Asymptotic expantion of covariant symbol on the complex unit sphere

TL;DR

This paper derives an asymptotic expansion for the Berezin transform on the complex unit sphere using a complete family of functions, and proves an Egorov-type theorem for covariant symbols of pseudo-differential operators.

Contribution

It introduces a new asymptotic expansion for the Berezin transform on the sphere and establishes an Egorov-type theorem for covariant symbols in this context.

Findings

Asymptotic expansion for Berezin transform derived

Egorov-type theorem proved for covariant symbols

Analysis involves asymptotic behavior of functions in a complete family

Abstract

Starting from a complete family (not defined by the reproducing kernel) for the unit sphere $\mathbf S^n$ in the complex $n$-space $\mathbb C^n$, we obtain an asymptotic expansion for the associated Berezin transform. The proof involves the computation of the asymptotic behaviour of functions in the complete family. Furthermore, we prove an Egorov-type theorem for the covariant symbol related to a pseudo-differential operator on $L^2(\mathbf S^n)$.

Equations

$$f(\mathbf{z})={\int }_{{\mathbb{C}}^n}f(\mathbf{w})k_{\mathbf{w}}^{\hslash }(\mathbf{z})\mathrm{d}{v}_n^{\hslash }(\mathbf{w})=⟨f,k_{\mathbf{z}}^{\hslash }⟩.$$

$$\mathbf{B}^{\hbar}(f)(\mathbf{z})=\frac{\langle fk_{\mathbf{z}}^{\hbar},k_{\mathbf{z}}^{\hbar}\rangle}{\langle k_{\mathbf{z}}^{\hbar},k_{\mathbf{z}}^{\hbar}\rangle}=\big{(}k_{\mathbf{z}}(\mathbf{z})\big{)}^{-1}\int_{\mathbb{C}^{n}}f(\mathbf{w})|k_{\mathbf{z}}(\mathbf{w})|^{2}\mathrm{d}v_{n}^{\hbar}(\mathbf{w}).$$

$${\mathbf{B}}^{\hslash }(f)(\mathbf{z})=f(\mathbf{z})+\hslash \frac{1}{1!4}{\partial }_{\mathbf{w}\overline{\mathbf{w}}}+{\hslash }^2\frac{1}{2!4^2}{\partial }_{\mathbf{w}\overline{\mathbf{w}}}+\cdots ,$$

$${\mathbf{B}}^{\hslash }(f)(\mathbf{z})={\int }_{\Omega }f(\mathbf{w})\frac{|K^{\hslash }(\mathbf{z},\mathbf{w}){|}^2}{K^{\hslash }(\mathbf{z},\mathbf{z})}{e}^{-\Phi (\mathbf{w})/\hslash }\mathrm{d}\mu (\mathbf{w}).$$

$$K^{\hslash }(\mathbf{z},\mathbf{z})$$

$${\mathbf{B}}^{\hslash }(f)$$

$$\mathbf{B}(A)(\alpha )=\frac{(A\mathbf{K}(\cdot ,\alpha ),\mathbf{K}(\cdot ,\alpha ))}{(\mathbf{K}(\cdot ,\alpha ),\mathbf{K}(\cdot ,\alpha ))},\alpha \in M,$$

$$⟨\varphi ,\psi {⟩}_{{\mathbf{S}}^n}={\int }_{{\mathbf{S}}^n}\varphi (\mathbf{x})\overline{\psi (\mathbf{x})}\mathrm{d}{\mathbf{S}}^n(\mathbf{x}),\varphi ,\psi \in L^2({\mathbf{S}}^n).$$

$$\mathbf{z}\cdot \mathbf{w}=\sum _{s=1}^k{z}_s{\overline{w}}_s,|\mathbf{z}|=\sqrt{\mathbf{z}\cdot \mathbf{z}},|\mathbf{\ell }|=\sum _{s=1}^k{\ell }_s,\mathbf{\ell }!=\prod _{s=1}^k{\ell }_s!,{\mathbf{z}}^{\mathbf{\ell }}=\prod _{s=1}^k{z}_s^{{\ell }_s}.$$

$${\mathbf{K}}_{n,p}^{\hslash }(\mathbf{x},\mathbf{z}):=\sum _{\ell =0}^{\infty }\frac{c_{\ell ,p}}{\ell !}{\left(\frac{\mathbf{x}\cdot \mathbf{z}}{\hslash }\right)}^{\ell },(c_{\ell ,p}{)}^2=\frac{(n{)}_{\ell }}{(n+p{)}_{\ell }},p>-n,\mathbf{x}\in {\mathbf{S}}^n$$

$$⟨\Phi ,\Psi {⟩}_{{\mathbf{S}}^n}$$

$$\mathrm{d}{m}_{n,p}^{\hslash }(\mathbf{z})=\frac{1}{\Gamma (n+p)}\frac{2}{(\pi {\hslash }^2{)}^n}{\left(\frac{|\mathbf{z}|}{\hslash }\right)}^p{\mathrm{K}}_p\left(2\frac{|\mathbf{z}|}{\hslash }\right)\mathrm{d}\mathbf{z}\mathrm{d}\overline{\mathbf{z}},p>-n$$

$${\mathbf{U}}_{n,p}\Psi (\mathbf{z})=⟨\Psi ,{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}){⟩}_{{\mathbf{S}}^n}$$

$$(f,g{)}_p={\int }_{\mathbf{z}\in {\mathbb{C}}^n}f(\mathbf{z})\overline{g(\mathbf{z})}\mathrm{d}{m}_{n,p}^{\hslash }(\mathbf{z}),f,g\in L^2({\mathbb{C}}^n,\mathrm{d}{m}_{n,p}^{\hslash }).$$

$${\mathbf{U}}_{n,p}\left({\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{w})\right)(\mathbf{z})={\mathbf{T}}_{n,p}(\mathbf{z},\mathbf{w}),$$

$${\mathbf{T}}_{n,p}(\mathbf{z},\mathbf{w})=\Gamma (n+p){\left(\frac{\mathbf{z}\cdot \mathbf{w}}{{\hslash }^2}\right)}^{\frac{1}{2}(-p-n+1)}{\mathrm{I}}_{n+p-1}\left(\frac{2\sqrt{\mathbf{z}\cdot \mathbf{w}}}{\hslash }\right),$$

$${\mathfrak{B}}_{\hslash ,p}(A)(\mathbf{z})=\frac{⟨A{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}),{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}){⟩}_{{\mathbf{S}}^n}}{⟨{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}),{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}){⟩}_{{\mathbf{S}}^n}},\mathbf{z}\in {\mathbb{C}}^n.$$

$${\mathfrak{B}}_{\hslash ,p}(A)(\mathbf{w},\mathbf{z}):=\frac{⟨A{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}),{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{w}){⟩}_{{\mathbf{S}}^n}}{⟨{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z}),{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{w}){⟩}_{{\mathbf{S}}^n}}.$$

$${\mathcal{T}}_{\Phi }(\psi )(\mathbf{x})=\mathbf{P}\left(\Phi \psi \right)(\mathbf{x}),\mathbf{x}\in {\mathbf{S}}^n$$

$${\mathbf{B}}_p^{\hslash }(\Phi )={\mathfrak{B}}_{\hslash ,p}({\mathcal{T}}_{\Phi })$$

$${\mathfrak{B}}_{\hslash ,p}(A)(\mathbf{w},\mathbf{z})={\mathfrak{B}}_{\hslash ,p}({\mathrm{T}}_{U^{-1}}A{\mathrm{T}}_U)(U^{-1}\mathbf{w},U^{-1}\mathbf{z}),$$

$${\mathrm{I}}_{\vartheta }(\omega )=\frac{{\mathrm{e}}^{\omega }}{\sqrt{2\pi \omega }}\sum _{k=0}^{\infty }\frac{(-1{)}^k}{(2\omega {)}^k}\frac{\Gamma (\vartheta +k+\frac{1}{2})}{k!\Gamma (\vartheta -k+\frac{1}{2})},|\mathrm{Arg}(\omega )|<\frac{\pi }{2}.$$

$${⟨{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{w}),{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z})⟩}_{{\mathbf{S}}^n}$$

$$\displaystyle=\frac{\Gamma(n+p)}{2\sqrt{\pi}}\left(\frac{\mathbf{z}\cdot\mathbf{w}}{\hbar^{2}}\right)^{\frac{1}{2}(-p-n+\frac{1}{2})}\mathrm{e}^{\frac{2}{\hbar}\sqrt{\mathbf{z}\cdot\mathbf{w}}}\biggl{[}1-$$

$$\frac{(n+p-\frac{3}{2})(n+p-\frac{1}{2})}{4\sqrt{\mathbf{z}\cdot \mathbf{w}}}\hslash +\mathrm{O}({\hslash }^2)].$$

$$||{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z})|{|}_{{\mathbf{S}}^n}^2$$

$$\displaystyle\hskip 85.35826pt\frac{(n+p-\frac{3}{2})(n+p-\frac{1}{2})}{4|\mathbf{z}|}\hbar+\mathrm{O}(\hbar^{2})\biggl{]}.$$

$${⟨{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{w}),{\mathbf{K}}_{n,p}^{\hslash }(\cdot ,\mathbf{z})⟩}_{{}_{{\mathbf{S}}^n}}=$$

$$\left[{\mathrm{e}}^{\frac{2}{\hslash }\sqrt{\mathbf{z}\cdot \mathbf{w}}}+{\mathrm{e}}^{-\frac{2}{\hslash }\sqrt{\mathbf{z}\cdot \mathbf{w}}}{\mathrm{e}}^{\pi ı(n+p-\frac{1}{2})}\right][1+\mathrm{O}(\hslash )].$$

$$g_a(z)=\sum _{\ell =0}^{\infty }\frac{\sqrt{a\ell +1}}{\ell !}{z}^{\ell }.$$

$$W_{\mathbf{z}}=\left\{\mathbf{x}\in {\mathbf{S}}^n|C\frac{\Re (\mathbf{x}\cdot \mathbf{z})}{|\mathbf{z}|}\ge 1\right\}\text{mbox}and{V}_{\mathbf{z}}={\mathbf{S}}^n-W_{\mathbf{z}}.$$

$$\frac{{\mathrm{d}}^s{g}_a(z)}{\mathrm{d}{z}^s}=\sqrt{a}{z}^{1/2}\exp (z)\left[1+\frac{a_{1,s}}{z}+\frac{a_{2,s}}{z^2}+...+\frac{a_{N,s}}{z^N}+\mathrm{O}(z^{-(N+1)})\right]$$

$$g_a(z)$$

Full text

1. Introduction and summary

Let $\mathcal{B}_{\mathbb{C}^{n}}$ be the Bargmann spaces of all entire functions on $\mathbb{C}^{n}$ square integrable with respect to the Gaussian measure $\mathrm{d}v_{n}^{\hbar}(\mathbf{z})=(\pi\hbar)^{-n}e^{-|\mathbf{z}|^{2}/\hbar}\mathrm{d}\mathbf{z}\mathrm{d}\overline{\mathbf{z}}$, $\hbar>0$, with $\mathbf{z}=(z_{1},\ldots,z_{n})$, $|\mathbf{z}|^{2}=|z_{1}|^{2}+\ldots+|z_{n}|^{2}$ and $\mathrm{d}\mathbf{z}\mathrm{d}\overline{\mathbf{z}}$ Lebesgue measure on $\mathbb{C}^{n}$.

