On the relationship between the semiclassical and standard pseudodifferential algebras

TL;DR

This paper clarifies the precise relationship between semiclassical and standard pseudodifferential algebras, exploring implications for spectral estimates and connecting second microlocalization to this relationship.

Contribution

It provides a detailed analysis of the relationship between semiclassical and standard pseudodifferential algebras and discusses their implications for spectral theory.

Findings

Established the relationship between semiclassical and standard pseudodifferential algebras.

Explored implications for large spectral parameter elliptic estimates.

Connected second microlocalization to the algebraic relationship.

Abstract

In this short paper we discuss the precise relationship between the semiclassical and standard pseudodifferential algebras and explore implications such as for large spectral parameter elliptic estimates, even in the case of pseudodifferential spectral familes. We also explain the connection between second microlocalization and this relationship.

Equations

$$(A-\lambda /h^m{)}^{-1}\in {\Psi }_{\infty ,\hslash }^{-m,-m,-m}={\Psi }_{\infty }^{-m,0}\cap {\Psi }_{\infty }^{0,-m}.$$

$$\mathrm{Op}(a)u(z)=(2\pi {)}^{-n}\int e^{i(z-z^{\prime })\cdot \zeta }a(z,\zeta )u(z^{\prime })d{z}^{\prime }d\zeta ,\mathrm{Op}(a)\in {\Psi }_{\infty }^m,$$

$$|(D_z^{\alpha }{D}_{\zeta }^{\beta }a)(z,\zeta )|\le C_{\alpha \beta }⟨\zeta {⟩}^{m-|\beta |};$$

$$|(D_z^{\alpha }{D}_{\zeta }^{\beta }a)(z,\zeta ,h)|\le C_{\alpha \beta }⟨\zeta {⟩}^{m-|\beta |};$$

$${\mathrm{Op}}_{\hslash }(a_h)u(z)=(2\pi h{)}^{-n}\int e^{i(z-z^{\prime })\cdot {\zeta }_{\hslash }/h}a(z,{\zeta }_{\hslash },h)u(z^{\prime })d{z}^{\prime }d\zeta ,{\mathrm{Op}}_{\hslash }(a_h)\in {\Psi }_{\hslash }^m,$$

$${\mathrm{Op}}_{\hslash }(a_h)u(z)=(2\pi {)}^{-n}\int e^{i(z-z^{\prime })\cdot \zeta }a(z,h\zeta ,h)u(z^{\prime })d{z}^{\prime }d\zeta =\mathrm{Op}({\tilde{a}}_h)u(z),$$

$$|((h{D}_h{)}^{\gamma }{D}_z^{\alpha }{D}_{\zeta }^{\beta }a)(z,\zeta ,h)|\le C_{\alpha \beta \gamma }{h}^{-k}⟨\zeta {⟩}^{m-|\beta |}.$$

$$|V_1\dots V_{N}a|\le C{\rho }_{\infty }^{-m}.$$

$$\overline{T^{\ast }}{\mathbb{R}}^n\times [0,1{]}_h={\mathbb{R}}_z^n\times \overline{({\mathbb{R}}^n{)}_{\zeta }^{\ast }}\times [0,1{]}_h,$$

$$|V_1\dots V_{N}a|\le C{\rho }_{\infty }^{-m}{h}^{-k};$$

$$[\overline{T^{\ast }}{\mathbb{R}}^n\times [0,1);\partial \overline{T^{\ast }}{\mathbb{R}}^n\times \{0\}]$$

$${\rho }_{\infty }/h=|h\zeta {|}^{-1}=|{\zeta }_{\hslash }{|}^{-1}.$$

$${\rho }_{\hslash ,\infty }=⟨{\zeta }_{\hslash }{⟩}^{-1}=(1+|{\zeta }_{\hslash }{|}^2{)}^{-1/2}=(1+h^2|\zeta {|}^2{)}^{-1/2}$$

$${\rho }_{\hslash ,\mathrm{ff}}=(h^2+{\rho }_{\infty }^2{)}^{1/2}=(h^2+(1+|\zeta {|}^2{)}^{-1}{)}^{1/2}=h(1+(h^2+|{\zeta }_{\hslash }{|}^2{)}^{-1}{)}^{1/2}$$

$${\rho }_{\hslash ,0}=\frac{(h^2+|{\zeta }_{\hslash }{|}^2{)}^{1/2}}{(1+|{\zeta }_{\hslash }{|}^2{)}^{1/2}};$$

$$\frac{(h^2+|{\zeta }_{\hslash }{|}^2{)}^{1/2}}{(1+|{\zeta }_{\hslash }{|}^2{)}^{1/2}}(1+(h^2+|{\zeta }_{\hslash }{|}^2{)}^{-1}{)}^{1/2}=\frac{(h^2+|{\zeta }_{\hslash }{|}^2+1{)}^{1/2}}{(1+|{\zeta }_{\hslash }{|}^2{)}^{1/2}}$$

$$[\overline{T^{\ast }}{\mathbb{R}}^n\times [0,1);\partial \overline{T^{\ast }}{\mathbb{R}}^n\times \{0\}],$$

