, Theorem 7.11], $\psi_{i}P\varphi_{i}=\mathcal{O}(1):L^{2}({\mathbb{R}}^{n})\rightarrow L^{2}({\mathbb{R}}^{n})$ . ∎

Another consequence of Lemma 3.4 is the following important fact.

Lemma 3.6.

$k^{-\infty}\Psi^{-\infty}(L)$ is a bilateral ideal of $\Psi_{\operatorname{Heis}}(L,\nabla)$ .

Proof.

Consider a pseudodifferential operator $A(h)$ of ${\mathbb{R}}^{n}$ with the Schwartz kernel $(2\pi h)^{-n}\int e^{ih^{-1}\xi(x-y)}a(h,x,y,\xi)d\xi$ where the amplitude $a(h,x,y,\xi)$ is zero if $|x|+|y|\geqslant C$ and satisfies $|\partial^{\alpha}_{x,y}a(h,x,y,\xi)|\leqslant h^{-|{\alpha}|}C_{{\alpha}}\langle\xi\rangle^{m}$ for any ${\alpha}$ . Then, with the usual regularisation of oscillatory integrals by integration by part, one proves that

h^{|{\alpha}|}\sup|\partial^{{\alpha}}A(h)u|\leqslant C^{\prime}_{{\alpha}}\max_{|{\beta}|\leqslant m+n+1+|{\alpha}|}\sup h^{|{\beta}|}|\partial^{\beta}u|,\qquad\forall h\in(0,1]

So for any family of operators $B(h):{\mathcal{C}}^{\infty}({\mathbb{R}}^{n})\rightarrow{\mathcal{C}}^{\infty}({\mathbb{R}}^{n})$ , $h\in(0,1]$ , if $B(h)$ has a compactly supported smooth kernel in $\mathcal{O}_{\infty}(h^{\infty})$ then the same holds for $A(h)\circ B(h)$ . Applying this to the $\psi_{i}P\varphi_{i}$ of (28) proves the result. ∎

To end this section, let us extend the mapping property to the Sobolev space. We denote by $\|\cdot\|_{m}$ the $m$ -th semiclassical Sobolev norm of sections of $L^{k}$ , defined as in Section 2.

Theorem 3.7.

For any $m,p\in{\mathbb{R}}$ and any $Q\in\Psi_{\operatorname{Heis}}^{m}(L)$ , there exists $C>0$ such that for any $k$ ,

\|Q_{k}u\|_{p}\leqslant Ck^{\frac{1}{2}m_{+}}\|u\|_{p+m},\qquad\forall u\in{\mathcal{C}}^{\infty}(M,L^{k}).

Since for any $k$ , $Q_{k}$ is a pseudodifferential operator of order $m$ of $L^{k}$ , we already know that $P_{k}$ is continuous $H^{p+m}(M,L^{k})\rightarrow H^{m}(M,L^{k})$ . Theorem 3.7 gives a uniform estimate with respect to $k$ .

Proof.

It suffices to prove that for any $E\in\Psi_{\operatorname{tsc}}^{p-m}(L)$ and $E^{\prime}\in\Psi_{\operatorname{tsc}}^{-p}(L)$ one has

\displaystyle E^{\prime}_{k}P_{k}E_{k}=\mathcal{O}(k^{\frac{1}{2}m_{+}}):L^{2}(M,L^{k})\rightarrow L^{2}(M,L^{k}).

(29)

For this it suffices to prove that for any chart domain $U$ of $M$ and functions $\rho_{j}\in{\mathcal{C}}^{\infty}_{0}(U)$ , $j=1,...,4$ , one has

\displaystyle\rho_{1}E^{\prime}_{k}\,\rho_{2}P_{k}\,\rho_{3}E_{k}\,\rho_{4}=\mathcal{O}(k^{\frac{1}{2}m_{+}}):L^{2}(M,L^{k})\rightarrow L^{2}(M,L^{k}).

(30)

To show that (30) implies (29), write $P$ on the form (28), $E^{\prime}\psi_{i}=\tilde{\psi}_{i}E^{\prime}\psi_{i}+(1-\tilde{\psi}_{i})E^{\prime}\psi_{i}$ with $\tilde{\psi}_{i}\in{\mathcal{C}}^{\infty}_{0}(U_{i})$ such that $\operatorname{supp}\psi_{i}\subset\operatorname{int}\{\tilde{\psi}_{i}=1\}$ and similarly for $E$ , and use that $k^{-\infty}\Psi^{-\infty}(L)$ is an ideal of both $\Psi_{\operatorname{Heis}}^{\infty}(L,\nabla)$ and $\Psi_{\operatorname{tsc}}^{\infty}(L)$ .

As in [10, Definition 7.5], for $\delta\in[0,1]$ and $m:{\mathbb{R}}^{n}\rightarrow[0,\infty)$ an order function, let $S_{\delta}(m)$ be the space of families $(a(h),\;h\in(0,1])$ of ${\mathcal{C}}^{\infty}({\mathbb{R}}^{n})$ such that $|\partial^{\alpha}a(h,x)|\leqslant C_{{\alpha}}h^{-\delta|{\alpha}|}m(x)$ . Identify $U$ with $\phi(U)$ and denote by $\operatorname{Op}_{k}(\tilde{b})$ the operator with kernel (25).

Then for any $\rho,\rho^{\prime}\in{\mathcal{C}}^{\infty}_{0}(U)$ , $\rho E^{\prime}_{k}\,\rho^{\prime}$ , $k^{-\frac{1}{2}m_{+}}\rho P_{k}\,\rho^{\prime}$ and $\rho E_{k}\,\rho^{\prime}$ are equal to $\operatorname{Op}_{k}(\tilde{b})$ with $\tilde{b}$ in $S_{0}(\langle\xi\rangle^{-p})$ , $S_{1/2}(\langle\xi\rangle^{m})$ and $S_{0}(\langle\xi\rangle^{p-m})$ respectively. By [10, Proposition 7.7, Theorem 7.9], their product is equal to $\operatorname{Op}_{k}(c)$ with $c\in S_{1/2}(1)$ , which proves (30) by [10, Theorem 7.11]. ∎

Actually, Theorem 3.7 can be improved if we use Sobolev norms associated to the covariant derivative $\nabla$ instead of the semiclassical Sobolev norms. For instance, for any $m\in{\mathbb{N}}$ , any $Q\in\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ and any vector fields $X_{1}$ , …, $X_{m}$ of $M$ , we will see in Proposition 5.1, that $P=(k^{-m/2}\nabla_{X_{1}}\ldots\nabla_{X_{m}}Q_{k})$ belongs to $\Psi_{\operatorname{Heis}}^{0}(L,\nabla)$ , so by Theorem 3.5,

\displaystyle k^{-m/2}\nabla_{X_{1}}\ldots\nabla_{X_{m}}Q_{k}=\mathcal{O}(1):L^{2}(M,L^{k})\rightarrow L^{2}(M,L^{k})

(31)

To compare, Theorem 3.7 only implies that the norm of $P_{k}$ in $\mathcal{L}(L^{2}(M,L^{k}))$ is in $\mathcal{O}(k^{m/2})$ . The generalisation of (31) to fractional exponents $m$ not necessarily nonnegative will be given in [5].

4 A product associated to an antisymmetric bilinear form

Let $E$ be a $n$ -dimensional real vector space and $A\in\wedge^{2}E^{*}$ . Later, we will choose $E=T_{x}M$ with $A={\omega}(x)$ . Introduce the covariant derivative of $E$

\displaystyle\nabla^{A}=d+\tfrac{1}{i}\beta,\qquad\text{ where }{\beta}\in\Omega^{1}(E,{\mathbb{R}}),\quad\beta(x)(Y)=\tfrac{1}{2}A(x,Y).

(32)

The curvature of $\nabla^{A}$ is $\frac{1}{i}A$ , that is $[\nabla_{X}^{A},\nabla_{Y}^{A}]=\frac{1}{i}A(X,Y)$ for any $X,Y\in E$ . We will define for any tempered distribution $g\in\mathcal{S}^{\prime}(E^{*})$ an operator $g(\frac{1}{i}\nabla^{A})$ .

We assume first that $E={\mathbb{R}}^{n}$ . For any $g\in\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ , we denote by $\widehat{g}$ and $g^{\vee}$ its Fourier transform and inverse Fourier transform, with the normalisation

\widehat{g}(\xi)=\int_{{\mathbb{R}}^{n}}e^{-ix\cdot\xi}g(x)\;dx,\qquad g^{\vee}(x)=(2\pi)^{-n}\,\widehat{g}(-x).