It is know that the Bargmann space $\mathcal{B}_{\mathbb{C}^{n}}$ enjoys the property of having a reproducing kernel $k_{\mathbf{w}}^{\hbar}(\mathbf{z})=e^{\mathbf{z}\cdot\mathbf{w}/\hbar}$, where $\mathbf{z}\cdot\mathbf{w}=z_{1}\overline{w}_{1}+\cdots+z_{n}\overline{w}_{n}$ denotes the usual inner product in $\mathbb{C}^{n}$, and for all $f\in\mathcal{B}_{\mathbb{C}^{n}}$ the following equation holds:

[TABLE]

Recall that for $f\in L^{\infty}(\mathbb{C}^{n})$, the Berezin transform $\mathbf{B}^{\hbar}(f)$ of $f$ is the function on $\mathbb{C}^{n}$ defined by

[TABLE]

Explicitly, $\mathbf{B}^{\hbar}(f)(\mathbf{z})=(\pi\hbar)^{-n}\int f(\mathbf{w})e^{-|\mathbf{z}-\mathbf{w}|^{2}/\hbar}\mathrm{d}\mathbf{w}\mathrm{d}\overline{\mathbf{w}}$ which is just the standard formula for the solution at time $t=\hbar/4$ of the heat equation on $\mathbb{C}^{n}=\mathbb{R}^{2n}$ with initial data $f$. It follows that for $f\in L^{\infty}(\mathbb{C}^{n})$ a smooth function in a neighbourhood of $\mathbf{z}$, its Berezin transform has the following asymptotic expansion when $h$ goes to zero

[TABLE]

with $\partial_{\mathbf{w}\overline{\mathbf{w}}}=\sum_{j=1}^{n}\partial^{2}/\partial w_{j}\partial\overline{w}_{j}$ denoting the Laplace operator.

It turns out that this kind of situation prevails in much greater generality. Namely, consider a domain $\Omega\in\mathbb{C}^{n}$ equiped with a Kähler form $\omega$, i.e. there must exist a strictly plurisubharmonic real-valued smooth function $\Phi$ such that $\omega=\partial\overline{\partial}\Phi$. For any $\hbar>0$, we take the weighted Bergman spaces $\mathcal{H}L^{2}(\Omega,e^{-\Phi/\hbar}\mathrm{d}\mu)$ of all holomorphic functions in $L^{2}(\Omega,e^{-\Phi/\hbar}\mathrm{d}\mu)$, where $\mathrm{d}\mu(\mathbf{z})=\mathrm{det}([g_{i\bar{j}}])\ \mathrm{d}\mathbf{z}\mathrm{d}\overline{\mathbf{z}}$ with $g_{i\bar{j}}=\partial^{2}\Phi/\partial z_{i}\partial\overline{z}_{j}$. These spaces enjoys the property of having a reproducing kernel $K^{\hbar}(\mathbf{z},\mathbf{w})$. The corresponding Berezin transform is given by

[TABLE]

Furthermore, for $f\in L^{\infty}(\Omega)$ one has the Toeplitz operator ${\mathcal{T}}_{f}$ with symbol $f$, namely, the operator on $\mathcal{H}L^{2}(\Omega,e^{-\Phi/\hbar}\mathrm{d}\mu)$ defined by ${\mathcal{T}}_{f}(\psi)=P_{h}(f\psi)$ , where $P_{h}:L^{2}(\Omega,e^{-\Psi/\hbar}\mathrm{d}\mu)\to\mathcal{H}L^{2}(\Omega,e^{-\Phi/\hbar}\mathrm{d}\mu)$ is the orthogonal projection.

Now, assume that $\Omega$ is a bounded symmetric domain and $e^{\Psi}$ is the Bergman kernel of $\Omega$; or that $\Omega$ is smoothly bounded and strictly pseudoconvex, and $e^{-\Psi}$ is a defining function for $\Omega^{2}$; or that $\Omega=\mathbb{C}^{n}$ and $\Psi(\mathbf{z})=|\mathbf{z}|^{2}$. Then as $\hbar\searrow 0$, there are asymptotic expansions [1, 2, 3, 4]

[TABLE]

for some functions $b_{\ell}\in C^{\infty}(\Omega)$, with $b_{0}=1$; some differential operators $Q_{\ell}$, with $Q_{0}$ the identity operator and $Q_{1}$ the Laplace–Beltrami operator with respect to the metric $g_{i\bar{j}}$.

On the other hand, in Ref. [1] Berezin defined the covariant symbol of a bounded linear operator $A$ on a Hilbert space $H$ endowed with the inner product $(\cdot,\cdot)$ as the complex-valued function $\mathbf{B}(A)$ on a set $M$ by

[TABLE]

where the family $\{\mathbf{K}(\cdot,\alpha)\in H\;|\;\alpha\in M\}$ forms a complete system for $H$.

Under this definition, the Berezin transform makes sense not only considering the complete family $\{\mathbf{K}(\cdot,\alpha)\}$ as the reproducing kernel with one freezing variable, as in the weighted Bergman spaces, but for any complete family. Hence, it is of interest to investigate if the spaces that have a complete family (not defined by the reproducing kernel) satisfy properties similar to (1) and (2).

One such candidate, namely, the Hilbert space $\mathcal{O}$ of all functions in $L^{2}(\mathbf{S}^{n})$ whose Poisson extension into the interior of $\mathbf{S}^{n}$ is holomorphic, where $\mathbf{S}^{n}=\{\mathbf{x}\in\mathbb{C}^{n}|\;|x_{1}|^{2}+\cdots+|x_{n}|^{2}=1\}$ and $L^{2}(\mathbf{S}^{n})$ denotes the Hilbert space of square integrable functions with respect to the normalized surface measure $\mathrm{d}\mathbf{S}^{n}(\mathbf{x})$ on $\mathbf{S}^{n}$ and endowed with the usual inner product

[TABLE]

In Ref. [5] it is defined a complete family $\mathcal{K}_{p}=\{{{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})|\mathbf{z}\in\mathbb{C}^{n}\}$ for $\mathcal{O}$ and the associated covariant symbol. The functions in $\mathcal{K}_{p}$ are not obtained by the reproducing kernel and its definition is in terms of a suitable power series of the inner product $\mathbf{x}\cdot\mathbf{z}/\hbar$ which is an infinite series that is not in a closed form like an exponential function.

The aim of the present paper is to show an analogue of the asymptotic expansions (1) and (2) for the functions ${{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})$, $\mathbf{z}\in\mathbb{C}^{n}$, and the associated Berezin transform, respectively.

The paper is organized as follows. Section 2 is devoted to give a brief description and some properties of the family $\mathcal{K}_{p}$ and the associated covariant symbol.

Analogues of the formula (1) for the asymptotic behaviour as $h\searrow 0$ of a function in $\mathcal{K}_{p}$ is established in Section 3. This formula is obtained following the work of Thomas and Wassell [6]. Moreover, to give rigorous proofs of subsequent theorems, we estimate the derivative of any order of the series defining ${{\bf K}_{n,p}^{\hbar}}(\mathbf{x},\mathbf{z})$.

Using the asymptotic expansion of functions in $\mathcal{K}_{p}$ and the stationary phase method, in Section 4 we get the asymptotic behaviour of the Berezin transform.

Moreover, since the covariant symbol is not only defined for Toeplitz operators but for any bounded linear operator, it is natural to ask if there exists an analogue of the asymptotic expansion obtained in Section 4 for the covariant symbol of more general operators than Toeplitz operators. For this reason, in Section 5 we prove an Egorov-type theorem which relates the principal symbol of a given semiclassical pseudo-differential operator of order zero acting on $L^{2}(\mathbf{S}^{n})$ (see Appendix A where we give a brief description of what we mean by semiclassical pseudo-differential operators on a manifold) with its covariant symbol in the semiclassical limit $\hbar\searrow 0$.

Throughout the paper, we will use the following basic notation. For every $\mathbf{z},\mathbf{w}\in\mathbb{C}^{k}$, $\mathbf{z}=(z_{1},\ldots,z_{k})$, $\mathbf{w}=(w_{1},\ldots,w_{k})$, and for every multi-index $\boldsymbol{\ell}=(\ell_{1},\ldots,\ell_{k})\in\mathbb{Z}_{+}^{k}$ of length $k$, where $\mathbb{Z}_{+}$ is the set of nonnegative integers, let

[TABLE]

Given $\omega\in\mathbb{C}$, let us denote its real and imaginary parts by $\Re(\omega)$ and $\Im(\omega)$ respectively.

Whenever convenient, we will abbreviate $\partial/\partial v_{j},\partial/\partial\overline{v}_{j}$, etc., to $\partial_{v_{j}},\partial_{\overline{v}_{j}}$, etc., respectively, $\partial_{v_{1}}\partial_{v_{2}}\ldots\partial_{v_{k}}$ to $\partial_{v_{1}v_{2}\cdots v_{k}}$, and $\partial_{\mathbf{z}}^{\mathbf{\ell}}=\partial_{z_{1}}^{\ell_{1}}\cdots\partial_{z_{k}}^{\ell_{k}}$ with $\partial_{z_{j}}^{\ell_{j}}=\underbrace{\partial_{z_{j}}\cdot\ldots\cdot{\partial_{z_{j}}}}_{{\ell_{j}}}$.

2. The covariant symbol on $\mathcal{O}$

In this section we introduce the covariant symbol of a bounded linear operator with domain in $\mathcal{O}$ and some of properties it satisfies. In order to define this symbol, let us start by defining the functions ${{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})$, $\mathbf{z}\in\mathbb{C}^{n}$, that form a complete family $\mathcal{K}_{p}$ in $\mathcal{O}$ that is not obtained by the reproducing kernel. See Ref. [5] for details.