$$|V_1\dots V_{N}a|\le C{\rho }_{\hslash ,\infty }^{-m}{\rho }_{\hslash ,\mathrm{ff}}^{-l}{\rho }_{\hslash ,0}^{-k}.$$

$${\rho }_{\infty }={\rho }_{\hslash ,\infty }{\rho }_{\hslash ,\mathrm{ff}},h={\rho }_{\hslash ,0}{\rho }_{\hslash ,\mathrm{ff}},$$

$$S_{\infty }^{m,k}=S_{\infty }^{m,m+k,k}$$

$$\overline{T^{\ast }}{\mathbb{R}}^n\times [0,1),\text{resp.}[\overline{T^{\ast }}{\mathbb{R}}^n\times [0,1);\partial \overline{T^{\ast }}{\mathbb{R}}^n\times \{0\}],$$

$${\rho }_{\infty }^{-m}{h}^{-k}={\rho }_{\hslash ,\infty }^{-m}{\rho }_{\hslash ,\mathrm{ff}}^{-m-k}{\rho }_{\hslash ,0}^{-k}.$$

$${\Psi }_{\infty }^{m,k}={\Psi }_{\infty ,\hslash }^{m,m+k,k}.$$

$$S_{\infty ,\hslash }^{m,l,k}\subset S_{\infty }^{m+\max (l-(m+k),0),k}$$

$${\rho }_{\hslash ,\infty }^{-m}{\rho }_{\hslash ,\mathrm{ff}}^{-l}{\rho }_{\hslash ,0}^{-k}\le C{\rho }_{\hslash ,\infty }^{-m}{\rho }_{\hslash ,\mathrm{ff}}^{-(m+k)}{\rho }_{\hslash ,0}^{-k}=C{\rho }_{\infty }^{-m}{h}^{-k},$$

$${\rho }_{\hslash ,\infty }^{-m}{\rho }_{\hslash ,\mathrm{ff}}^{-l}{\rho }_{\hslash ,0}^{-k}\le C{\rho }_{\hslash ,\infty }^{-(l-k)}{\rho }_{\hslash ,\mathrm{ff}}^{-l}{\rho }_{\hslash ,0}^{-k}=C{\rho }_{\infty }^{-(l-k)}{h}^{-k},$$

$${\Psi }_{\infty ,\hslash }^{m,l,k}\subset {\Psi }_{\infty }^{m+\max (l-(m+k),0),k}$$

$${\Psi }_{\hslash }^{m,k}\subset {\Psi }_{\infty ,\hslash }^{m,k,k};$$

$${\rho }_{\infty }^{-m}{h}^{-k}={\rho }_{\hslash ,\infty }^{-m}{\rho }_{\hslash ,\mathrm{ff}}^{-m-k}{\rho }_{\hslash ,0}^{-k}.$$

$$\tilde{E}=\tilde{B}A-I,\tilde{F}=A\tilde{B}-I\in {\Psi }_{\infty ,\hslash }^{-\infty ,-\infty ,0}={\Psi }_{\infty }^{-\infty ,0},$$

Full text

The author gratefully acknowledges support from the National Science Foundation under grant numbers DMS-1953987 and DMS-2247004. The author is also very grateful to Oliver Petersen and Jared Wunsch for comments on the manuscript, and to the anonymous referees for the careful reading of the manuscript and the comments which greatly improved the exposition.

1. Introduction

The purpose of this paper is to discuss the precise relationship between the semiclassical and standard pseudodifferential operator algebras and explain the connection to second microlocalization discussed in [1, 13, 11, 10, 12]. One immediate consequence is large spectral parameter resolvent estimates for elliptic pseudodifferential operators (as opposed to arguments which are specific to the standard differential case), and in addition a very simple approach to the functional calculus.

In order to state the main theorem, we need to introduce some notation that will be explained in detail below. First, there is flexibility in the overall context; for simplicity and definiteness we usually refer to the bounded geometry setting, thus including compact manifolds without boundary in particular, but for instance the same kind of arguments work in Melrose’s scattering pseudodifferential algebra [4]. With this in mind, $\Psi_{\infty}^{m,k}$ denotes the space of pseudodifferential operators depending on a parameter, which we consider small (say, in $[0,1)$) and denote by $h$; here the differential order is $m$ and parameter order is $k$. Note that here $h$ is purely a parameter on which symbols depend (smoothly or in a uniformly bounded way), distingushing this from the semiclassical pseudodifferential operator algebra. Next, $\Psi_{\infty,\hbar}^{m,l,k}$ denotes111Below we use $h$ for the actual semiclassical parameter, and $\hbar$ as a subscript to denote semiclassical objects. the combined semiclassical-classical algebra in this setting, with semiclassical differential order $m$, semiclassical parameter order $l$ and classical parameter order $k$. Then we have the following result:

Theorem 1.1**.**

Suppose that $A$ is an elliptic pseudodifferential operator in $\Psi_{\infty}^{m,0}$, $m>0$, with real principal symbol. Suppose $\lambda$ is in a compact subset of $\mathbb{C}$ disjoint from $\mathbb{R}$. Then $A-\lambda/h^{m}\in\Psi_{\infty,\hbar}^{m,m,m}$ and is elliptic there, invertible for $h$ sufficiently small, with

[TABLE]

We remark here that the issue for the large parameter spectral family for pseudodifferential operators $A\in\Psi_{\infty}^{m,0}$, $m>0$, is that it does not lie in Shubin’s class of large parameter pseudodifferential operators [6] since an expression like $a(z,\zeta)-\sigma^{m}$ (where $\sigma$ is the large parameter) is not jointly symbolic in $(\zeta,\sigma)$, unless $a$ is a polynomial, i.e. $A$ is a differential operator.