Let $g(\tfrac{1}{i}\partial)$ be the operator from $\mathcal{S}({\mathbb{R}}^{n})$ to $\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ such that $g(\tfrac{1}{i}\partial)u=v$ if and only if $g(\xi)\widehat{u}(\xi)=\widehat{v}(\xi)$ . The Schwartz kernel of $g(\tfrac{1}{i}\partial)$ is $g^{\vee}(x-y)$ .

Then for any antisymmetric bilinear form $A$ of ${\mathbb{R}}^{n}$ , define $g(\frac{1}{i}\nabla^{A})$ as the operator with Schwartz kernel

\displaystyle K_{g}(x,y)=e^{-\frac{i}{2}A(x,y)}g^{\vee}(x-y).

(33)

Since $g^{\vee}(x-y)$ is a tempered distribution of ${\mathbb{R}}^{n}_{x}\times{\mathbb{R}}^{n}_{y}$ , the same holds for $K_{g}$ , so $g(\tfrac{1}{i}\nabla^{A})$ is continuous from $\mathcal{S}({\mathbb{R}}^{n})$ to $\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ . We claim that this definition has an intrinsic meaning for $A\in\wedge^{2}E^{*}$ if we consider that $g\in\mathcal{S}^{\prime}(E^{*})$ and $g(\frac{1}{i}\nabla^{A})$ is an operator $\mathcal{S}(E)\rightarrow\mathcal{S}^{\prime}(E)$ . One way to see this is to write for $g$ and $u$ in $\mathcal{S}({\mathbb{R}}^{n})$

\displaystyle(g(\tfrac{1}{i}\nabla^{A})u)(x)=(2\pi)^{-n}\int_{{\mathbb{R}}^{n}\times{\mathbb{R}}^{n}}e^{-\frac{i}{2}A(x,y)+i\xi\cdot(x-y)}g(\xi)u(y)\;dy\,d\xi

(34)

and to notice that the product $\xi\cdot(x-y)$ is well-defined for $\xi\in E^{*}$ , $x,y\in E$ , and the measure $dy\,d\xi$ can be interpreted as the canonical volume form of $E\times E^{*}\simeq{\mathbb{R}}^{n}_{y}\times{\mathbb{R}}^{n}_{\xi}$ .

Assume again that $E\simeq{\mathbb{R}}_{x}^{n}$ , $E^{*}\simeq{\mathbb{R}}_{\xi}^{n}$ and let

\displaystyle\nabla_{j}^{A}:=\nabla_{\partial_{x_{j}}}^{A}=\partial_{x_{j}}+\tfrac{1}{2i}\textstyle{\sum_{k}}x_{k}A_{kj}

(35)

where $(A_{ij})$ is the matrix of $A$ , so $A_{ij}=A(e_{i},e_{j})$ with $(e_{i})$ the canonical basis of ${\mathbb{R}}^{n}$ .

Lemma 4.1.

For any $f\in\mathcal{S}^{\prime}(E^{*})$ , we have $\frac{1}{i}\nabla_{j}^{A}\circ f(\frac{1}{i}\nabla^{A})=(\xi_{j}\sharp_{A}f)(\frac{1}{i}\nabla^{A})$ where

\displaystyle\xi_{j}\sharp_{A}f=(\xi_{j}+\tfrac{i}{2}\textstyle{\sum_{k}}A_{jk}\partial_{\xi_{k}})f.

(36)

Proof.

Simply use the identity

\tfrac{1}{i}(\partial_{x_{j}}+\tfrac{1}{2i}\textstyle{\sum_{k}}x_{k}A_{kj})e^{-\frac{i}{2}A(x,y)+i\xi\cdot(x-y)}=(\xi_{j}+\frac{1}{2i}\sum_{k}A_{jk}\partial_{\xi_{k}})e^{-\frac{i}{2}A(x,y)+i\xi\cdot(x-y)}

in (34) and integrate by part with respect to the variables $\xi_{k}$ . ∎

The reason for the notation $g(\tfrac{1}{i}\nabla^{A})$ is that when $g$ is a monomial, $g(\tfrac{1}{i}\nabla^{A})$ is merely a symmetrization of covariant derivatives. The precise result is the following proposition which is not really needed in the sequel. Notice first that for $g\equiv 1$ , $g(\frac{1}{i}\nabla^{A})=\operatorname{id}$ as a direct consequence of the definition.

Proposition 4.2.

For any $N\geqslant 1$ and $X_{1},\ldots X_{N}\in E$ , if $g=\prod_{i=1}^{N}f_{i}$ with $f_{i}(\xi)=\xi\cdot X_{i}$ , then

g(\tfrac{1}{i}\nabla^{A})=\frac{(-i)^{N}}{N!}\sum_{{\sigma}\in\mathfrak{S}_{N}}\nabla_{X_{{\sigma}(1)}}^{A}\ldots\nabla_{X_{{\sigma}(N)}}^{A},

where $\mathfrak{S}_{N}$ is the group of permutations of $1,\ldots,N$ .

Proof.

For any $X\in E$ we have by (36) with $f(\xi)=\xi\cdot X$ that

\displaystyle\tfrac{1}{i}\nabla^{A}_{X}\circ g(\tfrac{1}{i}\nabla^{A})=(f\sharp_{A}\,g)(\tfrac{1}{i}\nabla^{A})

(37)

where $f\sharp_{A}\,g=(f+\tfrac{i}{2}A(X,\partial_{\xi}))g$ . Choosing $g=1$ , we obtain the result for $N=1$ . We now proceed by induction over $N$ and assume the result holds for $N-1$ with $N\geqslant 2$ . Thus

\frac{(-i)^{N}}{N!}\sum_{{\sigma}\in\mathfrak{S}_{N}}\nabla_{X_{{\sigma}(1)}}^{A}\ldots\nabla_{X_{{\sigma}(N)}}^{A}=\frac{1}{N}\sum_{j=1}^{N}\tfrac{1}{i}\nabla^{A}_{X_{j}}\circ g_{j}(\tfrac{1}{i}\nabla^{A}_{X})

with $g_{j}=g/f_{j}$ . By (37), $f\sharp_{A}g_{j}=fg_{j}+\frac{i}{2}\sum_{\ell\neq j}A(X,X_{\ell})g_{j\ell}$ where $g_{j\ell}=g/(f_{j}f_{\ell})$ . So we have

\frac{(-i)^{N}}{N!}\sum_{{\sigma}\in S_{N}}\nabla_{X_{{\sigma}(1)}}^{A}\ldots\nabla_{X_{{\sigma}(N)}}^{A}=g(\tfrac{1}{i}\nabla^{A})+\frac{1}{N}\sum_{j\neq\ell}A(X_{j},X_{\ell})g_{j\ell}(\tfrac{1}{i}\nabla^{A})

and the sum in the right-hand side is zero because $A$ is antisymmetric whereas $g_{j\ell}=g_{\ell j}$ . ∎

Let $\mathcal{D}_{\operatorname{is}}^{\infty}(A)$ be the filtered algebra generated by the covariant derivatives $\nabla_{X}^{A}$ where $X\in E$ . More explicitly, $\mathcal{D}_{\operatorname{is}}^{\infty}(A)=\cup_{m\in{\mathbb{N}}}\mathcal{D}_{\operatorname{is}}^{m}(A)$ with

\mathcal{D}_{\operatorname{is}}^{m}(A)=\operatorname{Span}(\nabla^{A}_{X_{1}}\ldots\nabla^{A}_{X_{\ell}}/\;0\leqslant\ell\leqslant m,X_{1},\ldots X_{\ell}\in E).

Let ${\mathbb{C}}_{\leqslant m}[E^{*}]$ be the space of complex polynomial functions of $E^{*}$ with degree less than or equal to $m$ . By Lemma 4.1 and Proposition 4.2,

\displaystyle\mathcal{D}_{\operatorname{is}}^{m}(A)=\bigl{\{}f(\tfrac{1}{i}\nabla^{A}),\;f\in{\mathbb{C}}_{\leqslant m}[E^{*}]\bigr{\}}.