2.1. The functions ${{\bf K}_{n,p}^{\hbar}}$

Follows Ref. [5], let us consider the set of functions $\mathcal{K}_{p}=\{{{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})|\mathbf{z}\in\mathbb{C}^{n}\}\subset\mathcal{O}$, with

[TABLE]

where $(a)_{\ell}$ stands for the Pochhammer symbol (raising factorial), $(a)_{\ell}=a(a+1)\cdots(a+\ell-1)=\frac{\Gamma(a+\ell)}{\Gamma(a)}$.

In Ref. [5] it is shown that the family $\mathcal{K}_{p}$ satisfies the conditions for defining a Berezin symbolic calculus on $\mathcal{O}$, i.e. the family $\mathcal{K}_{p}$ satisfies the following two properties:

(I) The family $\mathcal{K}_{p}$ forms a complete system for $\mathcal{O}$: for all $\Phi,\Psi\in\mathcal{O}$, Parseval’s identity is valid

[TABLE]

where

[TABLE]

with $\mathrm{d}\mathbf{z}\mathrm{d}\overline{\mathbf{z}}$ denoting Lebesgue measure on $\mathbb{C}^{n}$ and $\Gamma$, $\mathrm{K}_{\nu}$ denoting the Gamma and MacDonald-Bessel function of order $\nu$, respectively (see Sections 8.4 and 8.5 of Ref. [7] for definition and expressions for these special functions).

(II) The map ${\mathbf{U}_{n,p}}:L^{2}(\mathbf{S}^{n})\to L^{2}(\mathbb{C}^{n},\mathrm{d}m_{n,p}^{\hbar})$ defined by

[TABLE]

is an embedding, where $L^{2}(\mathbb{C}^{n},\mathrm{d}m_{n,p}^{\hbar})$ denotes the Hilbert space of square integrable functions on $\mathbb{C}^{n}$ with respect to the measure $\mathrm{d}m_{n,p}^{\hbar}$ and endowed with the inner product

[TABLE]

In fact, in Ref. [5] it is proved that the operator ${\mathbf{U}_{n,p}}$ is unitary onto the Hilbert space $\mathcal{E}_{n,p}$ of entire functions $f$ defined on $\mathbb{C}^{n}$ such that $||f||_{p}=\sqrt{(f,f)_{p}}$ is finite.

Moreover, for any $\mathbf{z},\mathbf{w}\in\mathbb{C}^{n}$,

[TABLE]

where

[TABLE]

with $\mathrm{I}_{k}$ denoting the modified Bessel function of the first kind of order $k$ (see Secs. 8.4 and 8.5 of Ref. [7] for definition and expressions for this special function).

2.2. The associated covariant symbol to the family $\mathcal{K}_{p}$

Since the functions in $\mathcal{K}_{p}$ satisfy the properties (I) and (II), the Berezin’s theory allow us to consider the following

Definition 2.1**.**

The covariant symbol ${{\mathfrak{B}}_{\hbar,p}}(A)$ of an operator $A$ with domain in $\mathcal{O}$ is defined as

[TABLE]

Note that this definition makes sense since the denominator is positive by the relation $||{{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})||_{\mathbf{S}^{n}}^{2}={\bf T}_{n,p}(\mathbf{z},\mathbf{z})$ (see Eqs. (6) and (7)) and Eq. (8). Moreover, since the functions in $\mathcal{K}_{p}$ are continuous, if $A:\mathcal{O}\to\mathcal{O}$ is an bounded operator, its covariant symbol can be extended uniquely to a function defined on a neighbourhood of the diagonal in $\mathbb{C}^{n}\times\mathbb{C}^{n}$ in such a way that it is holomorphic in the first factor and anti-holomorphic in the second. In fact, such an extension is given explicitly by

[TABLE]

On the other hand, to every $\Phi\in C^{\infty}(\mathbf{S}^{n})$, with $C^{\infty}(\mathbf{S}^{n})$ denoting the algebra of complex-valued $C^{\infty}$ functions on $\mathbf{S}^{n}$, is associated a linear operator ${\mathcal{T}}_{\Phi}$ -the Toeplitz operator with symbol $\Phi$- that is defined for $\psi\in\mathcal{O}$ by

[TABLE]

where ${{\mathbf{P}}}:L^{2}(\mathbf{S}^{n})\to\mathcal{O}$ is the orthogonal projection.

Starting from $\Phi\in C^{\infty}(\mathbf{S}^{n})$, we can assign to it its Toeplitz operator ${\mathcal{T}}_{\Phi}$ and then assign to ${\mathcal{T}}_{\Phi}$ the covariant symbol ${{\mathfrak{B}}_{\hbar,p}}({\mathcal{T}}_{\Phi})$. It is an element of $C^{\infty}(\mathbb{C}^{n})$. Altogether we obtain a map $\Phi\mapsto{{\mathbf{B}}^{\hbar}_{p}}(\Phi):={{\mathfrak{B}}_{\hbar,p}}({\mathcal{T}}_{\Phi})$.

Definition 2.2**.**

The map ${{\mathbf{B}}^{\hbar}_{p}}:C^{\infty}(\mathbf{S}^{n})\to C^{\infty}(\mathbb{C}^{n})$ defined for $\Phi\in C^{\infty}(\mathbf{S}^{n})$ by

[TABLE]

is called Berezin transform.

We end this section by showing some properties of the extended covariant symbol and the Berezin transform that we will use to obtain the asymptotic expansion of the covariant symbol.

Proposition 2.3**.**

Let $U\in\mathrm{SU}(n)$ (the group of $n\times n$ unitary matrices with unit determinant) and $\mathrm{T}_{U}:L^{2}(\mathbf{S}^{n})\to L^{2}(\mathbf{S}^{n})$ be the operator defined by $\mathrm{T}_{U}\Psi(\mathbf{x})=\Psi(U^{-1}\mathbf{x})$ with $\Psi\in L^{2}(\mathbf{S}^{n})$. Let $A$ be a bounded linear operator with domain in $L^{2}(\mathbf{S}^{n})$. Then:

1. (1)

For $\mathbf{w},\mathbf{z}\in\mathbb{C}^{n}$

[TABLE]

In particular, ${{\mathfrak{B}}_{\hbar,p}}(A)=\mathrm{T}_{U}{{\mathfrak{B}}_{\hbar,p}}(\mathrm{T}_{U^{-1}}A\mathrm{T}_{U})$ if $\mathbf{z}=\mathbf{w}$. 2. (2)

The Berezin transform ${{\mathbf{B}}^{\hbar}_{p}}$ is invariant under the orthogonal transformations $\mathrm{T}_{U}$, i.e ${{\mathbf{B}}^{\hbar}_{p}}\circ\mathrm{T}_{U}=\mathrm{T}_{U}\circ{{\mathbf{B}}^{\hbar}_{p}}$.

Proof.

Since the inner product in $\mathbb{C}^{n}$ is $\mathrm{SU}(n)$-invariant, we have ${{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})=\mathrm{T}_{U}{{\bf K}_{n,p}^{\hbar}}(\cdot,U^{-1}\mathbf{z})$. From the $\mathrm{SU}(n)$-invariance of $\mathrm{d}\mathbf{S}^{n}$ and definitions of the extended covariant symbol and the Berezin transform (see Eq. (10) and Definition 2.2) we conclude the proof of Proposition 2.3. ∎

3. Semiclassical properties of the functions in $\mathcal{K}_{p}$

Consider the set of functions $\mathcal{K}_{p}$ defined in subsection 2.1. The main goal of this section is to show semiclassical properties of the functions in $\mathcal{K}_{p}$, which in turn will allow us to obtain asymptotic expansions of the Berezin transform and the covariant symbol.

3.1. Estimate of the inner product of functions in the complete family $\mathcal{K}_{p}$

From Eqs. (6), (7), (8) and the fact that the modified Bessel function $\mathrm{I}_{\vartheta}$, $\vartheta\in\mathbb{R}$, has the following asymptotic expression when $|\omega|\to\infty$ (see formula 8.451-5 of Ref. [7])

[TABLE]

we can obtain the asymptotic expansion for the inner product of two functions in $\mathcal{K}_{p}$:

Proposition 3.1**.**

Let $n\geq 1$, $p>-n$ and $\mathbf{z},\mathbf{w}\in\mathbb{C}^{n}$. Assume $\mathbf{z}\cdot\mathbf{w}\neq 0$ and $|\mathrm{Arg}(\mathbf{z}\cdot\mathbf{w})|<\pi$, then for $\hbar\to 0$

[TABLE]

In particular, for $\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$:

[TABLE]

Remark 1**.**

We are mainly interested in using Proposition 3.1 for the cases $\mathbf{z}=\mathbf{w}$ (and then $\mathrm{Arg}(\mathbf{z}\cdot\mathbf{w})=0$) in this paper. The case when $|\mathrm{Arg}(\mathbf{z}\cdot\mathbf{w})|=\pi$ requires the use of an asymptotic expression valid in a different region than the one we are considering in Proposition 3.1. Thus if we take the branch of the square root function given by $\sqrt{z}=|z|^{1/2}\mathrm{exp}(\imath\mathrm{Arg}(z)/2)$ with $0<\mathrm{Arg}(z)<2\pi$ then by using formula 8.451-5 in Ref. [7] we obtain the asymptotic expression,

[TABLE]

Note that both asymptotic expressions in Eqs. (14) and (16) coincide up to an error of the order $\mathrm{O}(\hbar^{\infty})$, where $\mathrm{O}(\hbar^{\infty})$ denotes a quantity tending to zero faster than any power of $\hbar$ , in the common region where they are valid.

3.2. Asymptotic of the functions in $\mathcal{K}_{-1}$

In order to give a rigorous proof of Theorems 4.4 and 5.1 we need to estimate the derivative of any order of the series defining ${{\bf K}_{n,p}^{\hbar}}(\mathbf{x},\mathbf{z})$. The definition of our functions ${{\bf K}_{n,p}^{\hbar}}(\mathbf{x},\mathbf{z})$ in terms of an infinite series (not in a closed form like an exponential function) might look like it could be difficult to deal with them. However, from the work of Thomas and Wassell (see appendix B of Ref. [6]) we can obtain an asymptotic expansion of ${{\bf K}_{n,-1}^{\hbar}}(\mathbf{x},\mathbf{z})$, i.e. $n\geq 2$ and $p=-1$.