This theorem has standard generalizations as long as the symbol takes values in a closed cone not containing $\lambda$. Theorem 1.1 moreover provides a very simple approach to complex powers for self-adjoint $A$ as the $(A-\lambda/h^{m})^{-1}\in\Psi_{\infty}^{-m,0}$ statement indicates: along a contour, one integrates a family of standard pseudodifferential operators with uniform behavior in this class. In fact, after the preparation of the initial version of this paper, non-elliptic versions of the method (proceeding in the same setting, but via propagation estimates) have been used to analyze the spectral family of Lorentzian Dirac operators by Dang, Vasy and Wrochna [2] and this in turn has been utilized to construct complex powers even in this non-self-adjoint setting, extending the scalar (thus self-adjoint) Lorentzian work of Dang and Wrochna [3] which relied on the author’s earlier less precise results [9].

We now proceed to define these classes. Recall that standard pseudodifferential operators on $\mathbb{R}^{n}$ are of the form

[TABLE]

where $u$ is, say, Schwartz, $a\in S^{m}_{\infty}$ a symbol, and the integral is interpreted as an oscillatory integral. Here the symbol class demands

[TABLE]

we use this ‘bounded geometry on $\mathbb{R}^{n}$’ type class as an example for simplicity and as various versions are equivalent on compact manifolds. The subscript $\infty$ is inserted as a reminder that this is Hörmander’s uniform symbol class (in $z$), i.e. $z$ is not a symbolic variable, though again the discussion simply extends to the scattering class of Melrose, [4], for instance, which in this $\mathbb{R}^{n}$ case first arose in the works of Shubin [7] and Parenti [5]. Of course we may consider a family of operators depending on a parameter $h\in[0,1]$, so we have a family of symbols $a=(a_{h})$ with uniform estimates:

[TABLE]

this gives the class $\Psi_{\infty}^{m,0}$ in the above notation; if the right hand instead has $C_{\alpha\beta}h^{-k}\langle\zeta\rangle^{m-|\beta|}$, which we denote by $S_{\infty}^{m,k}$, we obtain the class $\Psi_{\infty}^{m,k}$. (The choice of the orders is such that the spaces become larger with each index.)

On the other hand, corresponding to a family $a=(a_{h})$, semiclassical pseudodifferential operators on $\mathbb{R}^{n}$ are of the form

[TABLE]

where $u$ is again, say, Schwartz, $a\in S^{m}_{\infty}$, and this expression is considered for $h\in(0,1]$. More generally, when $a\in S^{m,k}_{\infty}$, we define $\operatorname{Op}_{\hbar}(a_{h})\in\Psi_{\hbar}^{m,k}$ this way. Of course, for $h>0$, a simple change of variables connects these two:

[TABLE]

where $\tilde{a}(z,\zeta,h)=a(z,h\zeta,h)$.

This is an explicit, if singular in the limit $h\to 0$, connection between the two classes, but it is useful to interpret this more geometrically. This has a particular virtue if one considers a semiclassical family of operators but would like to have it act on non-semiclassical function spaces. A need for this arises, for instance, in the holomorphic functional calculus, in which functions of a, say, elliptic, positive operator (on say a compact manifold) are expressed as a contour integral involving the resolvent: the large spectral parameter behavior (as we go out to infinity along the contour) of the resolvent can be interpreted as a semiclassical problem, but if we want to get at mapping properties of the result directly, we need to work on a fixed function space (not on an $h$-dependent one).

We remark here that although we wrote the above formulae for $\mathbb{R}^{n}$, they immediately transfer essentially completely to compact manifolds via localization; of course due to far-from diagonal smoothing contributions, one needs to say a bit more in that case, namely add smooth (in the manifold variables) Schwartz kernels, with uniform in $h$ bounds on all derivatives, explicitly to $\Psi_{\infty}^{m,0}$, and similarly for the semiclassical case one needs to add smooth (in the manifold variables) Schwartz kernels that are rapidly decaying in $h$. Indeed, in order to simplify the geometric discussion (avoiding the need to discuss the fibration over $[0,1)_{h}$ in this context, and indeed b-fibration later after a blow-up, thus also avoiding the need to add tangency to its fibers conditions to the vector fields) it is convenient below to strengthen our classes; from now on we demand conormal to $h=0$ behavior for our symbol classes, i.e. we strengthen our symbol estimates to

[TABLE]