(38)

By Lemma 4.1 again, the left composition by any element of $\mathcal{D}_{\operatorname{is}}^{\infty}(A)$ preserves $\{g(\frac{1}{i}\nabla),\;g\in\mathcal{S}^{\prime}(E^{*})\}$ . This defines the product

\displaystyle\sharp_{A}:{\mathbb{C}}[E^{*}]\times\mathcal{S}^{\prime}(E^{*})\rightarrow\mathcal{S}^{\prime}(E^{*}),\qquad(f\sharp_{A}g)(\tfrac{1}{i}\nabla)=f(\tfrac{1}{i}\nabla)\circ g(\tfrac{1}{i}\nabla)

(39)

In the sequel we will use the basis $(\nabla^{{\alpha}},|{\alpha}|\leqslant m)$ of $\mathcal{D}_{\operatorname{is}}^{m}(A)$ , defined by $\nabla^{{\alpha}}:=(\nabla_{1}^{A})^{{\alpha}(1)}\ldots(\nabla_{n}^{A})^{{\alpha}(n)}$ , ${\alpha}\in{\mathbb{N}}^{n}$ . Clearly

\displaystyle i^{-|{\alpha}|}\nabla^{\alpha}=f(\tfrac{1}{i}\nabla)\qquad\text{ with }f=\xi^{\sharp{\alpha}}:=\xi_{1}^{\sharp{\alpha}(1)}\sharp\ldots\sharp\xi_{n}^{\sharp{\alpha}(n)},

(40)

where we have not written the $A$ dependence to lighten the notations. Furthermore, if $|{\gamma}|=m$ ,

\displaystyle\xi^{\sharp{\gamma}}\sharp_{A}f=\xi^{\gamma}f+\sum_{|{\alpha}|+|{\beta}|\leqslant m,\;|{\alpha}|\leqslant m-1}a_{{\alpha},{\beta},{\gamma}}\xi^{{\alpha}}\partial_{\xi}^{\beta}f

(41)

where the coefficients $a_{{\alpha},{\beta},{\gamma}}\in{\mathbb{C}}$ depends smoothly (even polynomially) on $A$ , which follows from Lemma 4.1 again. Actually there is a closed formula for $\sharp_{A}$ , cf. (42), but (41) is enough for our purpose.

Introduce the space $\Psi_{\operatorname{is}}^{m}(A):=\{f(\frac{1}{i}\nabla),\;f\in S^{m}_{\operatorname{ph}}(E^{*})\}$ . We have

\mathcal{D}_{\operatorname{is}}^{m}(A)\subset\Psi_{\operatorname{is}}^{m}(A),\qquad\mathcal{D}_{\operatorname{is}}^{m}(A)\circ\Psi_{\operatorname{is}}^{p}(A)\subset\Psi_{\operatorname{is}}^{m+p}(A),

the second assertion being a consequence of (41). This is all what we need to define in the next section the symbolic calculus corresponding to the composition of differential Heisenberg operators with Heisenberg pseudodifferential operators. In the case where $A=0$ , $\sharp_{A}$ is the usual pointwise product of functions. In Lemma 6.1, we will see that when $A$ is nondegenerate so that $n=2d$ , $\Psi_{\operatorname{is}}^{\infty}(A)$ is an algebra isomorphic to the Weyl algebra of ${\mathbb{R}}^{2d}$ .

In the companion paper [5], we will prove that for any $A$ , $\Psi_{\operatorname{is}}^{\infty}(A)$ is a filtered algebra, that is $\Psi_{\operatorname{is}}^{m}(A)\circ\Psi_{\operatorname{is}}^{p}(A)\subset\Psi_{\operatorname{is}}^{m+p}(A)$ . Moreover

\displaystyle(f\sharp_{A}\,g)(\xi)=\Bigl{[}e^{\frac{i}{2}A(\partial_{\xi},\partial_{\eta})}f(\xi)g(\eta)\Bigl{]}_{\xi=\eta}

(42)

So $\Psi_{\operatorname{is}}^{\infty}(A)$ is isomorphic with the algebra called the $A$ -isotropic algebra in [12, Chapter 4, section 2].

Recall the standard and Weyl quantization maps which associate to any $f\in\mathcal{S}^{\prime}({\mathbb{R}}^{2n})$ the operators $f(x,\frac{1}{i}\partial)$ and $f^{w}(x,\frac{1}{i}\partial)$ with Schwartz kernels

(2\pi)^{-n}\int e^{i\xi\cdot(x-y)}f(x,\xi)\;d\xi\quad\text{ and }\quad(2\pi)^{-n}\int e^{i\xi\cdot(x-y)}f(\tfrac{1}{2}(x+y),\xi)\;d\xi

respectively.

Lemma 4.3.

For any $g\in\mathcal{S}^{\prime}({\mathbb{R}}^{n})$ , we have

g(\tfrac{1}{i}\nabla^{A})=f(x,\tfrac{1}{i}\partial)=f^{w}(x,\tfrac{1}{i}\partial)

where $f(x,\xi)=g(\xi-\beta(x))$ and ${\beta}(x)$ is defined in (34), equivalently $f(x,\xi)=g(\xi_{1}-\frac{1}{2}A(x,e_{1}),\ldots,\xi_{n}-\frac{1}{2}A(x,e_{n}))$ .

Proof.

By the same computation as in (24),

\int e^{i\xi\cdot(x-y)}g(\xi-\beta(x))\;d\xi=e^{i{\beta}(x)(x-y)}\int e^{i\xi\cdot(x-y)}g(\xi)\;d\xi.

$A$ being antisymmetric, ${\beta}(x)(x-y)=-\frac{1}{2}A(x,y)$ , which proves that $g(\tfrac{1}{i}\nabla^{A})=f(x,\tfrac{1}{i}\partial)$ . The same proof by using this time that ${\beta}(\frac{1}{2}(x+y))(x-y)=-\frac{1}{2}A(x,y)$ shows that $g(\tfrac{1}{i}\nabla^{A})=f^{w}(x,\tfrac{1}{i}\partial)$ . ∎

5 Heisenberg differential operators

The algebra $\mathcal{D}_{\operatorname{Heis}}^{\infty}(L,\nabla)$ of Heisenberg differential operators consists of families of differential operators

\displaystyle P=(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}),\;k\in{\mathbb{N}}),

(43)

satisfying some conditions given below. It includes the multiplications by any $f$ in ${\mathcal{C}}^{\infty}(M)$ , the normalised covariant derivatives $k^{-1/2}\nabla_{X}$ where $X$ is any vector field of $M$ and the multiplication by $k^{-1/2}$ . It is actually generated by these operators but it will be easier to use the following definition.

For any $m\in{\mathbb{N}}$ , $\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla)$ consists of the families $P$ of differential operators of the form (43) such that for any coordinate chart $(U,x_{i})$ and frame $s\in{\mathcal{C}}^{\infty}(U,L)$ with $|s|=1$ , we have on $U$ ,

\displaystyle P_{k}=\sum_{\begin{subarray}{c}\ell\in{\mathbb{N}},\;{\alpha}\in{\mathbb{N}}^{n}\\ \ell+|{\alpha}|\leqslant m\end{subarray}}k^{-\frac{\ell}{2}}f_{\ell,{\alpha}}\tilde{\pi}^{\alpha}

(44)

where $f_{\ell,{\alpha}}\in{\mathcal{C}}^{\infty}(U)$ , $\tilde{\pi}^{{\alpha}}=\tilde{\pi}_{1}^{{\alpha}(1)}\ldots\tilde{\pi}_{n}^{{\alpha}(n)}$ and

\displaystyle\tilde{\pi}_{i}=\tfrac{1}{i\sqrt{k}}\nabla_{i}=\tfrac{1}{i\sqrt{k}}\partial_{i}-\sqrt{k}\beta_{i}\qquad\text{ with }\quad\nabla s=\tfrac{1}{i}\sum\beta_{i}dx_{i}\otimes s

(45)

Set

\mathcal{D}_{\operatorname{Heis}}^{\infty}(L,\nabla)=\bigcup_{m\in{\mathbb{N}}}\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla).

In the sequel to lighten the notations, we omit $(L,\nabla)$ and write $\mathcal{D}_{\operatorname{Heis}}^{m}$ , $\mathcal{D}_{\operatorname{Heis}}^{\infty}$ . Since $[\tilde{\pi}_{i},\tilde{\pi}_{j}]=\frac{1}{i}(\partial_{i}{\beta}_{j}-\partial_{j}\beta_{i})$ and $[\tilde{\pi}_{i},f]=\frac{1}{i\sqrt{k}}\partial_{i}f$ , we see that

\mathcal{D}_{\operatorname{Heis}}^{m}\circ\mathcal{D}_{\operatorname{Heis}}^{p}\subset\mathcal{D}_{\operatorname{Heis}}^{m+p}.