First, based on the definition of ${{\bf K}_{n,-1}^{\hbar}}$ (see Eq. (4)) let us define, for $a\in\mathbb{R}-\{0\}$, the function $g_{a}:\mathbb{C}\to\mathbb{C}$ by

[TABLE]

Note that for $\mathbf{z}\in\mathbb{C}^{n}$, ${{\bf K}_{n,-1}^{\hbar}}(\mathbf{x},\mathbf{z})=g_{\frac{1}{n-1}}\left(\frac{\mathbf{x}\cdot\mathbf{z}}{\hbar}\right)$, $\mathbf{x}\in\mathbf{S}^{n}$. In this subsection we obtain the main asymptotic term for the $s$th derivative of the function $g_{a}$ as a function of $z$ for $\Re(z)\to+\infty$ and $|\Im(z)|\leq C\Re(z)$ with $C$ a positive constant, which in turn will allow us to obtain asymptotics of the $s$th derivative of the function ${{\bf K}_{n,-1}^{\hbar}}(\cdot,\mathbf{z})$ for $\hbar$ small and $\mathbf{x}$ in either of the following regions on $\mathbf{S}^{n}$

[TABLE]

The proof of lemma below follows the work of Thomas and Wassel (see appendix B of Ref. [6] and lemma 10.1 of Ref. [8] for more details).

Lemma 3.2**.**

Let $z\in\mathbb{C}$, $s$ be any non-negative integer number and $a\in\mathbb{R}$ with $0<a\leq 2$. Suppose $\Re(z)>0$ and $|\Im(z)|\leq C\Re(z)$ with $C$ a positive constant and $\Re(z)\to+\infty$. Then the $s\mathrm{th}$ derivative of $g_{a}$ has the following asymptotic expansion:

[TABLE]

with $a_{1,s},a_{2,s},...,a_{N,s}$ some constants.

Proof.

In the proof of this Lemma we denote by $\mathfrak{K}^{(\ell)}(z)$, $\ell=0,1,\ldots$, the $\ell$st derivative of a given function $\mathfrak{K}:\mathbb{C}\to\mathbb{C}$ evaluated in $z\in\mathbb{C}$, i.e. $\mathfrak{K}^{(\ell)}(z)=\frac{\mathrm{d}^{\ell}\mathfrak{K}(z)}{\mathrm{d}z^{\ell}}$.

Using the fact $\frac{1}{\sqrt{a\ell+1}}=\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-(a\ell+1)t^{2}}\mathrm{d}t$ and the Taylor series for the exponential function we obtain

[TABLE]

Let us consider the change of variables $e^{-at^{2}}=1-w^{2}$ with $w\in[0,1]$. From Eq. (20)

[TABLE]

where

[TABLE]

Let us write the integral in Eq. (21) as an integral over the region $0\leq t<1/2$ plus an integral over the region $1/2\leq t\leq 1$ and let us call them $\mathrm{J}_{a}(\mathbf{z})$ and $\mathrm{I}_{a}(\mathbf{z})$ respectively.

Now we claim that for any $k\in\mathbb{Z}_{+}$, $\mathrm{I}_{a}^{(k)}(z)$ is $\mathrm{O}(z^{-\infty})$, where $\mathrm{O}(\hbar^{\infty})$ denotes a quantity tending to zero faster than any power of $\hbar$ . In order to estimate $\mathrm{I}_{a}^{(k)}(\mathbf{z})$, first write

[TABLE]

Since $0<a\leq 2$, then the numerator $(1-w^{2})^{\frac{1}{a}-\frac{1}{2}}$ in the last expression is a bounded function in the interval $[1/2,1]$. Notice also that the function $-1/\ln(1-w^{2})$ is bounded from above by $-1/\ln(3/4)$ in the same interval. Since the integral $\int_{1/2}^{1}1/\sqrt{1-w^{2}}\mathrm{d}w$ is finite and the absolute value of the derivative (respect to $z$) of any order of the integrand in Eq. (21) is bounded by the function $(1-w^{2})^{1/2}$, then by the dominated convergence theorem we have that for any natural number $r$

[TABLE]

with $M$ a constant number independent of $z$. Since $|\Im(z)|\leq C\Re(z)$, then $|z|\leq C_{1}\Re(z)$ for some constant $C_{1}$. Therefore $\left|z^{r}\mathrm{I}_{a}^{(k)}(z)\right|\leq M|\Re(z)|^{r+1}e^{-\Re(z)/4}$ which is a bounded function when $\Re(z)\to\infty$. Thus we have $\mathrm{I}_{a}^{(k)}(z)=\mathrm{O}(z^{-r})$ for all $r\in\mathbb{N}$, i.e.

[TABLE]

Let us now study the term $\mathrm{J}_{a}(z)$ and its $k$st derivative. First note that the function $\mathsf{m}$ has an even and $C^{\infty}$ extension to the interval $(-1/2,1/2)$ which implies that $\mathsf{m}$ has an asymptotic expansion in the interval $[0,1/2)$ in terms of even powers of the variable $w$ (see Ref. [8]). Namely, there exist numbers $b_{2j}$, $j=0,1,\ldots$ such that for $w\in[0,1/2)$

[TABLE]

Let us write $\mathrm{J}_{a}^{(k)}(z)=\mathrm{G}_{a}^{(k)}(z)+\mathrm{H}_{a}^{(k)}(z)$ with

[TABLE]

Let us study $\mathrm{G}_{a}^{(k)}$. Let $N$ be a natural number and define

[TABLE]

Notice that $\mathrm{G}_{a}^{(k)}(z)=\mathrm{G}_{a,1}^{(k)}(z)+\mathrm{G}_{a,2}^{(k)}(z)$. We claim that $\mathrm{G}_{a,2}^{(k)}(z)=\mathrm{O}(z^{-N-k-\frac{3}{2}})$. Since for any $\ell\in\mathbb{Z}_{+}$ the absolute value of the $\ell$st derivative of the integrand in Eq. (25) is bounded by the function $w^{2N+2+2\ell}$, which in turn is integrable, then by the dominated convergence theorem we have

[TABLE]

with $M$ a constant only dependents on $N$ and where to obtain the last inequality we have considered the change of variables $\eta=\sqrt{\Re(z)}w$ and that $|\Im(z)|\leq C\Re(z)$.

On the other hand, let us write $\mathrm{G}_{a,1}^{(k)}(z)$ as follows

[TABLE]

The second term of $\mathrm{G}_{a,1}^{(k)}(z)$ is $\mathrm{O}(z^{-\infty})$ because for any natural number $r$ we have

[TABLE]

with $M$, $C_{1}$ constants independent of $z$ and $C_{1}$ only depends on $N$, $r$ and $k$. To obtain the inequality (27) we have considered the changes of variables $v=w-1/2$ and $t=\sqrt{\mathrm{Re}(z)}v$. The inequality (28) is a consequence that the integrals $\int_{0}^{\infty}e^{-t^{2}}(t+\sqrt{\mathrm{Re}(z)}/2)^{2(k+j)}\mathrm{d}t$ are polynomials in the variable $\sqrt{\Re(z)}$.

On the other hand, since $\int_{0}^{\infty}e^{-zw^{2}}w^{2(k+j)}\mathrm{d}w=\frac{1}{z^{k+j}\sqrt{z}}\int_{0}^{\infty}e^{-t^{2}}t^{2(k+j)}\mathrm{d}t$, then from Eq. (26)

[TABLE]

with $\displaystyle d_{r}=\int_{0}^{\infty}e^{-t^{2}}t^{2r}\mathrm{d}t$. In particular $\displaystyle d_{0}=\frac{\sqrt{\pi}}{2}$.

We conclude that

[TABLE]

Let us now study $\mathrm{H}_{a}^{(k)}$, defined in Eq. (23). From the equality $\mathrm{H}_{a}(z)=az[\mathrm{G}_{a}(z)+\mathrm{G}_{a}^{(1)}(z)]$ and Eq. (30)

[TABLE]

where we are taking the convention that $kd_{k-1}=0$ when $k=0$.

Thus, from Eqs. (21), (22), (30) and (31) we obtain

[TABLE]

The asymptotic expansion of $g_{a}^{(s)}$ given in Eq. (19) follows from Eq. (32). ∎

Using Lemma 3.2 with $z=\mathbf{x}\cdot\mathbf{z}/\hbar$ we obtain the following asymptotic expansion of the $s$th derivative of $g_{a}$ evaluated at $z=\mathbf{x}\cdot\mathbf{z}/\hbar$ for $\hbar\to 0$:

Proposition 3.3**.**

Let $\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$, $C$ be a constant greater than one, $W_{\mathbf{z}}$, $V_{\mathbf{z}}$ be the regions defined in Eq. (18) and $s$ a non-negative integer number. Then for $\hbar\searrow 0$ we have

[TABLE]

with $C_{1}$ a constant and $\mu=\frac{1}{C}-1<0$. In particular, for $\mathbf{x}\in W_{\mathbf{z}}$

[TABLE]

with $a_{1}=a_{1,0}=\frac{1}{2}(n-\frac{5}{4})$.

Proof.

First we note that for $\mathbf{x}\in W_{\mathbf{z}}$, $|\Im(\mathbf{x}\cdot\mathbf{z})|\leq|\mathbf{z}|\leq C\Re(\mathbf{x}\cdot\mathbf{z})$, therefore we can use the asymptotic expression of the function $g_{a}^{(s)}$ given in Lemma 3.2. Thus, from Eq. (19)

[TABLE]

where the error term $\mathrm{E}_{s}(\mathbf{x},\hbar)$ is $\mathrm{O}(\hbar^{2})$ uniformly with respect to $\mathbf{x}\in W_{\mathbf{z}}$ because in such a region we have $1/|\mathbf{x}\cdot\mathbf{z}|\leq C/|\mathbf{z}|$.

To show Eq. (34), let us consider the integral expansion (21) to find that the $s$th derivative of $g$ evaluated at $z=\mathbf{x}\cdot\mathbf{z}/\hbar$ can be written as

[TABLE]

with $C_{1}$ a constant and $\mu=\frac{1}{C}-1<0$. ∎

4. Asymptotic expansion of the Berezin transform.

In this section we show an analogue of the asymptotic expansion (2) for the associated Berezin transform to the family $\mathcal{K}_{p}$ (see Eq. (11)).

It is possible to obtain all the asymptotic expansion of the Berezin transform ${{\mathbf{B}}^{\hbar}_{p}}(\phi)$, $\phi\in C^{\infty}(\mathbf{S}^{n})$, $p>-n$, when $\phi$ is a polynomial. For the general case, when $\phi$ is a smooth function, we consider only the case $n\geq 2$ and $p=0$ or $p=-1$.

Theorem 4.1**.**

Let ${\bf k}\in\mathbb{Z}_{+}^{n}$ be a multi-index and $\varphi_{\mathbf{k}}(\mathbf{x})=\mathbf{x}^{\mathbf{k}}$, with $\mathbf{x}\in\mathbf{S}^{n}$. Then for $\mathbf{w},\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$

[TABLE]

where $c_{\ell,p}$ was define in Eq. (4).