For our purpose, it is helpful to adopt a (partially) compactified perspective due to Melrose [4] in which the fibers of the cotangent bundle $(\mathbb{R}^{n})^{*}_{\zeta}$ are compactified to balls $\overline{\mathbb{R}^{n}}$, with boundary defining function $\rho_{\infty}=|\zeta|^{-1}$ (more precisely, $\langle\zeta\rangle^{-1}$; the two are equivalent outside a compact subset of $\mathbb{R}^{n}$): $\overline{T^{*}}\mathbb{R}^{n}=\mathbb{R}^{n}_{z}\times\overline{(\mathbb{R}^{n})^{*}_{\zeta}}$; see also [8]. More concretely, as $\{\zeta\in(\mathbb{R}^{n})^{*}:\ |\zeta|>1\}$ can be identified with $(1,\infty)_{|\zeta|}\times\mathbb{S}^{n-1}$ via ‘polar coordinates’, in this compactification a boundary at infinity is added by identifying this in turn with $(0,1)_{|\zeta|^{-1}}\times\mathbb{S}^{n-1}$, and regarding the latter as a subset of $[0,1)_{\rho_{\infty}}\times\mathbb{S}^{n-1}$, where now the ideal boundary $\{\rho_{\infty}=0\}$ has been added. The symbol estimates (locally in $z$, which suffices for compact manifold applications) are equivalent to the requirement that for all $N$ and vector fields $V_{1},\ldots,V_{N}$ tangent to $\partial\overline{T^{*}}\mathbb{R}^{n}$ there is $C>0$ such that

[TABLE]

Indeed, if each $V_{k}$ is either $D_{\zeta_{j}}$ or $\zeta_{i}D_{\zeta_{j}}$ or $D_{z_{j}}$ for some $i,j$, with $D_{\zeta_{j}}$ relevant only near $\zeta=0$, the estimates are easily seen to be equivalent to symbol estimates, and these $V_{k}$ span, over $C^{\infty}(\overline{T^{*}}\mathbb{R}^{n})$, the space of smooth vector fields tangent to $\partial\overline{T^{*}}\mathbb{R}^{n}$. As it is the fibers that are compactified in this manner, we name the boundary, $\{\rho_{\infty}=0\}$, fiber infinity. Adding a parameter $h$, we obtain the parameter dependent partially compactified cotangent bundle

[TABLE]

with a uniform in $h$ version of the membership statement; $\{\rho_{\infty}=0\}$ is still called fiber infinity. As already mentioned prior to (1.1), it is actually helpful to consider the conormal parameter dependent version of the class, i.e. when for all $N$ and vector fields $V_{1},\ldots,V_{N}$ tangent to $\partial(\overline{T^{*}}\mathbb{R}^{n}\times[0,1))$ (we suppress the uninteresting $h=1$ boundary by explicitly removing it) there is $C>0$ such that

[TABLE]

these vector fields include $hD_{h}$, and are spanned by $hD_{h}$ together with the ones listed above: $D_{z_{j}}$, $D_{\zeta_{j}}$, $\zeta_{i}D_{\zeta_{j}}$.

We now proceed to introduce the mixed semiclassical-classical pseudodifferential operators; see Figure 1 for an illustration of the phase spaces. Blowing up the corner, $\{h=0,\ \rho_{\infty}=0\}$, of $\overline{T^{*}}\mathbb{R}^{n}\times[0,1)$, to obtain

[TABLE]

introduces a new front face ${\mathrm{ff}}$. Away from the lift of $h=0$, i.e where $h$ is relatively large relative to $\rho_{\infty}$, i.e. $\rho_{\infty}<Ch$, projective coordinates are given by $h$, $\rho_{\infty}/h$ as well as tangential variables to the corner, while away from the lift of $\rho_{\infty}=0$, i.e. where $\rho_{\infty}$ is relatively large relative to $h$, i.e. $h<C\rho_{\infty}$, projective coordinates are given by $\rho_{\infty},h/\rho_{\infty}$ as well as tangential variables to the corner. But as observed above $\rho_{\infty}=|\zeta|^{-1}$, so

[TABLE]

Correspondingly near the lift of fiber infinity, i.e. $\rho_{\infty}=0$, this space is just the same as fiber infinity for the semiclassical space, so we may call it ‘semiclassical fiber infinity’ (and the earlier ‘classical fiber infinity’ for clarity)!

A different way of arriving at the same result is to consider the blow-up of the zero section at $h=0$ of the semiclassical cotangent bundle; Figure 1 illustrates why the resulting spaces are naturally diffeomorphic (i.e. the natural identification in the interior extends to a diffeomorphism). This second approach is called second microlocalization, in this case for the semiclassical symbols at the zero section.

Returning to the corner blow up perspective, we write $\rho_{\hbar,\infty}$ for the defining function of the lift of classical fiber infinity (i.e. semiclassical fiber infinity), which is (equivalent to) (1.2) locally near the lift of fiber infinity; globally one can for instance take it to be222An equivalent choice is $\rho_{\hbar,\infty}=\frac{\rho_{\infty}}{(h^{2}+\rho_{\infty}^{2})^{1/2}}=\frac{\langle\zeta\rangle^{-1}}{(h^{2}+(1+|\zeta|^{2})^{-1})^{1/2}}$, natural from the corner blow up perspective.