Notice that $\mathcal{D}_{\operatorname{Heis}}^{\infty}$ has two filtrations: one ascending $\mathcal{D}_{\operatorname{Heis}}^{m}\subset\mathcal{D}_{\operatorname{Heis}}^{m+1}$ and the other descending $k^{-\ell/2}\mathcal{D}_{\operatorname{Heis}}$ , $\ell\in{\mathbb{N}}$ . The generators $f$ , $k^{-1/2}\nabla_{X}$ and $k^{-1/2}$ have orders $0$ , $1$ , $1$ for the former and $0$ , $0$ , $1$ for the latter.

By the next proposition, $\mathcal{D}_{\operatorname{Heis}}^{\infty}$ is contained in $\Psi_{\operatorname{Heis}}^{\infty}(L,\nabla)$ and acts on it. Being Heisenberg pseudodifferential operators, the elements of $\mathcal{D}_{\operatorname{Heis}}^{\infty}$ have a principal symbol, cf Definition 3.2. As we will see, the product of symbols is the fiberwise product $\sharp$ of $T^{*}M$ defined from ${\omega}$ . Precisely, we denote by $\sharp_{x}$ the product

\displaystyle{\mathbb{C}}_{\leqslant m}[T^{*}_{x}M]\times S^{p}(T^{*}_{x}M)\rightarrow S^{m+p}(T^{*}_{x}M)

(46)

associated to ${\omega}(x)\in\wedge^{2}T^{*}_{x}M$ defined in (39). We will need as well the polynomials $\xi^{\sharp_{x}{\alpha}}$ defined in (40).

Proposition 5.1.

-

for any $m\in{\mathbb{N}}$ , $\mathcal{D}_{\operatorname{Heis}}^{m}\subset\Psi_{\operatorname{Heis}}^{m}$
-

the principal symbols of the operators of $\mathcal{D}_{\operatorname{Heis}}^{m}$ are the functions $f\in{\mathcal{C}}^{\infty}(T^{*}M)$ such that $f(x,\cdot)\in{\mathbb{C}}_{\leqslant m}[T^{*}_{x}M]$ for any $x$ . If (44) holds on $U$ , then ${\sigma}(P)(x,\xi)=\sum_{|{\alpha}|\leqslant m}f_{0,{\alpha}}(x)\xi^{\sharp_{x}{\alpha}}$ .
-

for any $P\in\mathcal{D}_{\operatorname{Heis}}^{m}$ , ${\sigma}(P)=0$ if and only if $P\in k^{-\frac{1}{2}}\mathcal{D}_{\operatorname{Heis}}^{m-1}$ .
-

for any $m\in{\mathbb{N}}$ and $p\in{\mathbb{R}}$ , $\mathcal{D}_{\operatorname{Heis}}^{m}\circ\Psi_{\operatorname{Heis}}^{p}\subset\Psi_{\operatorname{Heis}}^{m+p}$ . Furthermore

${\sigma}(P\circ Q)(x,\cdot)={\sigma}(P)(x,\cdot)\,\sharp_{x}\,{\sigma}(Q)(x,\cdot)$

for any $P\in\mathcal{D}_{\operatorname{Heis}}^{m}$ , $Q\in\Psi_{\operatorname{Heis}}^{p}$ .

Proof.

We start with the computation of $\tilde{\pi}_{i}\circ\operatorname{Op}_{\operatorname{Heis}}(a)$ where $\operatorname{Op}_{\operatorname{Heis}}(a)$ is the operator with Schwartz kernel (24). Using (45), we get first that $\tilde{\pi}_{i}\circ\operatorname{Op}_{\operatorname{Heis}}(a)=\operatorname{Op}_{\operatorname{Heis}}(b)$ with

b(h,x,y,\xi)=\bigl{(}h^{-1}\psi_{i}(x,y)+\xi_{i}+\tfrac{h}{i}\partial_{x_{i}}\bigr{)}\,a(h,x,y,\xi)

where $\psi_{i}(x,y)=(\partial_{i}{\beta})(\tfrac{1}{2}(x+y))\cdot(x-y)+\beta_{i}(\tfrac{1}{2}(x+y))-\beta_{i}(x)$ . Taylor expanding along $x=y$ , we get

\displaystyle\psi_{i}(x,y)=\tfrac{1}{2}\sum_{j}{\omega}_{ij}(x)(x_{j}-y_{j})+\sum_{i,j}r_{ij}(x,y)(x_{i}-y_{i})(x_{j}-y_{j})

with ${\omega}_{ij}=\partial_{x_{i}}\beta_{j}-\partial_{x_{j}}\beta_{i}$ . Integrating by part, $\tilde{\pi}_{i}\circ\operatorname{Op}_{\operatorname{Heis}}(a)=\operatorname{Op}_{\operatorname{Heis}}(c)$ with

c(h,x,y,\xi)=(\xi_{i}+\tfrac{i}{2}\sum_{j}{\omega}_{ij}(x)\partial_{\xi_{i}}+\tfrac{h}{i}\partial_{x_{i}}-h\sum_{ij}r_{ij}(x,y)\partial_{\xi_{i}}\partial_{\xi_{j}})a(h,x,y,\xi).

Notice that if $a\in S^{m}_{\operatorname{sc}}(U^{2},{\mathbb{R}}^{n})$ , then the same holds for $c$ . Furthermore, if $a(h,x,x,\xi)={\sigma}(x,\xi)+\mathcal{O}(h)$ , then $c(h,x,x,\xi)=(\xi_{i}\sharp{\sigma})(x,\xi)+\mathcal{O}(h)$ .

We claim that everything can be deduced easily from these preliminary observations. Starting from the fact that $\operatorname{Op}_{\operatorname{Heis}}(1)$ is the identity, we deduce by induction on $|{\alpha}|$ that $\tilde{\pi}^{{\alpha}}=\operatorname{Op}_{\operatorname{Heis}}(a_{\alpha})$ with $a_{{\alpha}}(h,x,x,\xi)=\xi^{\sharp{\alpha}}+\mathcal{O}(h)$ . The first two assertions follow. The third assertion is a consequence of the fact that the $\xi^{\sharp{\alpha}}|_{x}$ , ${\alpha}\in{\mathbb{N}}^{n}$ are linearly independent so that $\sum f_{0,{\alpha}}\xi^{\sharp{\alpha}}=0$ implies that $f_{0,{\alpha}}=0$ . Last assertion follows again from the preliminaries by induction on $m$ . ∎

6 Resolvent

Let $(F,{\lambda})$ be a real symplectic vector space with dimension $2d$ . The Weyl product of the Schwartz space $\mathcal{S}(F)$ is defined by

(a\circ_{\lambda}b)(\xi)=(\pi)^{-2d}\int e^{-2i{\lambda}(\eta,\zeta)}a(\xi+\eta)b(\xi+\zeta)\;d\mu_{F}(\eta)\;d\mu_{F}(\zeta)

where $\mu_{F}={\lambda}^{\wedge d}/d!$ is the Liouville measure of $F$ . For $F={\mathbb{R}}^{d}_{t}\times{\mathbb{R}}^{d}_{\tau}$ with ${\lambda}(t,\tau;s,{\varsigma})=\tau\cdot s-{\varsigma}\cdot t$ , it is the composition law of the Weyl symbols of pseudodifferential operators of ${\mathbb{R}}^{d}$ , cf. for instance [16, page 152].

This product extends continuously from $S^{m}(F)\times S^{p}(F)$ to $S^{m+p}(F)$ by preserving the subspaces of polyhomogeneous symbols. So the corresponding pseudodifferential operators, $f^{w}(x,\frac{1}{i}\partial)$ , with $f\in S^{\infty}({\mathbb{R}}^{2d})$ , form an algebra, called sometimes the Shubin class or isotropic class. This algebra is one of the most studied in microlocal analysis, cf. [24, Chapter IV], [15], [21, Chapter 4], [12, Chapter 4], [25, Appendix A] for lecture note references.

The Weyl product appears naturally in our context as the product of the operators $f(\frac{1}{i}\nabla^{A})$ defined in Section 4 when $A$ is nondegenerate.

Lemma 6.1.

If $A\in\wedge^{2}E^{*}$ is nondegenerate, then for any $f$ , $g$ in $S^{\infty}(E^{*})$ ,

f(\tfrac{1}{i}\nabla^{A})\circ g(\tfrac{1}{i}\nabla^{A})=(f\circ_{{\lambda}}g)(\tfrac{1}{i}\nabla^{A})

where ${\lambda}$ is the symplectic form of $E^{*}$ dual to $A$ .

Proof.