Proof.

From Eq. (10) and properties of the orthogonal projection

[TABLE]

From the dominated convergence theorem, Lemma A1 in Ref. [5] and Eqs. (6), (7) and (8) we conclude the proof of Theorem 4.1. ∎

Corollary 4.2**.**

Let $\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$, $\mathbf{k}\in\mathbb{Z}_{+}^{n}$ be a multi-index and $\varphi_{\mathbf{k}}$ as above. Then for $\hbar\searrow 0$

[TABLE]

Proof.

From Theorem 4.1 and the expression of the modified Bessel function $\mathrm{I}_{\nu}$, $\nu\in\mathbb{R}$, as a power series (see formula 8.445 of Ref. [7]) we have

[TABLE]

Using the asymptotic expression of the modified Bessel function $\mathrm{I}_{\nu}(\omega)$ when $|\omega|\to\infty$ (see Eq. (12)) we obtain the asymptotic expression given in Eq. (37). ∎

Theorem 4.3**.**

Let $\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$ and $\phi$ be a smooth function on $\mathbf{S}^{n}$. Then for $\hbar\searrow 0$

[TABLE]

where $\Delta_{\mathbf{x}\overline{\mathbf{x}}}=\sum_{j=1}^{n}\partial_{x_{j}\overline{x}_{j}}$ and $\mathcal{R}:=\sum_{j=1}^{n}\left(z_{j}\partial_{z_{j}}+\overline{z}_{j}\partial_{\overline{z}_{j}}\right)$ denote the Laplace operator and the radial derivative on $\mathbb{R}^{2n}\cong\mathbb{C}^{n}$, respectively.

Proof.

Let us first note that it is enough to show that (38) holds for all points $\mathbf{z}$ of the form $\mathbf{z}=r\hat{\mathbf{e}}_{1}$, with $r>0$, and $\hat{\mathbf{e}}_{1}=(1,0,\ldots,0)\in\mathbb{R}^{2n}$.

Indeed, let $U\in\mathrm{SU}(n)$ and $\mathrm{T}_{U}$ be the unitary transformation defined in Proposition 2.3. Since the Laplace operator $\Delta_{\mathbf{x}\overline{\mathbf{x}}}$ as well as the radial derivative $\mathcal{R}$ are clearly invariant under unitary transformation $\mathrm{T}_{U}$, the right hand side of Eq. (38) is likewise invariant under $\mathrm{T}_{U}$.

By (2) of Proposition 2.3, the validity of Eq. (38) for $\phi$ at $\mathbf{z}$ is therefore equivalent to its validity for $\mathrm{T}_{U^{-1}}\phi=\phi\circ U$ at $U^{-1}\mathbf{z}$.

Since any given $\mathbf{z}$ can be mapped by a suitable $U\in\mathrm{SU}(n)$ into a point of the form $r\hat{\mathbf{e}}_{1}$, with $r=|\mathbf{z}|$, it is indeed enough to prove (38) only for points $\mathbf{z}$ of the latter form.

For the rest of the proof of Theorem 4.3, we thus assume that $\mathbf{z}=r\hat{\mathbf{e}}_{1}$ with $r>0$.

Since ${{\bf K}_{n,0}^{\hbar}}(\mathbf{x},\mathbf{z})=e^{\mathbf{x}\cdot\mathbf{z}/\hbar}$ (see Eq. (4), with $p=0$). From Definition 2.2 and the norm estimate of the function ${{\bf K}_{n,0}^{\hbar}}(\cdot,\mathbf{z})$ (see Eq. (15)) we obtain

[TABLE]

Writing the coordinates $x_{j}$ of a point $\mathbf{x}\in\mathbf{S}^{n}$ in the form $x_{j}=y_{j}+\imath y_{n+j}$, with $y_{j}$ and $y_{n+j}$ real numbers. Note that the vector $\mathbf{y}=(y_{1},\ldots,y_{2n})\in S^{2n-1}=\{\mathbf{a}\in\mathbb{R}^{2n}\;|\;|\mathbf{a}|=1\}$.

In order to estimate (38), we define

[TABLE]

where $\mathrm{d}\Omega_{2n-1}$ is the normalized surface measure on $S^{2n-1}$ and f(\boldsymbol{\omega})=-2\imath r\big{[}\omega_{1}-1\big{]}.

Let us introduce spherical coordinates for the variables $(\omega_{1},\ldots,\omega_{2n})\in S^{2n-1}$:

[TABLE]

with $\theta_{1}\in(-\pi,\pi)$, $\theta_{2},\theta_{3},\cdots,\theta_{2n-1}\in(0,\pi)$.

The function $f$ appearing in the argument of the exponential function in Eq. (40) has a non-negative imaginary part and has only one critical point (as a function of the angles) $\boldsymbol{\theta}_{0}$ which contributes to the asymptotic given by $\theta_{1}=0$, $\theta_{j}=\pi/2$, $j=2,\ldots,2n-1$. In addition, since

[TABLE]

with $\delta_{ij}$ denoting the Kronecker symbol, then the Hessian matrix of $f$ evaluated at the critical point $\boldsymbol{\theta}_{0}$ is equal to $f^{\prime\prime}(\boldsymbol{\theta}_{0})=2\imath r\mathbf{I}_{2n-1}$, where $\mathbf{I}_{s}$ denotes the identity matrix of size $s$. Moreover, $\mathrm{det}(f^{\prime\prime}(\boldsymbol{\theta}_{0}))=(2\imath r)^{2n-1}$. From the stationary phase method (see Ref. [9]) we obtain that

[TABLE]

where $N$ is a normalization constant such that $\int_{\boldsymbol{\omega}\in S^{2n-1}}\mathrm{d}\Omega_{2n-1}(\boldsymbol{\omega})=1$ and

[TABLE]

with \mathfrak{g}\big{|}_{cp} denoting the evaluation at the critical point $\boldsymbol{\theta}_{0}$ of a given function $\mathfrak{g}$,

[TABLE]

and $\hat{D}$ the column vector of size $2n-1$ whose $j$ entry is $\partial_{\theta_{j}}$ (i.e. $(\hat{D})_{j}=\partial_{\theta_{j}}$).

Since $\left(-(f^{\prime\prime}(\boldsymbol{\theta}_{0}))^{-1}\right)\hat{D}\cdot\hat{D}=\frac{\imath}{2r}\sum_{j=1}^{2n-1}\partial_{\theta_{j}\theta_{j}}$ and $\partial_{\boldsymbol{\theta}}^{\mathbf{k}}\mathfrak{p}_{cp}(\boldsymbol{\theta}_{0})=0$ for any multi index $\mathbf{k}\in\mathbb{Z}_{+}^{2n-1}$ that satisfies $|\mathbf{k}|\leq 3$ or $|\mathbf{k}|$ is odd or $\theta_{j}$ is odd for some $j=1,\ldots,2n-1$, we obtain from Eq. (42) that

[TABLE]

where we have used the chain rule to obtain Eq. (45).

From Eqs. (41), (44) and (45)

[TABLE]

Thus, if we consider $\Psi(\Re(\mathbf{x}),\Im(\mathbf{x}))=\phi(\mathbf{x})$, $\mathbf{x}\in\mathbf{S}^{n}$, in Eq. (40), then from Eqs. (39), (46), the chain rule and the fact that $N=2\pi^{n}/\Gamma(n)$ we conclude the proof of this theorem. ∎

In subsection 3.2 we obtained an asymptotic expression of the functions ${{\bf K}_{n,-1}^{\hbar}}(\cdot,\mathbf{z})$, $\mathbf{z}\in\mathbb{C}^{n}$, this result will allow us to obtain an asymptotic expansion for the Berezin transform ${{\mathbf{B}}^{\hbar}_{-1}}$ similar to that obtained in Theorem 4.3.

Theorem 4.4**.**

Let $n\geq 2$, $\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$ and $\phi$ be a smooth function on $\mathbf{S}^{n}$. Then for $\hbar\searrow 0$

[TABLE]

where $\Delta_{\mathbf{x}\overline{\mathbf{x}}}=\sum_{j=1}^{n}\partial_{x_{j}\overline{x}_{j}}$ and $\mathcal{R}:=\sum_{j=1}^{n}\left(z_{j}\partial_{z_{j}}+\overline{z}_{j}\partial_{\overline{z}_{j}}\right)$ denote the Laplace operator and the radial derivative on $\mathbb{R}^{2n}\cong\mathbb{C}^{n}$, respectively.

Proof.

By the same argument that was used in the proof of Theorem 4.3, we only need to show this theorem for points $\mathbf{z}$ in the form $(r,0,\ldots,0)$, with $r>0$. For the rest of the proof of Theorem 4.4, we thus assume that $\mathbf{z}=(r,0,\ldots,0)$ with $r>0$.

From Definition 2.2

[TABLE]

Let $C$ the constant mentioned in Lemma 3.2 taken greater than one, and $W_{\mathbf{z}}$, $V_{\mathbf{z}}$ be the regions defined in Eq. (18).

The integral in Eq. (48) can be written as an integral over the region $W_{\mathbf{z}}$ plus an integral over $V_{\mathbf{z}}$ denoted by the letters $\mathbf{J}_{W}(\mathbf{z})$ and $\mathbf{J}_{V}(\mathbf{z})$ respectively.

The integral over $W_{\mathbf{z}}$ is the one that gives us the main asymptotic term and the integral over $V_{\mathbf{z}}$ is actually $\mathrm{O}(\hbar^{\infty})$.

The assertion $\mathbf{J}_{V}(\mathbf{z})=\mathrm{O}(\hbar^{\infty})$ follows from the following inequality

[TABLE]

which in turn is a consequence from Eq. (34) and the norm estimate for the function ${{\bf K}_{n,-1}^{\hbar}}(\cdot,\mathbf{z})$ (see Eq. (15)).

Let us now study the term $\mathbf{J}_{W}(\mathbf{z})$. From Eq. (35)

[TABLE]

where the error term $\mathbf{E}(\mathbf{z},\hbar)$ is $\mathrm{O}(\hbar^{2})$ uniformly with respect to $\mathbf{x}\in W_{\mathbf{z}}$, because in such a region we have $1/|\mathbf{x}\cdot\mathbf{z}|\leq C/|\mathbf{z}|$.

Furthermore, for each $\mathbf{x}\in V_{\mathbf{z}}$, $\Re(x_{1})-1=\frac{\Re(\mathbf{x}\cdot\mathbf{z})}{|\mathbf{z}|}-1<\frac{1}{C}-1<0$. Thus we can take the integral defining $\mathbf{J}_{W}(\mathbf{z})$ over the whole sphere with an error $\mathrm{O}(\hbar^{\infty})$.