[TABLE]

(which is immediate from the second approach). Next, we write $\rho_{\hbar,{\mathrm{ff}}}$ for the defining function of the front face, which is $h$ near the lift of fiber infinity and $\rho_{\infty}$ near the lift of $h=0$; globally we can take it to be

[TABLE]

as is immediate from the corner blow up perspective since $h=0$, $\langle\zeta\rangle^{-1}=0$ is being blown up. Finally, we write $\rho_{\hbar,0}$ for the defining function of the lift of $h=0$, which is locally given (up to equivalence) by $|\zeta_{\hbar}|$, and globally by333An equivalent choice is $\rho_{\hbar,0}=\frac{h}{(h^{2}+(1+|\zeta|^{2})^{-1})^{1/2}}$, again natural from the corner blow up perspective.

[TABLE]

we call this the parameter boundary. Notice that, as is necessarily the case, $\rho_{\hbar,0}\rho_{\hbar,{\mathrm{ff}}}$ is equivalent to $h$ as

[TABLE]

and the $h^{2}$ in the numerator is irrelevant in view of the term $1$ there. Similarly, $\rho_{\hbar,\infty}\rho_{\hbar,{\mathrm{ff}}}$ is equivalent to $\rho_{\infty}=\langle\zeta\rangle^{-1}$.

We write $S_{\infty}^{m,l,k}$ for the symbol class that arises on this resolved space:

Definition 1.2**.**

The space $S_{\infty}^{m,l,k}$ consists of conormal functions on the resolved space

[TABLE]

i.e. satisfy estimates which are stable under iterated application of vector fields tangent to all boundary hypersurfaces:

[TABLE]

Here it suffices to consider the $V$’s being $hD_{h}$, $D_{z_{j}}$, $D_{\zeta_{j}}$, $\zeta_{i}D_{\zeta_{j}}$ since over smooth functions on $[\overline{T^{*}}\mathbb{R}^{n}\times[0,1);\partial\overline{T^{*}}\mathbb{R}^{n}\times\{0\}]$ these span the space of vector fields tangent to all boundary hypersurfaces.

We call $m$ the semiclassical differential order, $l$ the semiclassical growth order, $k$ the standard parameter growth order at $h=0$ as a family of symbols, and we use the ordering differential, semiclassical, parameter decay at $h=0$ for the superscripts.

We already noted that (up to equivalence of boundary defining functions)

[TABLE]

and in fact correspondingly

[TABLE]

under the natural pullback identification. This identification breaks for the corresponding one-step polyhomogeneous symbols, with smooth dependence on $h$ (these are often called classical symbols, but we avoid this term due to our different use of the word ‘classical’), for which the corresponding left hand side is merely included in the corresponding right hand side. The reason is simple: for instance, if all orders are [math], locally both classes are simply smooth functions on

[TABLE]

and certainly these two spaces of smooth functions differ, unlike the spaces of conormal functions with all orders [math] on the two spaces.

This classical-semiclassical relationship is completely analogous to the relationship between the b- and sc- cotangent bundles: the sc-cotangent bundle arises by blowing up the corner of the fiber-compactified b-cotangent bundle, and conversely the b-cotangent bundle arises by blowing up the zero section of the sc-cotangent bundle at the boundary. While these statements are symmetric, there is a big difference: blowing up the corner does not change the regularity properties with respect to vector fields tangent to the boundary hypersurfaces, hence the sc-algebra is naturally included in the b-algebra (in a non-classical manner). On the other hand, blowing up the zero section at the boundary is highly singular with respect to the quantization map (one obtains an ill-behaved symbol class) which is the reason second microlocalization (which exactly adds b-algebra features to the scattering one) is considered delicate; see [11]. Thus, just as in the b-sc setting one should think of the process in the less symbolic b-terms, by blowing up the corner, in our semiclassical setting we should prefer blowing up the corner of the standard fiber-compactified (family) cotangent bundle at $h=0$ since this does not affect the symbolic properties; the standard family algebra is less symbolic (as it is not symbolic in $h$) than the semiclassical algebra. Note that in the semiclassical setting the more difficult, via second microlocalization, approach was carried out by Wunsch and the author in [13]. The present work of course only covers the model case of [13], namely second microlocalization at the zero section; this was transferred in that work to other Lagrangian submanifolds by conjugation by semiclassical Fourier integral operators, and an analogous process would be required here. (When doing this, there is substantial extra work if, as we do in this paper in the context of ‘full ellipticity below’, one wants to precisely describe the ‘residual’ behavior, which is global along the Lagrangian; this aspect was not covered in [13] either.)

One advantage of the blow-up procedure for even $\Psi^{m,k}_{\infty}$ is that one has a more refined notion of elliptic set and wave front set which are now subsets of semiclassical fiber infinity and the front face of $[\overline{T^{*}}\mathbb{R}^{n}\times[0,1);\partial\overline{T^{*}}\mathbb{R}^{n}\times\{0\}]$. Namely:

Definition 1.3**.**

Let $A\in\Psi^{m,k}_{\infty}$ be the left quantization444Right, Weyl, etc., quantizations would work equally well. of $a\in S^{m,k}_{\infty}$ modulo $\Psi^{-\infty,k}_{\infty}$. We say that a point $\alpha$ in semiclassical fiber infinity or the front face of $[\overline{T^{*}}\mathbb{R}^{n}\times[0,1);\partial\overline{T^{*}}\mathbb{R}^{n}\times\{0\}]$ is in the elliptic set of $A$ if $\alpha$ has a neighborhood $O$ in $[\overline{T^{*}}\mathbb{R}^{n}\times[0,1);\partial\overline{T^{*}}\mathbb{R}^{n}\times\{0\}]$ such that $|a|_{O}|$ is bounded below by a constant positive multiple of

[TABLE]

The wave front set is defined similarly, asking for infinite order vanishing of $a|_{O}$ at semiclassical fiber infinity and the front face.