Introduce a symplectic basis $(e_{i},f_{i})$ of $(E,A)$ and denote by $(x_{i},y_{i})$ the associated linear coordinates, so that $E={\mathbb{R}}^{d}_{x}\times{\mathbb{R}}^{d}_{y}$ . Then the operators

\tfrac{1}{i}\nabla_{e_{i}}=\tfrac{1}{i}\partial_{x_{i}}+\tfrac{1}{2}y_{i},\quad\tfrac{1}{i}\nabla_{f_{i}}=\tfrac{1}{i}\partial_{y_{i}}-\tfrac{1}{2}x_{i},\quad\ \tfrac{1}{i}\partial_{y_{i}}+\tfrac{1}{2}x_{i},\quad\tfrac{1}{i}\partial_{x_{i}}-\tfrac{1}{2}y_{i}

satisfy the same commutation relations as the operators $s_{i}$ , $\frac{1}{i}\partial_{s_{i}}$ , $t_{i}$ , $\frac{1}{i}\partial_{t_{i}}$ of ${\mathbb{R}}^{d}_{s}\times{\mathbb{R}}^{d}_{t}$ . So the linear isomorphism $\Phi:{\mathbb{R}}^{4d}\rightarrow{\mathbb{R}}^{4d}$ ,

\Phi(x,\xi,y,\eta)=(\xi+\tfrac{1}{2}y,\eta-\tfrac{1}{2}x,\eta+\tfrac{1}{2}x,\xi+\tfrac{1}{2}y).

is a symplectomorphism. Its metaplectic representation $U:L^{2}(E)\rightarrow L^{2}({\mathbb{R}}^{2d})$ satisfies $f^{w}=U(f\circ\Phi)^{w}U^{*}$ for any $f\in\mathcal{S}^{\prime}({\mathbb{R}}^{4d})$ , cf. [16, Theorem 18.5.9]. Applying this to $f(x,\xi,y,\eta)=g(\xi+\frac{1}{2}y,\eta-\frac{1}{2}x)=((g\boxtimes 1)\circ\Phi)(x,\xi,y,\eta)$ , we obtain

f^{w}(x,\tfrac{1}{i}\partial_{x},y,\tfrac{1}{i}\partial_{y})=U^{*}(g^{w}(s,\tfrac{1}{i}\partial_{s})\otimes\operatorname{id}_{{\mathbb{R}}^{d}_{t}})U

and by Lemma 4.3, $f^{w}(x,\tfrac{1}{i}\partial_{x},y,\tfrac{1}{i}\partial_{y})=g(\tfrac{1}{i}\nabla^{A})$ . The result follows. ∎

From now on, we assume that ${\omega}$ is nondegenerate. By Lemma 6.1, at any $x\in M$ , the product $\sharp_{x}$ defined in (46) extends continuously

S^{m}(T_{x}^{*}M)\times S^{p}(T_{x}^{*}M)\rightarrow S^{m+p}(T_{x}^{*}M).

Recall that $f\in S^{m}(M,T^{*}M)$ is elliptic if $|f(x,\xi)|\geqslant C^{-1}|\xi|^{m}$ when $|\xi|\geqslant C$ for some positive $C$ . We say that $f$ is invertible if at any $x\in M$ , $f(x,\cdot)$ is invertible in $(S^{\infty}(T_{x}^{*}M),\sharp_{x})$ .

Lemma 6.2.

1.

$S_{\operatorname{ph}}^{\infty}(M,T^{*}M)$ endowed with the fibered product $(f\sharp g)(x)=f(x)\sharp_{x}g(x)$ is a filtered algebra.
2.

For any $f\in S^{m}_{\operatorname{ph}}(M,T^{*}M)$ which is both elliptic and invertible, the pointwise inverse of $f$ belongs to $S^{-m}_{\operatorname{ph}}(M,T^{*}M)$ .

Proof.

This holds more generally for $S^{m}_{\operatorname{ph}}(N,E)$ where $E$ is any symplectic vector bundle with base $N$ . When $N$ is a point, $S^{\infty}_{\operatorname{ph}}(N,E)$ is isomorphic with the Weyl algebra $S^{\infty}_{\operatorname{ph}}({\mathbb{R}}^{2d})$ , and the result is well-known as we already mentioned it. In general, we can assume that $E$ is the trivial symplectic bundle ${\mathbb{R}}^{2d}$ over an open subset $U$ of ${\mathbb{R}}^{d}$ . Viewing symbols in $S^{m}(U,{\mathbb{R}}^{2d})$ as smooth maps from $U$ to the Fréchet space $S^{m}({\mathbb{R}}^{2d})$ , and using that the Weyl product is continuous $S^{m}({\mathbb{R}}^{2d})\times S^{p}({\mathbb{R}}^{2d})\rightarrow S^{m+p}({\mathbb{R}}^{2d})$ , we deduce with a little work that the fibered Weyl product $\sharp$ is continuous

\displaystyle S^{m}(U,{\mathbb{R}}^{2d})\times S^{p}(U,{\mathbb{R}}^{2d})\rightarrow S^{m+p}(U,{\mathbb{R}}^{2d}).

(47)

which proves the first assertion.

Let $f\in S_{\operatorname{ph}}^{m}(U,{\mathbb{R}}^{2d})$ be elliptic and invertible. Let us prove that its pointwise inverse $g$ is in $S^{-m}_{\operatorname{ph}}(M,T^{*}M)$ . Multiplying $f$ by $f(x_{0})^{-1}$ , we may assume that $m=0$ . Since $S^{\infty}_{\operatorname{ph}}(U,{\mathbb{R}}^{2d})$ is a filtered algebra, cf. (47), and by Borel lemma, $f$ has a parametrix $h\in S^{0}_{\operatorname{ph}}(U,{\mathbb{R}}^{2d})$ . Let us prove that $g=h+S^{-\infty}(U,{\mathbb{R}}^{2d})$ . We have $h\sharp f=1+r$ , $f\sharp h=1+s$ with $r$ and $s$ in $S^{-\infty}(U,{\mathbb{R}}^{2d})$ . So $g=h-r\sharp h+r\sharp g\sharp s$ . By (47) again, $r\sharp h\in S^{-\infty}(U,{\mathbb{R}}^{2d})$ . It remains to prove that $r\sharp g\sharp s\in S^{-\infty}(U,{\mathbb{R}}^{2d})$ .

By Calderon-Vaillancourt theorem, the Weyl quantization $\operatorname{Op}:S^{0}({\mathbb{R}}^{2d})\rightarrow\mathcal{L}(L^{2}({\mathbb{R}}^{d}))$ is continuous. $\operatorname{Op}(g(x))$ being the inverse of $\operatorname{Op}(f(x))$ for any $x$ , $\operatorname{Op}(g)\in{\mathcal{C}}^{\infty}(U,\mathcal{L}(L^{2}({\mathbb{R}}^{d}))$ . Moreover, the multilinear map

M:S^{-\infty}({\mathbb{R}}^{2d})\times\mathcal{L}(L^{2}({\mathbb{R}}^{d}))\times S^{-\infty}({\mathbb{R}}^{2d})\rightarrow S^{-\infty}({\mathbb{R}}^{2d}),

defined by $\operatorname{Op}(M({\sigma},A,\tau))=\operatorname{Op}({\sigma})\circ A\circ\operatorname{Op}(\tau)$ , being continuous, we obtain with a little work that $r\sharp g\sharp s=M(r,\operatorname{Op}(g),s)$ belongs to $S^{-\infty}(U,{\mathbb{R}}^{2d})$ . ∎

Consider now $P\in\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla)$ having an elliptic symbol. Then for any fixed $k$ , $P_{k}$ is an elliptic differential operator of ${\mathcal{C}}^{\infty}(M,L^{k})$ , so for any $s\in{\mathbb{R}}$ , $P_{k}$ extends to a Fredholm operator of $\mathcal{L}(H^{s}(M,L^{k}),H^{s-m}(M,L^{k}))$ . If we assume that the symbol of $P$ is invertible, then by the following Theorem, $P_{k}$ is invertible when $k$ is large, and its inverse is a Heisenberg pseudodifferential operator.

Theorem 6.3.

Assume that ${\omega}$ is nondegenerate. Let $P\in\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla)$ having an elliptic and invertible symbol ${\sigma}\in{\mathcal{C}}^{\infty}(T^{*}M)$ . Then there exists $Q\in\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ such that

-

$PQ-\operatorname{id}$ and $QP-\operatorname{id}$ are in $k^{-\infty}\Psi^{-\infty}(L)$
-

when $k$ is sufficiently large, $Q_{k}P_{k}=P_{k}Q_{k}=\operatorname{id}$
-

the symbol of $Q$ is the inverse of ${\sigma}$ for the product $\sharp$ .