Then, from Eqs. (49) and (40)

[TABLE]

where

[TABLE]

Following a procedure analogous to that performed in Theorem 4.3, we obtain from Eq.(46)

[TABLE]

Finally, from the norm estimate for the function ${{\bf K}_{n,-1}^{\hbar}}(\cdot,\mathbf{z})$ (see Eq. (15)), the chain rule and the fact that $N=2\pi^{n}/\Gamma(n)$ we conclude the proof of this theorem. ∎

5. Egorov-type theorem

The covariant symbol (see Eq. (2.2)) is not only defined for Toeplitz operators, but for any bounded linear operator. Hence it is natural to ask if there exists an analogue of the asymptotic expansion obtained in Theorems 4.3 and 4.4 for a more general operator than a Toeplitz operator. For this reason, in this section we prove an Egorov-type theorem which states that, for a pseudo-differential operator $A_{\hbar}$ acting on $L^{2}(\mathbf{S}^{n})$ with principal symbol $\wp(A_{\hbar})$ (see Appendix A where we give a brief description of what we mean by semiclassical pseudo-differential operators on a manifold), the covariant symbol of $A_{\hbar}$ goes to the composition of $\wp(A_{\hbar})$ with the map $\mathbf{z}\mapsto(\mathbf{z}/|\mathbf{z}|,-\imath\mathbf{z})$ when goes $\hbar$ to zero. Namely,

Theorem 5.1**.**

Let $p=0,-1$ and $A_{\hbar}$ be a semiclassical pseudo-differential operator of order zero with domain in $L^{2}(\mathbf{S}^{n})$, $n\geq 2$. Then for $\mathbf{z}\in\mathbb{C}^{n}-\{\mathbf{0}\}$ and $\hbar\searrow 0$ we have

[TABLE]

where $\wp(A_{\hbar})$ is the principal symbol of the semiclassical pseudo-differential operator $A_{\hbar}$.

Proof.

Let $U\in\mathrm{SU}(n)$ and $\mathrm{T}_{U}$ be the unitary transformation defined in Proposition 2.3. Let us first note that by Propositions 2.3 and A.6, the validity of Eq. (50) for $A_{\hbar}$ at $\mathbf{z}$ is equivalent to its validity for $\mathrm{T}_{U^{-1}}A_{\hbar}\mathrm{T}_{U}$ at $U^{-1}\mathbf{z}$. Since any given $\mathbf{z}$ can be mapped by a suitable $U\in\mathrm{SU}(n)$ into a point of the form $\mathbf{z}=\lambda\hat{\mathbf{u}}_{1}$ with $\lambda=|\mathbf{z}|$ and $\hat{\mathbf{u}}_{1}=(1,0,\ldots,0)$ a canonical unit vector in $\mathbb{R}^{n}$, it is indeed enough to prove (50) only for points $\mathbf{z}$ of the latter form.

For the rest of the proof of Theorem 5.1, we thus assume that $\mathbf{z}=\lambda\hat{\mathbf{u}}_{1}$ with $\lambda>0$.

Let us identify $\mathbf{S}^{n}$ with $S^{2n-1}$ via the map $\Upsilon$, which sends $\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbf{S}^{n}$ to $(\Re(x_{1}),\Im(x_{1}),\ldots,\Re(x_{n}),\Im(x_{n}))\in S^{2n-1}$.

Let us consider the following particular charts $(U_{\mathfrak{c}},\kappa_{\mathfrak{c}}\circ\Upsilon)$ of $\mathbf{S}^{n}$, with $U_{\mathfrak{c}}=\Upsilon^{-1}\circ\kappa_{\mathfrak{c}}^{-1}\mathfrak{A}$, $\mathfrak{c}=1,2,\ldots,2n-1$, where

[TABLE]

and $\kappa_{\mathfrak{c}}^{-1}:\mathfrak{A}\to\Upsilon(U_{\mathfrak{c}})$ are defined by

[TABLE]

with $\mathbf{v}=(v_{3},\ldots,v_{2n})$, $\mathfrak{a}=2,\ldots,2n-1$ and $\mathcal{R}_{\mathfrak{a}}\in\mathrm{SO}(2n)$ denoting the matrix which rotates the $(2n-1)$-sphere on the $(x_{1},x_{2})$ plane by $\pi$ and then followed by another rotation on the $(x_{2},x_{\mathfrak{a}})$ plane by $\pi/2$. More specifically, $\mathcal{R}_{\mathfrak{a}}$ is given by the matrix

[TABLE]

where $\{\hat{\mathbf{e}}_{1},\ldots,\hat{\mathbf{e}}_{2n}\}$ is the canonical basis of $\mathbb{R}^{2n}$ regarding $\hat{\mathbf{e}}_{\mathfrak{a}}$, $\mathfrak{a}=1,\ldots 2n$, as a column vectors. Note that $\hat{\mathbf{u}}_{1}=\Upsilon^{-1}(\hat{\mathbf{e}}_{1})\notin U_{\mathfrak{a}}$ for $\mathfrak{a}=2,\ldots,2n-1$, and that $\mathbf{S}^{n}=\bigcup_{\mathfrak{c}=1}^{2n-1}U_{\mathfrak{c}}$ (see Appendix B for details).

Let $\{\mathfrak{t}_{\mathfrak{c}}\}$ be a partition of the unity associated to the open cover $\{U_{\mathfrak{c}}\}$. Since $\hat{\mathbf{u}}_{1}$ only belongs to $U_{1}$ then $\mathfrak{t}_{1}(\hat{\mathbf{u}}_{1})=1$. Consider a set of functions $\{\varrho_{\mathfrak{c}}\}$ with $\varrho_{\mathfrak{c}}\in C_{0}^{\infty}(U_{\mathfrak{c}})$ such that $\mathfrak{t}_{\mathfrak{c}}\varrho_{\mathfrak{c}}=\mathfrak{t}_{\mathfrak{c}}$, for $\mathfrak{c}=1,\ldots,2n-1$. Then

[TABLE]

Moreover, since $\mathrm{supp}(\mathfrak{t}_{\mathfrak{c}})\cap\mathrm{supp}(1-\varrho_{\mathfrak{c}})=\varnothing$ and condition $(\mathbf{B})$ in the definition of a semiclassical pseudo-differential operator (see Eq (64)) we have

[TABLE]

The basic idea of the proof is to show that the main contribution to the left hand side of Eq. (50) comes from the term $\frac{\langle\mathfrak{t}_{1}A_{\hbar}\varrho_{1}{{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z}),{{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})\rangle_{\mathbf{S}^{n}}}{||{{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})||^{2}_{\mathbf{S}^{n}}}$. The stationary phase method will be used to obtain such a main contribution.

From the definition of a semiclassical pseudo-differential operator given in Appendix A, there exist $m\in\mathbb{R}$, $k\in\mathbb{Z}_{+}$ larger than $m+n$ and symbols $a_{\kappa_{\mathfrak{c}}}\in\mathrm{S}_{4n-2}(\langle p_{\theta},p_{\mathbf{v}}\rangle^{m})$ such that

[TABLE]

where we have used the coordinates $(\theta,\mathbf{v})$ in $\mathfrak{A}$ and where $|\mathbf{S}^{n}|$ denotes the surface area of the complex $n$-sphere, the Jacobian $J(\theta,\mathbf{v})=1$ and the operator $\mathrm{M}$ is defined by

[TABLE]

Notice that on the right hand side of Eq. (54) we have

[TABLE]

For $q\geq 1$, the action of the operator $\mathrm{M}^{q}$ on the function ${{\bf K}_{n,p}^{\hbar}}(\cdot,\mathbf{z})$ can be written as (see formula 0.430-2 of Ref. [7]))

[TABLE]

with $g=g_{\frac{1}{n-1}}$ the function defined in (17) if $p=-1$ or $g$ the exponential function if $p=0$, $\;\tilde{\mathbf{x}}_{\mathfrak{c}}=\tilde{\mathbf{x}}_{\mathfrak{c}}(\tilde{\theta},\tilde{\mathbf{v}})=(\kappa_{\mathfrak{c}}\circ\Upsilon)^{-1}(\tilde{\theta},\tilde{\mathbf{v}})$ and

[TABLE]

where the summation stands for non-negative numbers $p_{1},\ldots,p_{\ell}$ such that $p_{1}+2p_{2}+\ldots+\ell p_{\ell}=q$ and $d=p_{1}+p_{2}+\ldots+p_{\ell}$.

Let us first suppose that $p=-1$. Let $W_{\mathbf{z}}$, $V_{\mathbf{z}}$ be the regions defined in Eq. (18). Notice that if $\tilde{\mathbf{x}}_{\mathfrak{c}}\in W_{\mathbf{z}}$, $\mathfrak{c}=1,\ldots,2n-1,$ then we can use the asymptotic expression of the derivative of any order of $g$ evaluated at $\tilde{\mathbf{x}}_{\mathfrak{c}}\cdot\mathbf{z}/\hbar$ (see Eq. (33)). For this reason, let us regard the integration with respect to the variable $(\tilde{\theta},\tilde{\mathbf{v}})$ in Eq. (54) over the regions $\kappa_{\mathfrak{c}}\circ\Upsilon(W_{\mathbf{z}}\cap U_{\mathfrak{c}})$ and $\kappa_{\mathfrak{c}}\circ\Upsilon(V_{\mathbf{z}}\cap U_{\mathfrak{c}})$ separately and let us call them $\mathfrak{I}_{W,\mathfrak{c}}(\mathbf{z})$ and $\mathfrak{I}_{V,\mathfrak{c}}(\mathbf{z})$, respectively. Thus we have

[TABLE]

Now we claim that $\mathfrak{I}_{V,\mathfrak{c}}(\mathbf{z})=\mathrm{O}(\hbar^{\infty})$. Let $(\tilde{\theta},\tilde{\mathbf{v}})\in\kappa_{\mathfrak{c}}\circ\Upsilon(V_{\mathbf{z}}\cap U_{\mathfrak{c}})$ and $\tilde{\mathbf{x}}_{\mathfrak{c}}=(\kappa_{\mathfrak{c}}\circ\Upsilon)^{-1}(\tilde{\theta},\tilde{\mathbf{v}})$.