This refines the standard elliptic and wave front sets of the family which would lie in classical fiber infinity; the difference is that classical fiber infinity at $h=0$ is replaced by the whole new front face.

Another advantage of this blow-up procedure is that we can introduce a collection of pseudodifferential operators $\Psi_{\infty,\hbar}^{m,l,k}$ that has an additional order based on quantizing $S^{m,l,k}_{\infty,\hbar}$ using the same ‘standard’ (i.e. non-semiclassical) quantization map, $\operatorname{Op}$. Before stating the details, if the operator arises from a standard symbol of order $m,k$, i.e. is in $\Psi_{\infty}^{m,k}$, one necessarily has $l=m+k$ corresponding to (1.3), and indeed the converse relationship also holds for symbols so

[TABLE]

Thus, standard pseudodifferentials are in this joint algebra, even though typically they are not, unlike differential operators, in the regular semiclassical algebra. Note, however, that for the one-step polyhomogeneous, with smooth dependence on $h$ subalgebra of $\Psi_{\infty}^{m,k}$, and the analogous one-step polyhomogeneous subalgebra of $\Psi_{\infty,\hbar}^{m,m+k,k}$, the analogue of the equality (1.4) does not hold, unlike in the symbolic algebra; we only have the inclusion of the corresponding left hand side in the corresponding right hand side.

One way to give the precise definition of $\Psi^{m,l,k}_{\infty,\hbar}$ in general is via noting that

[TABLE]

since if $l\leq m+k$ then $\rho_{\hbar,{\mathrm{ff}}}^{l}\geq C\rho_{\hbar,{\mathrm{ff}}}^{m+k}$ hence

[TABLE]

while if $l\geq m+k$ then $\rho_{\hbar,\infty}^{m}\geq C\rho_{\hbar,\infty}^{l-k}$, so

[TABLE]

and the derivative estimates are equivalent on the two spaces. Thus, one can simply define:

Definition 1.4**.**

The space $\Psi^{m,l,k}_{\infty,\hbar}$ of semiclassical-classical pseudodifferential operators

[TABLE]

is the image of the quantization map $\operatorname{Op}$ on $S^{m,l,k}$, plus elements of $\Psi^{-\infty,k}_{\infty}$.

Notice that with this definition applied in the compact manifold setting, Schwartz kernels which are conormal to $h=0$, with $h^{-k}$ bounds, are automatically included in $\Psi_{\infty,\hbar}^{m,l,k}$.

Membership in this subspace $\Psi^{m,l,k}_{\infty,\hbar}$ is characterized by purely symbolic properties, namely finite order vanishing conditions within this class, concretely of order $\max(l-(m+k),0)$ at semiclassical fiber infinity and $m+k-l+\max(l-(m+k),0)=\max(m+k-l,0)$ at the front face. Therefore, using the standard left and right reduction, hence adjoint and composition, formulae, $\Psi_{\hbar,\infty}$ is easily seen to form a tri-filtered $*$-algebra.

In addition to (1.4), directly from the symbol level, where the underlying space can be identified as a blow up of the semiclassical phase space at the zero section at $h=0$, we also have the inclusion

[TABLE]

this is a proper inclusion since for instance any smooth Schwartz kernel with smooth dependence on $h$ (and bounded with all derivatives) lies in $\Psi_{\infty,\hbar}^{-\infty,-\infty,0}$, but is not a semiclassical pseudodifferential operator (composition is not even commutative to leading order in $h$!).

A key point is that just as $\Psi^{m,k}_{\infty}$ only has a principal symbol at fiber infinity, i.e. it captures $\Psi^{m,k}_{\infty}$ modulo $\Psi^{m-1,k}_{\infty}$, not gaining any decay at $h=0$, $\Psi_{\infty,\hbar}^{m,l,k}$ inherits this principal symbol at semiclassical fiber infinity and the front front face, so the principal symbol captures it modulo $\Psi_{\infty,\hbar}^{m-1,l-1,k}$ which is a gain at semiclassical fiber infinity and the front face but not at the parameter boundary. Namely:

Definition 1.5**.**

The principal symbol of $A\in\Psi_{\infty,\hbar}^{m,l,k}$, written as the left quantization555Again, right or Weyl quantizations could be used equally well. of $a\in S_{\infty,\hbar}^{m,l,k}$ modulo $\Psi_{\infty,\hbar}^{-\infty,-\infty,k}$, is the class of $a$ in $S_{\infty,\hbar}^{m,l,k}/S_{\infty,\hbar}^{m-1,l-1,k}$.