Proof.

This follows merely from the previous results, by the standard techniques for elliptic operators. First, using Lemma 6.2, we construct a parametrix $Q\in\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ of $P$ , so $PQ=\operatorname{id}+R$ and $QP=\operatorname{id}+S$ with $R,S$ in the residual algebra $k^{-\infty}\Psi^{-\infty}(L)$ . Then, by the Sobolev continuity (19), $R_{k}$ and $S_{k}$ belongs to $\mathcal{L}(L^{2}(M,L^{k}))$ and their operator norms are in $\mathcal{O}(k^{-\infty})$ . So when $k\geqslant k_{0}$ , $P_{k}$ is invertible from $H^{m}(M,L^{k})$ to $H^{0}(M,L^{k})$ , which implies by the Fredholm properties of elliptic operators [24, Theorem 8.1], that $P_{k}$ is an invertible operator of the distribution space $\mathcal{D}^{\prime}(M,L^{k})$ .

Its inverse satisfies

\displaystyle P^{-1}_{k}=Q_{k}-R_{k}Q_{k}+R_{k}(Q_{k}P_{k})^{-1}Q_{k}S_{k}

(48)

By Lemma 3.6, $(R_{k}Q_{k})$ and $(Q_{k}S_{k})$ are in $k^{-\infty}\Psi^{-\infty}(L)$ . It is a classical fact that if $(A_{k})$ , $(B_{k})$ are in $k^{-\infty}\Psi^{-\infty}(L)$ and $C_{k}=\mathcal{O}(1):L^{2}(M,L^{k})\rightarrow L^{2}(M,L^{k})$ , then $(A_{k}C_{k}B_{k})$ is in $k^{-\infty}\Psi^{-\infty}(L)$ . So the last term in (48) belongs to $k^{-\infty}\Psi^{-\infty}(L)$ . So by adding to $Q_{k}$ an element of the residual algebra, we have that $Q_{k}=P_{k}^{-1}$ when $k$ is large. ∎

Assume $m>0$ and consider $P\in\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla)$ having an elliptic symbol ${\sigma}$ such that for some $z_{0}\in{\mathbb{C}}$ , ${\sigma}-z_{0}$ is invertible. Then by Theorem 6.3, when $k$ is sufficiently large, $P_{k}-z_{0}$ has an inverse, which is continuous $L^{2}(M,L^{k})\rightarrow H^{m}(M,L^{k})$ . So the restriction of $P_{k}$ to $H^{m}(M,L^{k})$ is a closed unbounded operator of $L^{2}(M,L^{k})$ having a compact resolvent. So its spectrum is a discrete subset of ${\mathbb{C}}$ and it consists only of eigenvalues with finite multiplicity, the generalised eigenvectors being smooth [24, Theorem 8.4].

To state the next theorem, we need some spectral properties of the symbols themselves. Later we will explain these properties in terms of Weyl quantization, but since this quantization is only auxiliary in what we do, we prefer first to discuss everything intrinsically in terms of the algebra $(S^{\infty}(F),\circ_{{\lambda}})$ where $(F,{\lambda})$ is a symplectic vector space as above.

The spectrum of $a\in S^{\infty}(F)$ is defined by: $z\notin\operatorname{sp}(a)$ if and only if $z-a$ is invertible in $(S^{\infty}(F),\circ_{\sigma})$ . A family $(b(z),z\in\Omega)$ of $S^{m}(F)$ is holomorphic if $\Omega$ is an open set of ${\mathbb{C}}$ , $b\in S^{m}(\Omega,F)$ and $\partial_{\overline{z}}b=0$ . By the analytic Fredholm theory for the isotropic algebra exposed in [21, Chapter 3], for any elliptic $a\in S^{m}_{\operatorname{ph}}(F)$ with $m>0$ , the spectrum of $a$ is ${\mathbb{C}}$ or a discrete subset of ${\mathbb{C}}$ . In the latter case, the resolvent $((a-z)^{(-1)_{\circ_{\lambda}}},\;z\in{\mathbb{C}}\setminus\operatorname{sp}(a))$ is a holomorphic family of $S^{-m}(F)$ and for any $z_{0}\in\operatorname{sp}(a)$ , we have on a neighborhood of $z_{0}$ for some $N\in{\mathbb{N}}$

\displaystyle(a-z)^{(-1)_{\circ_{\lambda}}}=h(z)+\frac{r_{1}}{z-z_{0}}+\ldots+\frac{r_{N}}{(z-z_{0})^{N}}

(49)

where $(h(z))$ is a holomorphic family of $S_{\operatorname{ph}}^{-m}(F)$ and $r_{1}$ , …, $r_{N}$ are in $S^{-\infty}(F)$ .

Theorem 6.4.

Assume that ${\omega}$ is nondegenerate. Let $P\in\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla)$ be elliptic with $m\geqslant 1$ and symbol ${\sigma}$ . Let $\Sigma$ be the closed set $\bigcup_{x\in M}\operatorname{sp}({\sigma}(x))$ . Then

1.

if $K$ is a compact subset of ${\mathbb{C}}$ disjoint from $\Sigma$ , then the spectrum of $P_{k}$ does not intersect $K$ when $k$ is large enough.
2.

if $\Omega$ is an open bounded subset of ${\mathbb{C}}$ with a smooth boundary disjoint from ${\Sigma}$ , then there exists $\Pi\in\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ such that $\Pi_{k}=(1_{\Omega}(P_{k}))$ when $k$ is large. Furthermore the principal symbol of $\Pi$ is at $x$

$\displaystyle\pi(x)=(2i\pi)^{-1}\int_{\partial\Omega}({\sigma}(x)-z)^{(-1)_{\sharp_{x}}}dz.$ (50)
3.

if for any $k$ , $P_{k}$ is formally self-adjoint for some volume element of $M$ , then for any $E_{-},E_{+}\in{\mathbb{R}}\setminus{\Sigma}$ with $E_{-}<E_{+}$ , $(1_{[E_{-},E_{+}]}(P_{k}))$ belongs to $\Psi_{\operatorname{Heis}}^{-\infty}(L,\nabla)$ .

Observe that the symbol $\pi(x)$ is the sum of the residues of the poles in $\Omega$ of the resolvent of ${\sigma}(x)$ . As we will see in the proof, the third assertion is a particular case of the second one, the symbol being the sum of the residues of the poles in $[E_{-},E_{+}]$ .

In [5], we will prove that $\Psi_{\operatorname{Heis}}^{m}\circ\Psi_{\operatorname{Heis}}^{p}\subset\Psi_{\operatorname{Heis}}^{m+p}$ . So in the second assertion, $\Pi$ being idempotent, it belongs to $\Psi_{\operatorname{Heis}}^{-\infty}(L,\nabla)$ .

Proof.

First, ${\Sigma}$ is closed because the Weyl quantization is continuous from $S^{m}({\mathbb{R}}^{d})$ to $\mathcal{L}(H^{m}_{\operatorname{iso}}({\mathbb{R}}^{d}),H^{0}_{\operatorname{iso}}({\mathbb{R}}^{d}))$ , so that the characterization of the spectrum given below implies that if $z_{0}\notin\operatorname{sp}({\sigma}(x_{0}))$ then $z\notin\operatorname{sp}({\sigma}(x))$ when $(z,x)$ is sufficiently close to $(z_{0},x_{0})$ .

Assume that $K$ is a compact subset of ${\mathbb{C}}$ disjoint from ${\Sigma}$ . When $z\in K$ , $(P_{k}-z)$ satisfies the assumptions of Theorem 6.3, so there exists $Q(z)\in\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ such that $Q_{k}(z)=(z-P_{k})^{-1}$ when $k\geqslant k_{0}(z)$ . This proves at least that $P_{k}$ has a compact resolvent as explained above when $k$ is large. Moreover we claim that everything in the proof of Theorem 6.3 can be done continuously with respect to $z\in K$ (even holomorphically with respect to $z$ in a neighborhood of $K$ ). More precisely, the Schwartz kernel of $Q_{k}(z)$ is locally of the form (25) where the dependence in $z$ in only in the symbol $b$ , which is continuous in $z$ . This proves first that we can choose $k_{0}(z)$ independent of $z$ , which shows the first assertion. Second, if $\Omega$ satisfies the assumptions of the second assertion of the Theorem, we can apply the previous consideration to $K=\partial\Omega$ and it follows that $\Pi_{k}:=(2i\pi)^{-1}\int_{\partial\Omega}Q_{k}(z)\;dz$ belongs to $\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ with a symbol given by (50). When $k$ is large enough, $Q_{k}(z)$ is the resolvent, so by Cauchy formula, $\Pi_{k}=1_{\Omega}(P_{k})$ . This concludes the proof of the second assertion.