From definition of the operator $\mathrm{M}$ (see Eq. (55)) we have $\left|\mathrm{M}^{b}\tilde{\mathbf{x}}_{\mathfrak{c}}\cdot\mathbf{z}\right|\leq(1+p_{\theta}^{2}+p_{\mathbf{v}}^{2})^{\frac{b}{2}}\left|\mathrm{N}^{b}\;\tilde{\mathbf{x}}_{\mathfrak{c}}\cdot\mathbf{z}\right|$, where $\mathrm{N}=\frac{\partial}{\partial\tilde{\theta}}+\sum_{j=3}^{2n}\frac{\partial}{\partial\tilde{v}_{j}}$. Note that the action of the operator $\mathrm{N}$ on the function $\tilde{\mathbf{x}}_{\mathfrak{c}}\cdot\mathbf{z}$ involve inverse powers of the variable $\tilde{r}=(1-|\tilde{\mathbf{v}}|^{2})^{1/2}$ which are bounded in the support of $\varrho_{\mathfrak{c}}$ and therefore do not create any singularities; in fact $\left|\mathrm{N}^{b}\;\tilde{\mathbf{x}}_{\mathfrak{c}}\cdot\mathbf{z}\right|\leq C_{1}\tilde{r}^{-b}\leq C_{1}(2\sqrt{2n-2})^{b}$ for some constant $C_{1}$. Then, from Eq. (58)

[TABLE]

On the other hand, since $a_{\kappa_{\mathfrak{c}}}\in S_{4n-2}(\langle(p_{\theta},p_{\mathbf{v}})\rangle^{m})$

[TABLE]

Then, from Eqs. (54), (56), (57), (59), (60), Proposition 3.3 (specifically Eq. (34)) and the estimate of the norm of ${{\bf K}_{n,-1}^{\hbar}}(\cdot,\mathbf{z})$ (see Eq. (15)) we have

[TABLE]

with $\mu<0$. Using the Cauchy-Schwartz inequality and that $n<k-m$ we conclude that $\mathfrak{I}_{V,\mathfrak{c}}(\mathbf{z})=\mathrm{O}(\hbar^{\infty})$.

Let us now study the terms $\mathfrak{I}_{W,\mathfrak{c}}(\mathbf{z})$. From Eqs. (54), (56), (57), (58), the estimate of the norm of ${{\bf K}_{n,-1}^{\hbar}}(\cdot,\mathbf{z})$ (see Eq. (15)) and Proposition 3.3 (specifically Eq. (33) with $a=\frac{1}{n-1}$) we have

[TABLE]

with $\tilde{\mathbf{x}}_{\mathfrak{c}}=\tilde{\mathbf{x}}_{\mathfrak{c}}(\tilde{\theta},\tilde{\mathbf{v}})=(\kappa_{\mathfrak{c}}\circ\Upsilon)^{-1}(\tilde{\theta},\tilde{\mathbf{v}})$, $\mathbf{x}_{\mathfrak{c}}=\mathbf{x}_{\mathfrak{c}}(\theta,{\mathbf{v}})=(\kappa_{\mathfrak{c}}\circ\Upsilon)^{-1}(\theta,{\mathbf{v}})$.

Then, as $\mathbf{z}=\lambda\hat{\mathbf{u}}_{1}$ we have

[TABLE]

where

[TABLE]

with $\mathfrak{f}^{1}(\theta,\mathbf{v})=re^{\imath\theta}$ and $\mathfrak{f}^{\mathfrak{c}}(\theta,\mathbf{v})=-r\cos\theta+\imath v_{\mathfrak{c}+1}$ for $\mathfrak{c}=2,\ldots,2n-1$.

From Eqs. (61), (59), (60) and using the arguments given above to study $\mathfrak{I}_{V,\mathfrak{c}}(\mathbf{z})$, it can be show that $\mathfrak{I}_{W,\mathfrak{c}}(\mathbf{z})=\mathrm{O}(\hbar^{\infty})$ for $\mathfrak{c}=2,\ldots,n$.

Let us now study the term $\mathfrak{I}_{W,1}$. Since $a_{\kappa_{1}}\in\mathrm{S}_{4n-2}(\langle(p_{\theta},p_{\mathbf{v}})\rangle^{m})$ is a classical symbol, there exist a sequence $(a_{\kappa}^{(j)})\subset\mathrm{S}_{4n-2}\left(\langle(p_{\theta},p_{\mathbf{v}})\rangle^{m}\right)$ (independent of $\hbar$) such that if $N>3n/2$ and $\Psi_{N}=\sum_{\ell=0}^{N}\hbar^{\ell}a_{\kappa}^{(\ell)}$, then for any $\mathbf{b}\in\mathbb{Z}_{+}^{2n-1}$, there exist $\hbar_{N,\mathbf{b}}>0$ and $C_{N,\mathbf{b}}$ such that

[TABLE]

uniformly in $\mathbb{R}^{4n-2}\times(0,\hbar_{N,\mathbf{b}})$.

Let us denote by $\mathfrak{I}_{W,1}^{1}(\mathbf{z})$ and $\mathfrak{I}_{W,1}^{2}(\mathbf{z})$ the right side of Eq. (61) with $a_{\kappa_{1}}-\Psi_{N}$ and $\Psi_{N}$ instead of $a_{\kappa_{1}}$ respectively. Note that $\mathfrak{I}_{W,1}(\mathbf{z})=\mathfrak{I}_{W,1}^{1}(\mathbf{z})+\mathfrak{I}_{W,1}^{2}(\mathbf{z})$. Furthermore, from Eqs. (61), (59), (60) and (62) it can be shown that $\lim_{\hbar\to 0}\mathfrak{I}_{W,1}^{1}(\mathbf{z})=0$

Finally, let us use the stationary phase method to study the term $\mathfrak{I}_{W,\mathfrak{1}}^{2}(\mathbf{z})$.

The function $\mathfrak{f}_{1}$ has a non-negative imaginary part and one critical point $\boldsymbol{\vartheta}_{0}$ that contributes to the main asymptotic given by

[TABLE]

Since $\det(\mathfrak{f}_{1}^{\prime\prime}(\boldsymbol{\vartheta}_{0})/2\pi\imath\hbar)=\lambda^{2n-1}/2^{4n-2}(\pi\hbar)^{6n-3}$ (with $\mathfrak{f}_{1}^{\prime\prime}$ the Hessian matrix of $\mathfrak{f}_{1}$), $\mathfrak{f}_{1}(\boldsymbol{\vartheta_{0}})=0$, $\mathfrak{t}_{1}((\kappa_{1}\circ\Upsilon)^{-1}(\theta_{0},\mathbf{v}_{0}))=\mathfrak{t}_{1}(\hat{\mathbf{u}}_{1})=1$ and $\varrho_{1}(\hat{\mathbf{u}}_{1})=1$ , then we obtain from the stationary phase method (see Ref. [9])

[TABLE]

where we have used that $F_{s,s}^{1}(\tilde{\theta}_{0},\tilde{\mathbf{v}}_{0})=(\lambda^{2})^{s}$ and the definition of the principal symbol. Thus, we have proved Eq. (50) if $p=-1$.

The case $p=0$ is proved in the same way. ∎

Appendix A Pseudo-differential operators

In this appendix, we give a brief description of what we mean by semiclassical pseudo-differential operators on a manifold. See [10] y [11] for details on definitions of semiclassical pseudo-differential operators on $\mathbb{R}^{n}$ and manifolds, respectively.

Definition A.1**.**

For $\boldsymbol{\eta}\in\mathbb{R}^{n}$, let us define $\langle\boldsymbol{\eta}\rangle:=\sqrt{1+|\boldsymbol{\eta}|^{2}}$, where $|\boldsymbol{\eta}|^{2}=\eta_{1}^{2}+\ldots+\eta_{n}^{2}$.

Definition A.2**.**

Let $\hbar_{0}>0$ and $m\in\mathbb{R}$. The space of symbols $\mathrm{S}_{2n}(\langle\boldsymbol{\xi}\rangle^{m})$ denotes the set of functions $p=p(\mathbf{b},\boldsymbol{\xi};\hbar)$, $\mathbf{b},\boldsymbol{\xi}\in\mathbb{R}^{n}$, $\hbar\in(0,\hbar_{0}]$ which are smooth in the variable $\mathbf{z}=(\mathbf{b},\boldsymbol{\xi})$ and for any $\mathbf{k}\in\mathbb{Z}_{+}^{2n}$

[TABLE]

uniformly with respect to $(\mathbf{z},\hbar)\in\mathbb{R}^{2n}\times(0,\hbar_{0}]$.

Definition A.3**.**

Let $m\in\mathbb{R}$, $a\in\mathrm{S}_{2n}(\langle\boldsymbol{\xi}\rangle^{m})$ and $(a^{(j)})$ a sequence of symbols of $\mathrm{S}_{2n}(\langle\boldsymbol{\xi}\rangle^{m})$ independent of $\hbar$, with $a^{(0)}$ non identically zero. Then we say that $a$ is asymptotically equivalent to the formal sum $\sum_{j=0}^{\infty}\hbar^{j}a^{(j)}$ in $\mathrm{S}_{2n}(\langle\boldsymbol{\xi}\rangle^{m})$ if and only if for any $N\in\mathbb{N}$ and for any $\mathbf{k}\in\mathbb{Z}_{+}^{n}$ there exist $\hbar_{N,\mathbf{k}}>0$ and $C_{N,\mathbf{k}}>0$ such that

[TABLE]

uniformly on $\mathbb{R}^{2n}\times(0,\hbar_{N,\mathbf{k}}]$. In this case we say that $a$ is a classical symbol.

Now we will give the definition of semiclassical pseudo-differential operator on $\mathbb{R}^{n}$. In order

Definition A.4**.**

Let $m\in\mathbb{R}$, $a\in\mathrm{S}_{2n}(\langle\boldsymbol{\xi}\rangle^{m})$ and $t\in[0,1]$. The action of a semiclassical pseudo-differential operator in $\mathbb{R}^{n}$, denoting by $\mathrm{Op}_{\hbar}^{t}(a)$, on a smooth function $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ is defined by

[TABLE]

where $a_{t}(\mathbf{b},\mathbf{c},\boldsymbol{\xi};\hbar)=a((1-t)\mathbf{b}+t\mathbf{c},\boldsymbol{\xi};\hbar)$, $k$ is any non-negative integer number such that $m+n<k$ and

[TABLE]

If $a$ is a classical symbol, the function $a^{(0)}$ will be called principal symbol of the semiclassical pseudo-differential operator $\mathrm{Op}_{\hbar}^{t}(a)$ and will be denoted by $\wp(\mathrm{Op}_{\hbar}^{t}(a))$.