At the classical face, i.e. the lift of $h=0$, elements of $\Psi^{m,k,l}_{\infty,\hbar}$ have, just like elements of $\Psi^{m,k}_{\infty}$, an operator valued symbol. Namely, for $\Psi^{m,k}_{\infty}$, if the element is actually smooth in $h$ (rather than conormal), this operator valued symbol, or normal operator, is the $h=0$ value of $h^{k}$ times the family. For one-step polyhomogeneous symbols in $S_{\infty,\hbar}^{m,l,k}$ this normal operator can still be considered as the restriction (after multiplying by $h^{k}$) of the symbol family to the parameter boundary and quantized in the standard (non-semiclassical) manner; for the general class $\Psi_{\infty,\hbar}^{m,l,k}$ it is best to simply consider it as a representative in $\Psi_{\infty,\hbar}^{\infty,l,k}$ modulo $\Psi_{\infty,\hbar}^{\infty,l,k-1}$; the first order is irrelevant here since semiclassical fiber infinity does not intersect the parameter boundary.

Ellipticity corresponds to the ellipticity at the principal symbol level, as in Definition 1.3:

Definition 1.6**.**

Let $A\in\Psi^{m,l,k}_{\infty,\hbar}$ and let $a\in S_{\infty,\hbar}^{m,l,k}$ be a representative of its principal symbol. We say that a point $\alpha$ in semiclassical fiber infinity or the front face of $[\overline{T^{*}}\mathbb{R}^{n}\times[0,1);\partial\overline{T^{*}}\mathbb{R}^{n}\times\{0\}]$ is in the elliptic set of $A\in\Psi^{m,l,k}_{\infty,\hbar}$ if $\alpha$ has a neighborhood $O$ in $[\overline{T^{*}}\mathbb{R}^{n}\times[0,1);\partial\overline{T^{*}}\mathbb{R}^{n}\times\{0\}]$ such that $|a|_{O}|$ is bounded below by a constant positive multiple of

[TABLE]

The wave front set is defined similarly, asking for infinite order vanishing of $a|_{O}$, where $A$ now is, say, the left quantization of $a$ modulo $\Psi^{m,l,k}_{\infty,\hbar}$, at semiclassical fiber infinity and the front face.

If $A\in\Psi_{\infty,\hbar}^{m,l,k}$ is elliptic then by the standard symbolic construction there is a parametrix $\tilde{B}\in\Psi_{\infty,\hbar}^{-m,-l,-k}$ such that

[TABLE]

and $\tilde{E}=\tilde{E}(h)$ (and similarly $\tilde{F}$) then gives a uniformly bounded family of operators between any (standard) Sobolev spaces. In particular, if the underlying manifold is compact or the pseudodifferential operators are symbolic also at infinity (in which case there would be also order $-\infty$ in that sense, and the Sobolev spaces would have arbitrary decay), such as the scattering algebra, these operators $\tilde{E},\tilde{F}$ are in addition compact, giving a uniform (in $h$) Fredholm theory.

On the other hand full ellipticity in addition includes the invertibility of the normal operator:

Definition 1.7**.**

We say that $A\in\Psi_{\infty,\hbar}^{m,l,k}$ if fully elliptic if it is elliptic and there exists a representative $A_{0}\in\Psi_{\infty,\hbar}^{\infty,l,k}$ of $A\in\Psi_{\infty,\hbar}^{m,l,k}$ modulo $\Psi_{\infty,\hbar}^{\infty,l,k-1}$ and an element $B_{0}$ of $\Psi_{\infty,\hbar}^{\infty,-l,-k}$ such that $B_{0}A_{0}=A_{0}B_{0}=I\in\Psi_{\infty,\hbar}^{0,0,0}$.

Full ellipticity guarantees invertibility of the family $A$ for small $h$:

Proposition 1.8**.**

If $A\in\Psi_{\infty,\hbar}^{m,l,k}$ is fully elliptic then there exists $B\in\Psi_{\infty,\hbar}^{-m,-l,-k}$ such that

[TABLE]

and similarly $AB=I+F$, $F\in\Psi_{\infty,\hbar}^{-\infty,-\infty,-\infty}$.

See below for an example involving spectral families of positive order operators.

Proof.

Suppose that $A\in\Psi_{\infty,\hbar}^{m,l,k}$ is fully elliptic. Then, as mentioned above, by the standard symbolic construction there is a parametrix $\tilde{B}\in\Psi_{\infty,\hbar}^{-m,-l,-k}$ such that

[TABLE]

Let $A_{0},B_{0}$ be as above using the full ellipticity of $A$. Let

[TABLE]

then

[TABLE]

with $E_{1}\in\Psi_{\infty,\hbar}^{-\infty,-\infty,-1}$, and a standard iteration and asymptotic summation improves this to $B\in\Psi_{\infty,\hbar}^{-m,-l,-k}$ such that

[TABLE]

and similarly for $AB$. Equality of left and right parametrices modulo $\Psi_{\infty,\hbar}^{-\infty,-\infty,-\infty}$ follows as usual. ∎

Corollary 1.9**.**

If $A\in\Psi_{\infty,\hbar}^{m,l,k}$ is fully elliptic then there is $h_{0}>0$ such that $A=A_{h}$ is invertible for $h<h_{0}$.