For the last assertion, by assumption, for any fixed $k$ , $P_{k}$ is a formally self-adjoint elliptic differential operator on a compact manifold, so its spectrum is a discrete subset of ${\mathbb{R}}$ and $1_{[E_{-},E_{+}]}(P_{k})$ is a finite rank projector onto a subspace of ${\mathcal{C}}^{\infty}(M,L^{k})$ , [24, Theorem 8.3]. Moreover ${\sigma}$ is real valued so ${\Sigma}\subset{\mathbb{R}}$ . So there exists $\Omega$ satisfying the previous assumptions and such that $\Omega\cap{\mathbb{R}}=[E_{-},E_{+}]$ . So $\Pi=(1_{[E_{-},E_{+}]}(P_{k}),\;k\in{\mathbb{N}})$ belongs to $\Psi_{\operatorname{Heis}}^{-m}(L,\nabla)$ .

For any odd $N\in{\mathbb{N}}$ , $\Pi=1_{[E^{N}_{-},E^{N}_{+}]}(P^{N})$ and $P^{N}\in\mathcal{D}_{\operatorname{Heis}}^{mN}(L,\nabla)$ , which implies by the previous argument that $\Pi$ belongs to $\Psi_{\operatorname{Heis}}^{-Nm}(L,\nabla)$ , so $\Pi\in\Psi_{\operatorname{Heis}}^{-\infty}(L,\nabla)$ . ∎

Let us discuss briefly the convertibility and resolvent of elliptic elements of $(S^{\infty}(F),\circ_{\lambda})$ from the point of view of Weyl quantization. Let $\Psi_{\operatorname{iso}}^{\infty}({\mathbb{R}}^{d})$ be the space of pseudodifferential operators of ${\mathbb{R}}^{d}$ with a symbol in $S^{\infty}_{\operatorname{ph}}({\mathbb{R}}^{2d})$ . Any $A\in\Psi_{\operatorname{iso}}^{m}({\mathbb{R}}^{d})$ acts continuously $\mathcal{S}({\mathbb{R}}^{d})\rightarrow\mathcal{S}({\mathbb{R}}^{d})$ , $\mathcal{S}^{\prime}({\mathbb{R}}^{d})\rightarrow\mathcal{S}^{\prime}({\mathbb{R}}^{d})$ and $H^{s}_{\operatorname{iso}}({\mathbb{R}}^{d})\rightarrow H^{s-m}_{\operatorname{iso}}({\mathbb{R}}^{d})$ , where $H^{s}_{\operatorname{iso}}({\mathbb{R}}^{d})$ is the isotropic Sobolev spaces

H^{s}_{\operatorname{iso}}({\mathbb{R}}^{d})=\{u\in\mathcal{S}^{\prime}({\mathbb{R}}^{d}),\;Au\in L^{2}({\mathbb{R}}^{d}),\;\forall A\in\Psi_{\operatorname{iso}}^{s}({\mathbb{R}}^{d})\},\quad s\in{\mathbb{R}}.

When $A$ is elliptic, the following Fredholm property holds: $\ker A$ and $\ker A^{*}$ are finite dimensional subspaces of $\mathcal{S}({\mathbb{R}}^{d})$ ,

\mathcal{S}^{\prime}({\mathbb{R}}^{d})=A(\mathcal{S}^{\prime}({\mathbb{R}}^{d}))\oplus\ker A^{*}=A^{*}(\mathcal{S}^{\prime}({\mathbb{R}}^{d}))\oplus\ker A

and the generalised inverse $B:\mathcal{S}^{\prime}({\mathbb{R}}^{d})\rightarrow\mathcal{S}^{\prime}({\mathbb{R}}^{d})$ such that $BA-\operatorname{id}$ and $AB-\operatorname{id}$ are the orthogonal projectors onto $\ker A$ and $\ker A^{*}$ respectively, belongs to $\Psi_{\operatorname{iso}}^{-m}({\mathbb{R}}^{d})$ . So $A$ is invertible in the algebra $\Psi_{\operatorname{iso}}^{\infty}({\mathbb{R}}^{d})$ if and only if $\ker A=\ker A^{*}=0$ if and only if $A$ is invertible as an operator in $\mathcal{S}^{\prime}$ if and only if $A$ is invertible in $\mathcal{L}(H^{s}_{\operatorname{iso}}({\mathbb{R}}^{d}),H^{s-m}_{\operatorname{iso}}({\mathbb{R}}^{d}))$ .

When $m>0$ , any elliptic $A\in\Psi_{\operatorname{iso}}^{m}({\mathbb{R}}^{d})$ defines by restriction a closed unbounded operator of $L^{2}({\mathbb{R}}^{d})$ with domain $H^{m}_{\operatorname{iso}}({\mathbb{R}}^{d})$ . By the previous characterization of invertibility, the spectrum of $A$ is the same as the spectrum of its symbol $a$ defined above. Assume it is not empty, then $A$ has a compact resolvent, and as it was already explained, $\operatorname{sp}(A)$ is a discrete subset of ${\mathbb{C}}$ and the resolvent $(A-z)^{-1}$ is a holomorphic family of $\Psi_{\operatorname{iso}}^{-m}({\mathbb{R}}^{d})$ . Furthermore, for $A=a^{w}(x,\frac{1}{i}\partial_{x})$ , the residues $r_{\ell}^{w}(x,\frac{1}{i}\partial_{x})$ defined in (49) have finite rank and $r_{1}^{w}(x,\frac{1}{i}\partial_{x})$ is a projector onto the space of generalised eigenvectors of $A$ for the eigenvalue $z_{0}$ , which is a subspace of $\mathcal{S}({\mathbb{R}}^{d})$ .

7 Auxiliary bundles

Let us first define symbols taking values in an auxiliary bundle. Recall the spaces $S^{m}_{*}(N,E)$ introduced in Section 2 for a real vector bundle $p:E\rightarrow N$ and $*=\emptyset$ , $\operatorname{ph}$ , $\operatorname{sc}$ . Let $B$ be a complex vector bundle over $N$ . By definition $S^{m}(N,E;B)$ is the space of sections $s\in{\mathcal{C}}^{\infty}(E,p^{*}B)$ such that for any frame $(u_{{\alpha}})$ of $B$ over an open set $U$ of $N$ , we have over $p^{-1}(U)$ ,

s(x,\xi)=\sum f_{{\alpha}}(x,\xi)u_{{\alpha}}(x),\qquad x\in N,\;\xi\in E_{x}

with coefficients $f_{{\alpha}}$ in $S^{m}(U,E)$ . Since $S^{m}(U,E)$ is a ${\mathcal{C}}^{\infty}(U)$ -submodule of ${\mathcal{C}}^{\infty}(U,E)$ , this definition is compatible with the frame changes. Similarly, we define $S^{m}_{*}(M,E;B)$ for $*=\operatorname{ph}$ or $\operatorname{sc}$ by requiring that the coefficients $f_{{\alpha}}$ belong to $S^{m}_{*}(U,E)$ . More precisely, in the case of semiclassical symbols where the section $s$ and its local coefficients depend on $h$ , we only choose frames $(u_{{\alpha}})$ independent of $h$ .

Let $A_{1}$ and $A_{2}$ be two complex vector bundles over $M$ and let us define the pseudodifferential operator spaces $\Psi^{m}_{\operatorname{sc}}(M;A_{1},A_{2})$ , $\Psi_{\operatorname{tsc}}^{m}(L;A_{1},A_{2})$ and $\Psi_{\operatorname{Heis}}^{m}(L,\nabla;A_{1},A_{2})$ . For $A_{1}$ , $A_{2}$ being both the trivial line bundle, these are the spaces we introduced previously. In general, set $B=A_{2}\boxtimes A_{1}^{*}$ . Then

•

$\Psi^{m}_{\operatorname{sc}}(M;A_{1},A_{2})$ consists of the families $(P_{h}:{\mathcal{C}}^{\infty}(M,A_{1})\rightarrow{\mathcal{C}}^{\infty}(M,A_{2})$ , $h\in(0,1])$ satisfying the same conditions as before except that the amplitude $a$ appearing in (15) belongs to $S_{\operatorname{sc}}^{m}(U^{2},{\mathbb{R}}^{n};B)$ .