Remark 2**.**

In the previous definition, the operator $\mathrm{L}(\boldsymbol{\xi},\hbar\mathrm{D}_{\mathbf{c}})$ is introduced in order to make sense of the possibly undefined integral

[TABLE]

Remark 3**.**

For the values $t=0$, $t=\frac{1}{2}$ and $t=1$, $\mathrm{Op}_{\hbar}^{t}(a)$ is called standard quantization or left quantization, Weyl quantization and right quantization respectively.

Notation. Let $\mathrm{M}$ be a smooth $n$-dimensional manifold with atlas $\mathcal{F}$. The elements of $\mathcal{F}$ are called charts and will be denoted by $(U_{\kappa},\kappa)$ with $\kappa:U_{\kappa}\to V_{\kappa}$ and $\kappa$ an homeomorphism between the open sets $U_{\kappa}\subset\mathrm{M}$ and $V_{\kappa}\subset\mathbb{R}^{n}$.

Now we will give a brief description of the concept of semiclassical pseudo-differential operator on $\mathrm{M}$.

Definition A.5**.**

Let $A_{\hbar}:C^{\infty}(\mathrm{M})\to C^{\infty}(\mathrm{M})$ be a linear operator. We say that $A_{\hbar}$ is a semiclassical pseudo-differential operator of order zero iff there exist $m\in\mathbb{R}$ and $0\leq t\leq 1$ such that the following three condition $\mathrm{(\mathbf{A})}$, $\mathrm{(\mathbf{B})}$ and $\mathrm{(\mathbf{C})}$ are satisfied:

$\mathrm{(\mathbf{A})}$* For each chart $(U_{\kappa},\kappa)$, there exist a classical symbol $a_{\kappa}\in\mathrm{S}_{2n}(\langle\boldsymbol{\xi}\rangle^{m})$, such that for all $u\in C^{\infty}(\mathrm{M})$ we have*

[TABLE]

where we are extending $(\kappa^{-1})^{*}(\Psi u)$ as zero outside of the set $V_{\kappa}$.

$\mathrm{(\mathbf{B})}$* If $\Phi_{j}\in C_{0}^{\infty}(\mathrm{M})$, $j=1,2$, with disjoint supports, then*

[TABLE]

$\mathrm{(\mathbf{C})}$* Given two charts $(U_{\kappa},\kappa)$, $(U_{\nu},\nu)$ such that $U_{\kappa}\cap U_{\nu}\neq\varnothing$ we require the condition*

[TABLE]

The last condition implies that we can define the principal symbol $\wp(A_{\hbar})$ of the pseudo-differential operator $A_{\hbar}$ by

[TABLE]

where $(U_{\kappa},\kappa)$ is any chart such that $\mathbf{x}\in U_{\kappa}$.

Since $\mathbf{S}^{n}$ can be identified with $S^{2n-1}$ via the map $\Upsilon$, which sends $\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbf{S}^{n}$ to $(\Re(x_{1}),\Im(x_{1}),\ldots,\Re(x_{n}),\Im(x_{n}))\in S^{2n-1}$, we can consider $\mathbf{S}^{n}$ as a smooth $2n-1$ dimensional manifold.

Let $A_{\hbar}$ be a semiclassical pseudo-differential operators on $\mathbf{S}^{n}$. Based on relating corresponding properties of functions and $A_{\hbar}$ on a given chart and its rotated chart under the action of $\mathrm{SU}(n)$ on the complex $n$-sphere $\mathbf{S}^{n}$, we can show the relationship between the principal symbols of $A_{\hbar}$ and $\mathrm{T}_{U^{-1}}A_{\hbar}\mathrm{T}_{U}$ (with $U\in\mathrm{SU}(n)$ and $\mathrm{T}_{U}$ defined as in Proposition 2.3) as the following proposition establish it. This proposition will be useful to prove Theorem 5.1.

Proposition A.6**.**

Let $U\in\mathrm{SU}(n)$ and $A_{\hbar}$ be a semiclassical pseudo-differential operator on $\mathbf{S}^{n}$. Then $\mathrm{T}_{U^{-1}}A_{\hbar}\mathrm{T}_{U}$ is a semiclassical pseudo-differential operator on $\mathbf{S}^{n}$ with principal symbol

[TABLE]

Proof.

Let $(\mathbf{x},\boldsymbol{\eta})\in T^{*}(\mathbf{S}^{n})$ and $(X_{\tau},\tau)$ be any chart such that $\mathbf{x}\in X_{\tau}$. Let us denote by $(X_{\tilde{\tau}},\tilde{\tau})$ the chart given by $X_{\tilde{\tau}}=UX_{\tau}$ and $\tilde{\tau}=\tau\circ U^{-1}$.

Let $v\in C^{\infty}(\mathbf{S}^{n})$, and $\phi,\psi\in C_{0}^{\infty}(X_{\tau})$. Let us denote by $\tilde{v}=\mathrm{T}_{U}v$, $\tilde{\psi}=\mathrm{T}_{U}\psi$ and $\tilde{\phi}=\mathrm{T}_{U}\phi$. Note that $\tilde{\psi},\tilde{\phi}\in C_{0}^{\infty}(X_{\tilde{\tau}})$. From Eq. (63)

[TABLE]

which establishes that condition $(\mathbf{A})$ holds for the operator $\mathrm{T}_{U^{-1}}A_{\hbar}\mathrm{T}_{U}$.

Condition $(\mathbf{B})$ follows from Eq. (64) and the $\mathrm{SU}(n)$-invariance of $\mathrm{d}\mathbf{S}^{n}$.

Let us now consider another chart $(X_{\tau^{\prime}},\tau^{\prime})$ such that $X_{\tau}\bigcap X_{\tau^{\prime}}=\varnothing$. Condition $(\mathbf{C})$ follows from the equalities $\tau(X_{\tau})=\tilde{\tau}({X}_{\tilde{\tau}})$, $\tau(X_{\tau^{\prime}})=\tilde{\tau^{\prime}}(X_{\tilde{\tau^{\prime}}})$ and $\tilde{\tau^{\prime}}\circ(\tilde{\tau})^{-1}=\tau^{\prime}\circ\tau^{-1}$.

Thus we conclude that $\mathrm{T}_{U^{-1}}A_{\hbar}\mathrm{T}_{U}$ is a pseudo-differential operator with principal symbol (see Eq. (65)).

[TABLE]

∎

Appendix B The complex n-sphere $\mathbf{S}^{n}$ as the union of the charts $U_{\mathfrak{c}}$

In this appendix we prove that $\mathbf{S}^{n}=\bigcup_{\mathfrak{c}=1}^{2n-1}U_{\mathfrak{c}}$ and that $\hat{\mathbf{u}}_{1}\notin U_{\mathfrak{a}}$ for $\mathfrak{a}=2,\ldots,2n-1$, where $\hat{\mathbf{u}}_{1}=(1,0,\ldots,0)$ is a canonical vector in $\mathbb{R}^{n}$.

To prove that $\mathbf{S}^{n}=\bigcup_{\mathfrak{c}=1}^{2n-1}U_{\mathfrak{c}}$, let us first note that

[TABLE]

where $\Upsilon$ is the map that identify $\mathbf{S}^{n}$ with $S^{2n-1}$ (see its definition at the beginning of the proof of Theorem 5.1).

Consider $\mathbf{x}\in\mathbf{S}^{n}$ and write it as \mathbf{x}=\Upsilon^{-1}\big{(}(r\cos\theta,r\sin\theta,\mathbf{v})\big{)}. Let us assume that $\mathbf{x}\notin U_{1}$. Then we have the following two cases:

1. i)

$\displaystyle 1-\frac{1}{8(n-1)}\leq|\mathbf{v}|^{2}\leq 1$. 2. ii)

$\theta=\pi$ and $\displaystyle|\mathbf{v}|^{2}<1-\frac{1}{8(n-1)}$.

Case i) Let $v_{j}=\max\{v_{3},\ldots,v_{2n}\}$, then $v_{j}>\frac{1}{2\sqrt{n-1}}$, because otherwise $|\mathbf{v}|^{2}\leq\frac{1}{2}<1-\frac{1}{8(n-1)}$.

Let us define $\tilde{\mathbf{v}}=(\tilde{v}_{3},\ldots,\tilde{v}_{2n})\in\mathbb{R}^{2n-2}$ such that $\tilde{v}_{i}=v_{i}$, $i\neq j$ and $\tilde{v}_{j}=r\sin(\theta)$. Since

[TABLE]

then there exist $-\pi\leq\tilde{\theta}<\pi$ such that $\cos(\tilde{\theta})=-\frac{r}{\tilde{r}}\cos(\theta)$ and $\sin(\tilde{\theta})=\frac{v_{j}}{\tilde{r}}$. Actually $\tilde{\theta}\neq\pi$ because otherwise $v_{j}$ would have to be equal to zero.

Since $|\tilde{\mathbf{v}}|^{2}=1-r^{2}\cos^{2}(\theta)-v_{j}^{2}<1-\frac{1}{8(n-1)}$ then $(\tilde{\theta},\tilde{\mathbf{v}})\in\mathfrak{A}$. Moreover, using the explicit expression of the matrix $\mathcal{R}_{j-1}$ (see Eq. (51)), one can check that $\mathbf{x}=(\kappa_{j+1}\circ\Upsilon)^{-1}(\tilde{\theta},\tilde{\mathbf{v}})\in U_{j+1}$.

Case ii) Let us define $v_{j}$ and the vector $\tilde{\mathbf{v}}$ as above. Note that $\tilde{r}^{2}=1-|\tilde{\mathbf{v}}|^{2}=r^{2}+v_{j}^{2}\geq r^{2}=1-|\mathbf{v}|^{2}>0$.

Consider $-\pi<\tilde{\theta}<\pi$ such that $\cos(\tilde{\theta})=\frac{r}{\tilde{r}}$ and $\sin(\tilde{\theta})=\frac{v_{j}}{\tilde{r}}$. Moreover, $|\tilde{\mathbf{v}}|^{2}=1-r^{2}-v_{j}^{2}\leq|\mathbf{v}|^{2}<1-\frac{1}{8(n-1)}$. Therefore, $(\tilde{\theta},\tilde{\mathbf{v}})\in\mathfrak{A}$ and one can check that $\mathbf{x}\in U_{j-1}$.

Let us now prove that $\hat{\mathbf{u}}_{1}$ only belongs to $U_{\mathfrak{1}}$. Let us assume that $\hat{\mathbf{u}}_{1}\in U_{\mathfrak{a}}$ for some $\mathfrak{a}=2,\ldots,2n-1$, then there exist $(\theta,\mathbf{v})\in\mathfrak{A}$ such that

[TABLE]

which is impossible since $-\pi<\theta<\pi$.