Proof.

By Proposition 1.8, $E$ is $O(h^{\infty})$ on any reasonable space (such as $L^{\infty}_{h}L^{2}$) hence invertibility of $I+E$ for small $h$ follows, and then

[TABLE]

together with the regularizing property of $E$ shows that $(I+E)^{-1}-I\in\Psi_{\infty,\hbar}^{-\infty,-\infty,-\infty}$ as well. ∎

It is simple to read off mapping properties of the joint standard-semiclassical spaces $H_{\infty,\hbar}^{s,r,p}$, which are defined, e.g. if all the indices are $\geq 0$, as the space of those $u\in L^{\infty}_{h}L^{2}$ such that $Au\in L^{\infty}_{h}L^{2}$ for some (hence all) fully elliptic $A\in\Psi_{\infty,\hbar}^{s,r,p}$. (One can also use spaces that are $L^{2}_{h}L^{2}$, which are often more natural!) Equivalently, using only ellipticity (rather than full ellipticity), if all the indices are $\geq 0$ and $r\geq p$, $H_{\infty,\hbar}^{s,r,p}$ is the space of those $u\in h^{p}L^{\infty}_{h}L^{2}$ such that $Au\in L^{\infty}_{h}L^{2}$ for some (hence all) fully elliptic $A\in\Psi_{\infty,\hbar}^{s,r,p}$. Namely, the mapping property is

[TABLE]

Moreover, $H_{\infty}^{s,p}=H_{\infty,\hbar}^{s,s+p,p}$ (with the left hand side defined using $\Psi_{\infty}^{s,p}$ in place of $\Psi_{\infty,\hbar}^{s,r,p}$, so this is an immediate consequence of the definitons and the relationship between the pseudodifferential algebras), and $H_{\hbar}^{s,r}=H_{\infty,\hbar}^{s,r,r}$ (with the left hand side defined using the semiclassical algebra); this different character of the identification of the classical and semiclassical Sobolev spaces as a joint space is what makes mapping properties of semiclassical operators on classical spaces more subtle.

Now, the typical issue with semiclassical estimates on non-semiclassical spaces is that the orders of the operator do not conform to the standard case. For instance, the spectral family $\Delta-z$, where $z$ runs to infinity in a cone disjoint from the positive reals, can be written as $h^{-2}(h^{2}\Delta-\lambda)$, so $z=\lambda/h^{2}$, and now $\lambda$ is bounded away from $[0,\infty)$ and is bounded. As $\Delta\in\Psi_{\infty}^{2,0}$, $h^{-2}\lambda\in\Psi_{\infty}^{0,2}$, we have

[TABLE]

and it is elliptic in this class since its principal symbol is $h^{-2}(|\zeta_{\hbar}|^{2}_{g}-\lambda)$, where the second term is only relevant at the front face and it being bounded away from $[0,\infty)$ assures the appropriate lower bound for this principal symbol. Moreover, its normal operator is $h^{-2}\lambda\in\Psi_{\infty,\hbar}^{0,2,2}$, i.e. a non-zero multiple of the identity, which is certainly invertible. Correspondingly $h^{-2}(h^{2}\Delta-\lambda)$ is fully elliptic, and thus is invertible for small $h$ with

[TABLE]

Hence,

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Of course, by (1.5), in this case we have the stronger statement

[TABLE]

is elliptic in this semiclassical algebra, hence the inverse lies in

[TABLE]

and then we can simply use the final inclusion here to obtain the mapping properties above. However, as we point out below, proceeding with the joint algebra from the start immediately extends the argument to the spectral family of pseudodifferential operators.

Now, if we take $r=s+p$ so that the domain is $H_{\infty}^{s,p}$, then the output is in $H_{\infty,\hbar}^{s+2,s+p+2,p+2}$, but this is not $H_{\infty}^{s+2,p+2}$ as the first and last orders may suggest! Correspondingly, in order to work with purely non-semiclassical spaces one has to give up something and have, with $t\in[0,2]$,

[TABLE]

Since the output space is $H_{\infty,\hbar}^{s+t,s+p+2,p+2-t}$, this is actually sharp in the second order sense, but one is juggling whether to give up differentiability or decay in the standard sense as $h\to 0$. Note that the extreme cases, which immediately imply (1.6), are the well-known

[TABLE]

the former embodies the large parameter decay, the second standard ellipticity, but in any case there is a compromise.

Taking advantage of the joint standard/semiclassical algebra we can similarly prove Theorem 1.1:

Proof of Theorem 1.1.

The large parameter spectral family for pseudodifferential operators $A\in\Psi_{\infty}^{m,0}$, $m>0$, is included in the joint standard/semiclassical algebra, namely with $z=\lambda/h^{m}\in\Psi_{\infty,\hbar}^{0,m,m}$,

[TABLE]

Thus, completely analogous arguments as for the Laplacian apply (with $2$ replaced by $m$). In particular, if $A$ has a real elliptic symbol and $\lambda$ is in a compact set disjoint from $\mathbb{R}$ (and either $h$ is small or $A$ is self-adjoint), we have

[TABLE]

All of these arguments go through in the bounded geometry setting. They also go through in other operator algebras like the scattering algebra. ∎