•

$\Psi_{\operatorname{tsc}}^{m}(L;A_{1},A_{2})$ consists of the families

\displaystyle P=(P_{k}:{\mathcal{C}}^{\infty}(M,L^{k}\otimes A_{1})\rightarrow{\mathcal{C}}^{\infty}(M,L^{k}\otimes A_{2}),\;k\in{\mathbb{N}})

(51)

satisfying the conditions of Definition 2.1 with $a\in S_{\operatorname{sc}}^{m}(U^{2},{\mathbb{R}}^{n};B)$

•

$\Psi_{\operatorname{Heis}}^{m}(L;A_{1},A_{2})$ consists of the families $P$ of the form (51) satisfying the conditions of Definition 3.2 with $Q_{h}$ an operator of $S^{m}_{\operatorname{sc}}(M;A_{1},A_{2})$

The symbol of $P$ is defined as before. Since the restriction of $B$ to the diagonal is isomorphic with $\operatorname{Hom}(A_{1},A_{2})$ , in the three cases, the symbol identifies with an element of $S^{m}_{\operatorname{ph}}(M,T^{*}M;\operatorname{Hom}(A_{1},A_{2}))$ .

The space $\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla;A_{1},A_{2})$ of Heisenberg differential operators consists of the families (51) of differential operators such that for any coordinate chart $(U,x_{i})$ of $M$ , we have on $U$

\displaystyle P_{k}=\sum_{\ell\in{\mathbb{N}},\;{\alpha}\in{\mathbb{N}}^{n},\;\ell+|{\alpha}|\leqslant m}k^{-\frac{\ell}{2}}f_{\ell,{\alpha}}\tilde{\pi}^{\alpha}

(52)

where $f_{\ell,{\alpha}}\in{\mathcal{C}}^{\infty}(U,\operatorname{Hom}(A_{1},A_{2}))$ , $\tilde{\pi}^{{\alpha}}=\tilde{\pi}_{1}^{{\alpha}(1)}\ldots\tilde{\pi}_{n}^{{\alpha}(n)}$ with $\tilde{\pi}_{i}=\tfrac{1}{i\sqrt{k}}\nabla_{\partial_{x_{i}}}^{L^{k}\otimes A_{2}}$ . Here we use a connection of $A_{2}$ , which induces with the connection of $L$ a covariant derivative of $A_{2}\otimes L^{k}$ . Proposition 5.1 still holds: Heisenberg differential operators are Heisenberg pseudodifferential operators, the symbol of (52) is $\sum_{|{\alpha}|\leqslant m}f_{0,{\alpha}}(x)\xi^{\sharp_{x}{\alpha}}$ ,

\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla;A_{2},A_{3})\circ\Psi_{\operatorname{Heis}}^{p}(L,\nabla;A_{1},A_{2})\subset\Psi_{\operatorname{Heis}}^{m+p}(L,\nabla;A_{1},A_{3}),

and the product of symbols is the fiberwise product $\sharp_{x}$ tensored by the composition $\operatorname{Hom}(A_{2,x},A_{3,x})\times\operatorname{Hom}(A_{1,x},A_{2,x})\rightarrow\operatorname{Hom}(A_{1,x},A_{3,x}).$ It is easy to see that the definition of the Heisenberg differential operators and of their symbols do not depend on the choice of the connection of $A_{2}$ .

In the sequel we assume that $A_{1}=A_{2}=A$ and is equipped with a Hermitian metric. We use the notation $\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla;A)$ instead of $\mathcal{D}_{\operatorname{Heis}}^{m}(L,\nabla;A,A)$ and similarly for the other operator spaces. Our goal is to generalize Theorem 6.4 for $P\in\mathcal{D}_{\operatorname{Heis}}^{2}(L,\nabla;A)$ having a symbol ${\sigma}$ of the form

\displaystyle{\sigma}(x,\xi)=\tfrac{1}{2}|\xi|_{x}^{2}+V(x)

(53)

where $|\cdot|$ is the norm of $T^{*}M$ for a Riemannian metric of $M$ not necessarily compatible with ${\omega}$ and $V\in{\mathcal{C}}^{\infty}(M,\operatorname{End}A)$ is Hermitian at each point. Example of such operators include Schrödinger operators with magnetic field and electric potential, holomorphic Laplacians or semiclassical Dirac operators, cf. [7, Section 3]. Besides of the numerous examples, the interest of these operators is that we can compute explicitly the spectrum of the symbols ${\sigma}(x,\cdot)$

\operatorname{sp}({\sigma}(x,\cdot))=\Bigl{\{}\sum_{i=1}^{n}B_{i}(x)({\alpha}(i)+\tfrac{1}{2})+V_{j}(x)/\;{\alpha}\in{\mathbb{N}}^{d},j=1,\ldots,r\Bigr{\}}

where $0<B_{1}(x)\leqslant\ldots\leqslant B_{d}(x)$ are the eigenvalues of ${\omega}(x)$ with respect to $g_{x}$ and $V_{1}(x)\leqslant\ldots\leqslant V_{r}(x)$ are the eigenvalues of $V(x)$ . Moreover, we have

\tfrac{1}{2}|\xi|^{2}_{x}=\sum_{i=1}^{d}B_{i}(x)h(s_{i},{\sigma}_{i}),\qquad h(y,\eta)=\tfrac{1}{2}(y^{2}+\eta^{2})

where $s_{i}$ and ${\sigma}_{i}$ are the linear coordinates of $T_{x}^{*}M$ associated to a symplectic basis. So the analysis of ${\sigma}(x,\cdot)$ boils down to the standard quantum harmonic oscillator $h^{w}$ or the Landau Hamiltonian $h(\frac{1}{i}\nabla)$ .

Theorem 7.1.

Let $P\in\mathcal{D}_{\operatorname{Heis}}^{2}(L,\nabla;A)$ having a symbol ${\sigma}$ of the form (53) and such that for each $k$ , $P_{k}$ is formally selfadjoint for a volume element of $M$ . Assume ${\omega}$ is nondegenerate and let ${\Sigma}=\bigcup_{x\in M}\operatorname{sp}({\sigma}(x,\cdot))$ . Then

-

For any $z\in{\mathbb{C}}\setminus{\Sigma}$ , there exists $Q(z)\in\Psi_{\operatorname{Heis}}^{-2}(L,\nabla;A)$ such that $(P_{k}-z)Q_{k}(z)=\operatorname{id}$ and $Q_{k}(z)(P_{k}-z)=\operatorname{id}$ when $k$ is large.
-

For any $E\in{\mathbb{R}}\setminus{\Sigma}$ , $(1_{(-\infty,E]}(P_{k}))$ belongs to $\Psi_{\operatorname{Heis}}^{-\infty}(L,\nabla;A)$ .

The proof is the same as the one of Theorem 6.4. The symbols $\tau(z)$ and $p_{E}$ of $Q(z)$ and $1_{(-\infty,E]}$ respectively are such that for any $x\in M$ ,

\tau(z)(x,\cdot)^{w}=({\sigma}(x,\cdot)^{w}-z)^{-1},\qquad p_{E}(x,\cdot)^{w}=1_{(-\infty,E]}(({\sigma}(x,\cdot)^{w}).

In the case where $B_{i}=1$ and $V=0$ , they have been studied for themselves in [9], [26], and given by the formulas (12) and (13) respectively.

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	$\displaystyle\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}\tilde{a}_{\epsilon}(x,\xi)\|$	$\displaystyle\leqslant C\sum_{{\alpha}^{\prime},{\beta}^{\prime}}\epsilon^{-\|{\beta}^{\prime}\|}\\|a\\|_{m,\ell,K}\langle\epsilon^{-1}(\xi-{\beta}(x))\rangle^{m-\|{\beta}^{\prime}\|}\quad\text{ by }\eqref{eq:e2}$
		$\displaystyle\leqslant C\sum_{{\alpha}^{\prime},{\beta}^{\prime}}\epsilon^{-\|{\beta}^{\prime}\|}\\|a\\|_{m,\ell,K}\langle\xi\rangle^{m-\|{\beta}^{\prime}\|}\epsilon^{-m_{+}}\quad\text{ by }\eqref{eq:e1}$
		$\displaystyle\leqslant C\epsilon^{-(\|{\alpha}\|+\|{\beta}\|)}\\|a\\|_{m,\ell,K}\langle\xi\rangle^{m-\|{\beta}\|}\epsilon^{-m_{+}}$